Operations on Graphs Increasing Some Graph Parameters

OPERA TIONS ON GRAPHS INCREASING SOME GRAPH P ARAMETERS Alexander Kelmans Univ ersit y of Puerto Rico, San Juan, Puerto Rico Rutgers Univ ersit y , New Brunswic k, New Jersey Abstract In this partly exp ository pap er we discuss and describ e some of our old and recen t results on partial orders on the set G m n of graphs with n vertices and m edges and some op erations on graphs within G m n that are monotone with resp ect to these partial orders. The partial orders under consideration include those related with some Laplacian characteristics of graphs as w ell as with some probabilistic c haracteristics of graphs with randomly deleted edges. Section 3 con tains some basic facts on the Laplacian p olynomial of a graph. Section 4 describ es v arious graph op eration and their properties. In Section 5 we in tro duce some partial orders  on G m n related with the graph Laplacian and the graph reliabilit y ( L aplacian p osets and r eliability p osets ). Section 6 con tains some old and recent results on the  -monotonicity of some graph operations with resp ect to Laplacian p osets. Section 7 includes some old and recen t results on the  -monotonicit y of some graph op erations with resp ect to reliability p osets and some op en problems. In Section 8 w e consider some other parameters of graphs and establish some results on  -monotonicity of our graph op erations with resp ect to the linear orders  on G m n related with these parameters. The list of these parameters includes the n umbers of Hamiltonian cycles or paths and the num b ers of forests of sp ecial type. Section 9 contains some generalizations of the describ ed results to w eighted graphs. 1 In tro duction All notions and facts on graphs, that are used but not described here, can be found in [1, 6]. Let ¯ G m n ( G m n ) denote the set of graphs (resp ectiv ely , simple graphs) with n v ertices and m edges. Replacing in the ab o v e notations G by C results in the notations of the corresp onding sets of connected graphs. If G, F ∈ G m n , we also sa y simply that G and F are of the same size . In a series of pap ers (see, for example, [18, 20, 22 – 27, 32, 35, 36]) we considered v arious asp ects of the following combinatorial optimization problems related with the syn thesis of reliable net w orks. Let G b e a graph. Supp ose that every edge of G has probability p to exist and that all the edge even ts are mutually indep enden t. Let R ( p, G ) denote the probabilit y that the random graph ( G, p ) is connected. W e call R ( p, G ) the r eliability function (or just the r eliability ) of G . The problem R max is to ﬁnd a most reliable graph M ( p ) in ¯ G m n , i.e. suc h that R ( p, M ( p )) = max { R ( p, G ) : G ∈ ¯ G m n } . 1 The corresp onding dual problem R ∗ max is to ﬁnd a graph M ∗ ( p ) ∈ ¯ G n suc h that e ( M ∗ ( p )) = min { e ( G ) : G ∈ ¯ G n , R ( p, G ) ≥ α ( p ) } . It is also interesting to consider the problem R min of ﬁnding a least reliable graph L ( p ) in ¯ G m n . If G is not connected, then R ( p, G ) = 0. F or that reason, the non-trivial problem R min is to ﬁnd a graph L ( p ) ∈ ¯ C m n suc h that R ( p, L ( p )) = min { R ( p, G ) : G ∈ ¯ C m n } . Let t ( G ) denote the num b er of spanning trees of G . F or p close to zero, the problem R max is equiv alen t to problems T max of ﬁnding a graph M ∈ ¯ G m n suc h that t ( M ) = max { t ( G ) : G ∈ ¯ G m n } and R min is equiv alen t to the problem T min of ﬁnding a graph L in ¯ G m n suc h that t ( L ) = min { t ( G ) : G ∈ ¯ C m n } , where m ≥ n − 1. The corresp onding dual problem T ∗ max is to ﬁnd a graph M ∗ ∈ ¯ G n suc h that e ( M ∗ ) = min { e ( G ) : G ∈ ¯ G n , t ( G ) ≥ α } . Although in general these problems are probably N P -hard, it turns out that they can b e solved in some non-trivial particular cases and, in addition, their analysis leeds to some interesting mathematical results, ideas, and questions. In particular, it is not hard to solve problems R max and R min for graphs of relatively small cyclomatic n um b er or corank (see, for example, 7.9 b elo w). In [35, 36] we w ere able to give an asymptotically optimal solution to problem R ∗ max as w ell as to problem R ∗ max ( k ) on the probabilit y that ( G, p ) is k -connected. Since problems R max and R min ha v e parameter p ∈ [0 , 1], it is natural to introduce the follo wing relation on G m n : giv en G, F ∈ G m n , let G  r F if R ( p, G ) ≥ R ( p, F ) for ev ery p ∈ [0 , 1]. In [35] we observed that there are graphs G and F of the same size that are not  r -comparable as w ell as non-isomorphic graphs G and F of the same size suc h that R ( p, G ) ≡ R ( p, F ). Therefore  r is a partial quasi-order relation on G m n . No w the following natural question on problem R max is in order: Do es every G m n ha v e a  r -maxim um graph ? In other w ords, do es a most reliable graph M ( p ) in some G m n dep end on p ? W e gav e a construction pro viding inﬁnitely man y pairs ( n, m ) for whic h G m n has no  r -maxim um [22, 24] (see more details in 7.14 b elo w), i.e. for each of these pairs ( n, m ) there are non-isomorphic graphs M ( p 1 ) and M ( p 2 ) for some 0 < p 1 < p 2 < 1. Some further interesting questions along this line are describ ed b elo w in Section 7. Ob viously , if G is not connected, then R ( p, G ) ≡ 0. Therefore, a similar non-trivial question ab out problem R min is: Do es ev ery C m n ha v e a  r -minim um graph ? In other w ords, do es a least reliable graph L ( p ) in some C m n dep end on p ? The answ er to this question is not known. More detailed ab out this question are given in Section 7. The Matrix T ree Theorem (see, for example, [1, 5] and 3.1 b elo w) provides a b eau- tiful algebraic form ula for t ( G ) and allows to ﬁnd it in p olynomial time. On the other hand, the problem of ﬁnding R ( p, G ) is # P -hard. F or that reason the ab o v e problems on R ( p, G ) are m uch harder to analyze than the problems on t ( G ). Therefore it w as natural to exp ect that the Matrix T ree Theorem could b e useful in dev eloping adequate 2 approac hes to attack problem T max on ﬁnding graphs of given size with the maximum n um b er of spanning trees. In early 60’s motiv ated by this idea, we disco v ered that the c haracteristic p olynomial L ( λ, G ) = P { ( − 1) s c s ( G ) λ n − s : s ∈ { 0 , . . . , n }} of the Laplacian matrix from the Matrix T ree Theorem should play an imp ortan t role in this regard. Accordingly , in [34, 37 – 39] w e underto ok some study and established v arious com binatorial prop erties of L ( λ, G ), its co eﬃcien ts, the Laplacian sp ectrum of G , and their relation with the ab o v e mentioned problems. P ap ers [34, 37, 38] were published in Russian in the So viet Union, [37, 38] were trans- lated in to English in 1966, the results of our manuscript [39] of 1963 were describ ed b y Cv etk ovi ´ c in one of his pap ers in 1971 (see also [5]), and later pap ers [9, 17, 18, 23 – 29, 31] w ere published in the W est, where according to [5] the interest to this topic b ecame apparen t in 70’s. Unfortunately , till no w some researchers w orking in this area are not a w are of certain basic facts from those pap ers and quite a few results from these pap ers ha v e later b een published again and again. The following simple facts on L ( λ, G ) turned out to b e pretty useful. 1.1 [34, 38] L et G ∈ ¯ G m n and ( λ 0 ( G ) ≤ · · · ≤ λ n − 1 ( G )) the list of al l n r o ots of L ( λ, G ) (i.e. the sp ectrum of the Laplacian matrix of G ) . Then ( a 1) 0 = λ 0 ( G ) ≤ · · · ≤ λ n − 1 ( G ) ≤ n and ( a 2) c 1 ( G ) = P { λ i ( G ) : i ∈ { 1 , . . . , n − 1 }} = 2 e ( G ) = 2 m and c n − 1 ( G ) = nt ( G ) = Q { λ i ( G ) : i ∈ { 1 , . . . , n − 1 }} . Let K n b e the complete graph with n vertices. F rom 1.1 w e ha ve: 1.2 [34] L et G ∈ ¯ G m n . Then ( a 1) t ( G ) ≤ n − 1 (2 m/ ( n − 1)) n − 1 , and so ( a 2) if m = ( n 2 ) , then t ( G ) ≤ n n − 2 = t ( K n ) . Th us, ( a 2) in 1.2 gives a solution of problem T max for m = ( n 2 ). Moreo v er, it turns out that t ( G ) = n n − 2 if and only if G = K n . In [41] this solution w as extended to problem R max for m = ( n 2 ). Here is another useful fact on L ( λ, G ). 1.3 [38] L et G ∈ G m n . Then t ( K n + s − E ( G )) = ( n + s ) r − 2 L ( n + s, G ) for every s ≥ 0 . It is in teresting that the formula in 1.3 turns out to b e the inclusion-exclusion formula for the num ber of spanning trees in K n + s a v oiding the edges of its subgraph G [29] (see more details in 3.19 b elo w). Let d ( G, n + s ) = t ( K n + s ) − t ( K n + s − E ( G ). Then d ( G, n + s ) is the n um b er of spanning trees in K n + s that are destro yed when the edges of G are remo ved from K n + s . F or that reason we call d ( G, n + s ) the destr oying ability of an n -vertex graph G in K n + s . 3 No w by 1.3 , comparing simple n -vertex graphs by their n um b er of spanning trees is equiv alen t to comparing the destroying ability of their complements in K n . F urthermore, comparing the destroying abilities of G in ev ery complete graph K n + s is equiv alen t to comparing the p olynomials L ( λ, G ) in ev ery integer p oin t λ = n + s . This suggests the follo wing partial quasi-order relation on G m . Given G, F ∈ G m , let G  τ F if L ( ν + s, G ) ≥ L ( ν + s, F ) for ev ery integer s ≥ 0, where ν = max { v ( G ) , v ( F ) } . No w it is clear that the study of prop erties of relation  τ on G m n ma y help to solv e problems T max and T min for some classes of graphs. In [25] w e found some  τ -increasing and  r -increasing op erations on graphs in G m n . Using these op erations w e were able to solv e problems R max for m ≥ ( n 2 ) − b n/ 2 c and R min for m ≥ ( n 2 ) − n + 2 [25] as w ell as problems T max for m ≥ ( n 2 ) − b n/ 2 c and T min for m ≥ ( n 2 ) − n + 2 [29]. In all these cases there exist a  r -minim um graph C m n and a unique  r -maxim um graph in G m n . Later we hav e found some more delicate  τ -increasing op- erations for some sp ecial classes of graphs [17] which allo w ed us to also solve problem T max for m ≥ ( n 2 ) − n + 2 [23]. In this partly exp ository pap er w e discuss and describ e some partial orders on the set G m n of graphs with n v ertices and m edges and some op erations on graphs within G m n that are monotone with resp ect to these partial orders. The partial orders under consideration include those related with some Laplacian characteristics of graphs as well as with some probabilistic c haracteristics of graphs having randomly deleted edges. In Section 2 we giv e necessary notions and notation, as well as some simple obser- v ations. In Section 3 we describ e some basic results on the Laplacian p olynomial of a graph. In Section 4 we deﬁne v arious graph op eration preserving the size of the graph and describ e some simple and useful prop erties of these op erations. In Section 5 we in tro duce v arious partial orders on G m n related, in particular, with the Laplacian p oly- nomial L ( λ, G ) and with the graph reliabilit y R ( p, G ) ( L aplacian p osets and r eliability p osets ) and establish some m utual prop erties of these relations. Section 6 contains some old and recent results on the  -monotonicity of some graph op erations with resp ect to Laplacian p osets. Section 7 includes some old and recen t results on the  -monotonicity of some graph op erations with resp ect to reliabilit y p osets and some op en problems. In Section 8 we consider linear orders  on G m n related with some other parameters of graphs and establish some results on  -monotonicity of some graph op erations with resp ect to these  -orders. The list of parameters considered in this section includes the n um b ers of Hamiltonian cycles or paths and the num bers of forests of sp ecial types, for example, the n um b er of matchings of a giv en size. Section 9 contains some generaliza- tions of the describ ed results to weigh ted graphs. Man y results describ ed in this pap er w ere included in our lectures on Algebra and Com binatorics in Rutgers Universit y , 1992 - 1993 and in Universit y of Puerto Rico, 1995 - 2009. 4 2 Notions, notation, and simple observ ations All notions and facts on graphs that are used but not describ ed here can b e found in [1, 6]. 2.1 Let G = ( V , E , ϕ ) b e a graph, where V = V ( G ) is the set of vertic es of G , E = E ( G ) is the set of e dges of G , and ϕ = ϕ G is the function from E to the set of unordered pairs of vertices of G (the incident function of G ). A graph is called simple if it has no lo ops and no parallel (or, the same, multiple) edges. Let v ( G ) = | V ( G ) | and e ( G ) = | E ( G ) | . W e say that graphs G and F are of the same size if v ( G ) = v ( F ) and e ( G ) = e ( F ). Let C mp ( G ) denote the set of comp onen ts of G and cmp ( G ) = | C mp ( G ) | . Let r ( G ) = v ( G ) − cmp ( G ) and r ∗ ( G ) = e ( G ) − r ( G ) = e ( G ) − v ( G ) + cmp ( G ). The parameter r ( G ) is called the r ank of G and r ∗ ( G ) is called the c or ank (or the cyclomatic numb er ) of G . Let isl ( G ) denote the n um b er of isolated v ertices of G . 2.2 Let ¯ G , ¯ G n and ¯ G m denote the sets of all graphs, graphs with n v ertices, and graphs with m edges, resp ectiv ely , and let ¯ G m n = ¯ G n ∩ ¯ G m . Replacing in the ab o v e notation ¯ G b y G ( C ) gives the corresp onding sets of simple graphs (resp ectiv ely , connected simple graphs). Let T n denote the set of trees with n vertices. Let F ( G ) and T ( G ) denote the sets of spanning forests and spanning trees of G , and accordingly , f ( G ) = |F ( G ) | and t ( G ) = |T ( G ) | . 2.3 F or X , Y ⊆ V ( G ) let [ X , Y ] denote the set of edges of G with one end-vertex in X and the other end-v ertex in Y . F or x ∈ V ( G ), let N ( x, G ) = { v ∈ V ( G ) : xv ∈ E ( G ) } , D ( x, G ) = [ x, V ( G )], and d ( x, G ) = | D ( x, G ) | . W e call d ( x, G ) the de gr e e of a vertex x in G . Let ∆( G ) = max { d ( x, G ) : x ∈ V ( G ) } and δ ( G ) = min { d ( x, G ) : x ∈ V ( G ) } . 2.4 Let K = ( V , E ) b e the graph such that E =  V 2  . This graph is called the simple c omplete gr aph with the vertex set V . W e put K = K n if | V | = n . If G is a subgraph of K , then [ G ] c = K − E ( G ) is called the simple c omplement of G . In particular, [ K ] c is the ( e dge ) empty gr aph with the v ertex set V . 2.5 Given t wo disjoint graphs G and F , let G + F = G ∪ F and G × F b e the graph obtained from G ∪ F by adding the set of edges { g f : g ∈ V ( G ) , f ∈ V ( F ) } [37, 38]. Ob viously , [ G + F ] c = [ G ] c × [ F ] c . If G consists of k disjoin t copies of a graph F , we write G = k F . A graph G is called de c omp osable if G = A + B or G = A × B for some disjoint graphs A and B . W e call a graph G total ly de c omp osable [37, 38] (see also [9]) if G can b e obtained from one v ertex graphs by a series of op erations + and × . The notion of a totally decomp osable graph turned out to b e so natural that it w as later rein tro duced again and again under diﬀerent names: a cograph in [40], a her e ditary Dac ey gr aph in [45], a D ∗ -gr aph in [14], etc. A totally decomp osable graph G can b e naturally describ ed b y the so called (+ , × ) -de c omp osition tr e e of G introduced in [37, 38] (see also [9]) and later in [4] under the name the c otr e e of G . Similar notions for w eight ed graphs were introduced in [31]. 5 In [37] we gav e a very simple pro cedure for ﬁnding form ulas for the Laplacian p oly- nomial and sp ectrum (and, in particular the n um b er of spanning trees) of a totally decomp osable graph G in terms of the parameters of the (+ , × )-decomp osition tree of G (see also [9]) . F rom this pro cedure we ha v e, in particular: 2.6 [38] Every total ly de c omp osable gr aph has an inte gr al sp e ctrum. Similar results for multigraphs and weigh ted graphs (and their Laplacian matrices) can b e found in [31]. 2.7 W e call a graph G vertex c omp ar able [19, 20, 33] if N ( x, G ) − y ⊆ N ( y ) or N ( y , G ) − x ⊆ N ( x ) for every pair ( x, y ) of v ertices x and y in G . A graph G is called thr eshold [3] if G has no induced subgraph isomorphic to P 3 , O 4 , and 2 P 1 . Let F m n denote the set of threshold graphs with n vertices and m edges. It is easy to prov e that the following is true. 2.8 G is vertex c omp ar able if and only if G is thr eshold. It is also easy to pro v e the following: 2.9 A thr eshold is total ly de c omp osable. Mor e over, if H is a thr eshold gr aph with n ≥ 2 vertic es, then ther e exists a thr eshold gr aph H 0 with n − 1 vertic es such that H = H 0 + g or H = H 0 × g , wher e g is a one vertex gr aph. The ab o v e property pro vides a simple r e cursive description of a threshold graph with at least tw o vertices. The Laplacian sp ectrum and Laplacian p olynomial of a threshold graph ha ve some sp ecial properties. F or example, by 2.6 , ev ery threshold graph has an integral sp ectrum. 2.10 Now w e will deﬁne some sp ecial threshold graphs whic h we call extr eme [19, 20]. Let ( k , r, s ) be a triple of non-negativ e in tegers suc h that r < s . Let F ( k , r , s ) denote the graph obtained from the complete graph K s with s vertices as follows: ﬁx in K s a set A of r vertices and a vertex a in A , add to K s a new v ertex c and the set { cx : x ∈ A } of new edges to obtain graph C ( r, s ), add to C ( r, s ) the set B of k new vertices and the set { az : z ∈ B } of new edge to obtain graph F ( k , r , s ). Let H ( k, r, s ) denote the set of all graphs H obtained from C ( r, s ) ∪ B by adding a tree on the vertex set B ∪ { a } (and so this tree has k edges). Clearly , F ( k , r, s ) is a threshold graph and F ( k , r, s ) ∈ H ( k , r , s ). Ob viously , H ( k, r , s ) = { F ( k , r , s ) } if and only if k = 0. Let, as ab o v e, C m n b e the set of simple connected graphs with n vertices and m edges. It is easy to pro ve the following. 2.11 F or every p air ( n, m ) of inte gers such that C m n 6 = ∅ ther e exists a unique triple ( k , r, s ) of non-ne gative inte gers such that r < s and F ( k , r , s ) ∈ C m n . 6 Figure 1: Connected threshold graphs with m ≤ n + 3 2.12 If F ( k , r , s ) ∈ C m n , we put F ( k , r, s ) = F m n and H ( k , r , s ) = H m n , and so H m n ⊂ C m n . W e call F m n the extr eme thr eshold gr aph in C m n . Ob viously , F n − 1 n = S n for n ≥ 2, F 3 3 = ∆, F n n with n ≥ 4 is obtained from disjoint triangle ∆ and the ( n − 3)-edge star S by identifying its center with a vertex in ∆ (and so F n n with n ≥ 4 is W n ), F 5 4 = K − 4 , F n +1 n with n ≥ 5 is obtained from disjoint K − 4 and the ( n − 4)-edge star S b y identifying its center with a vertex of degree three in K − 4 , F 6 4 = K 4 , F n +2 n with n ≥ 5 is obtained from disjoint K 4 and the ( n − 4)-edge star S by iden tifying its center with a vertex in K 4 (see Figure 1). It is easy to pro ve the following: 2.13 L et n and m b e natur al numb ers. Supp ose that n − 1 ≤ m ≤ 2 n − 3 . Then ther e exists only one c onne cte d thr eshold gr aph with n vertic es and m e dges, and so F m n = { F m n } . 2.14 W e need notation for some sp ecial graphs: P is a path, O is a cycle, K 1 ,n is called a star S (if n ≥ 2, then a vertex of degree n is the c enter of S and every other v ertex is a le af of S ; if n = 1, then every of tw o vertices of S is a le af and a c enter ), Z is obtained from a star S with e ( S ) ≥ 2 b y adding a new vertex x and a new edge b et w een x and a leaf of S , K − 4 is obtained from K 4 b y remo ving one edge, ∆ is the triangle, and W is obtained from a star S with e ( S ) ≥ 3 by adding an edge b et w een t w o lea v es of S . Using the ab ov e op erations “+” and ‘‘ × ” on graphs (see 2.5 ) we hav e, in particular: mP 1 is a matc hing with m edges and P 2 + ( m − 2) P 1 is the disjoin t union of the t wo-edge path and a matching with m − 2 edges. 7 Figure 2: K = K n ( r ), where v ( K ) = n = 12 and diam ( K ) = r = 6. 2.15 Let D ( r ) denote the sets of trees having diameter r . W e call a graph F a star- for est if ev ery comp onen t of F is a star with at least one edge. Ob viously , T ∈ D (3) if and only if T is obtained from a star-forest with tw o comp o- nen ts b y connecting their centers b y an edge. Also, T ∈ D (4) if and only if T is obtained from a star-forest F with at least tw o comp onen ts having t w o or more edges by sp ecifying a leaf for every star and iden tify- ing all sp eciﬁed leav es with a new vertex. Let S 1 , . . . , S k b e the comp onen ts of F with e ( S 1 ) ≤ . . . ≤ e ( S k ) and let u ( T ) = ( e ( S 1 ) , . . . , e ( S k )). Obviously , every tree T in D (4) is uniquely deﬁned (up to isomorphism) by u ( T ). Let P b e a path with r ≥ 2 edges and F a star-forest with at most r − 1 comp onen ts. Then there is an injection ξ from C mp ( F ) to the set of non-leaf vertices of P . Now let Y b e a tree obtained from disjoint P and F by iden tifying the center of each comp onen t C of F with v ertex ξ ( C ) in P . Ob viously , Y ∈ D ( r ). A tree Y obtained this w a y is called a c aterpil lar . Let K ( r ) denote the set of caterpillars having diameter r , and so K ( r ) ⊆ D ( r ). Let K ( r ) b e the graph obtained from disjoint path P with r ≥ 2 edges and a star S b y identifying a center vertex of P and a center of S . Clearly , K ( r ) is a caterpillar and K ( r ) ∈ D ( r ). Let K n ( r ) b e graph K ( r ) with n v ertices (see Figure 2). Let D n ( r ) denote the set of n -vertex graphs in D ( r ), and K n ( r ) the set of n -v ertex caterpillars ha ving diameter r . 2.16 Let L ( r ) denote the sets of trees having r leav es. Let S ( r ), r ≥ 3, denote the set of trees T suc h that T has exactly one vertex of degree r and every other vertex in T has degree at most tw o, and so S ( r ) ⊆ L ( r ). If r ≥ 2, then we call the v ertex of degree r in T the r o ot of T . In other words, a tree T ∈ S ( r ) if and only if it can b e obtained from r disjoint paths P i , 1 ≤ i ≤ r , having at least t w o vertices by sp ecifying one end-v ertex of eac h path and iden tifying these sp eciﬁed end-v ertices of all paths. Let e ( P 1 ) ≤ . . . ≤ e ( P r ) and put w ( T ) = ( e ( P 1 ) , . . . , e ( P r )). Ob viously , every tree T in S ( r ) is uniquely deﬁned (up to isomorphism) by w ( T ). Let M ( r ) denote the tree T in S ( r ) such that every e ( P i ) in T , except p ossibly for e ( P r ), equals one. Let L ( r ) denote the tree T in S ( r ) with the prop ert y: | e ( P i ) − e ( P j ) | ≤ 1 for every 8 Figure 3: M = M n ( r ), where v ( M ) = n = 11 and l v ( M ) = r = 6. Figure 4: L = L n ( r ), where v ( L ) = n = 12 and l v ( L ) = r = 6. 1 ≤ i, j ≤ r . Let S n ( r ) denote the set of n -vertex graphs in S ( r ). W e denote the n -v ertex trees M ( r ) and L ( r ) b y M n ( r ) and L n ( r ), resp ectiv ely . F or t w o trees T and D in S n ( r ), let T > w D if w ( T ) lexicographically less than w ( D ). Ob viously , > w is a linear order on S n ( r ). In particular, if M n ( r ) > w T n > w L n ( r ) for ev ery T n ∈ S n ( r ) \ { M n ( r ) , L n ( r ) } . Examples of graphs M n ( r ) and L n ( r ) are giv en on Figures 2 and 4. 2.17 Given x = ( x 1 , . . . , x n ) ∈ R n and a p erm utation σ : { 1 , . . . , n } → { 1 , . . . , n } , let σ [ x ] = ( x σ (1) , . . . , x σ ( n ) ). A function f : R n → R is called symmetric if f ( x ) = f ( σ [ x ]) for every x ∈ R and every p erm utation σ : { 1 , . . . , n } → { 1 , . . . , n } . Let S F denote the set of symmetric functions. Let z and n b e p ositiv e in tegers, z ≤ n , and X = { x 1 , . . . , x n } , where each x i is a real num b er. Let σ z ( X ) = P { Q { x i : i ∈ Z } : Z ⊆ { 1 , . . . , n } , | Z | = z } . 9 F unction σ z ( X ) is called the elementary symmetric p olynomial of de gr e e z in the variables fr om X . Let σ ( ∅ ) = 1. 2.18 A symmetric functions f : R n → R is said to b e c onc ave if it has the following prop ert y ( ∩ ): for ev ery r , s ∈ { 1 , . . . , n } , r 6 = s , and ε ≥ 0, if x r ≤ x s and each x i ≥ 0, then f ( { x i : i ∈ { 1 , . . . , n } \ { x r } ∪ { x r + ε }} ) ≥ f ( { x i : i ∈ { 1 , . . . , n } \ { x s } ∪ { x s + ε }} ). It is easy to see that every elementary symmetric p olynomial is concav e. A symmetric functions f : R n → R is said to be c onvex if it has the follo wing prop erty ( ∪ ): for ev ery r , s ∈ { 1 , . . . , n } , r 6 = s , and ε ≥ 0, if x r ≥ x s and each x i ≥ 0, then f ( { x i : i ∈ { 1 , . . . , n } \ { x r } ∪ { x r + ε }} ) ≥ f ( { x i : i ∈ { 1 , . . . , n } \ { x s } ∪ { x s + ε }} ). F or x = ( x 1 , . . . , x n ) ∈ R n , let δ k ( x ) = P { x k i : i ∈ { 1 , . . . , n }} . Obviously , function δ k : R n → R is con vex. If f ≡ c for c ∈ R , then f is b oth con vex and concav e. 3 Preliminaries on Laplacian parameters of graphs Let G b e a graph with p ossible parallel edges but with no lo ops and let V = V ( G ) = { v 1 , . . . , v n } . Let A ( G ) b e the the symmetric n × n -matrix ( a ij ), where each a ii = 0 and eac h a ij , i 6 = j , is the num b er of parallel edges with the end-vertices v i and v j . Let D ( G ) b e the (diagonal) n × n -matrix ( d ij ), where each d ii = d ( v i , G ) and d ij = 0 for i 6 = j . Let L ( G ) = D ( G ) − A ( G ). Matrix L ( G ) is called the L aplacian matrix of G . Let X ⊂ V . W e need the following notation: • G X is the graph obtained from G b y identifying all vertices from X and removing all lo ops (that ma y app ear as a result of such iden tiﬁcation), • L X ( G ) denotes the matrix obtained from L ( G ) by remo ving the rows and columns corresp onding to ev ery vertex x ∈ X , and so in particular, L x ( G ) is the matrix obtained from L ( G ) b y remo ving the row and column corresp onding to v ertex x of G , and • f ( G, X ) denotes the n umber of spanning forests F of G suc h that every comp onent of F has exactly one v ertex in X , and s o the num b er t ( G ) of spanning trees of G is equal to f ( G, x ) for ev ery x ∈ V . W e start with the following classical Matrix Three Theorem (see, for example, [1, 5]). 3.1 L et G b e a gr aph with p ossible p ar al lel e dges. Then t ( G ) = det ( L x ( G )) for every vertex x in G . F rom 3.1 w e ha v e the follo wing generalization. 10 3.2 [19, 34] L et G b e a gr aph, V = V ( G ) , and X ⊆ V ( G ) . Then t ( G X ) = f ( G, X ) = det ( L X ( G )) . Here is a more general version of the Matrix T ree Theorem. Let L ( i,j ) ( G ) denote the matrix obtained from L ( G ) b y remo ving i -th row and j -th column. 3.3 [19] L et G b e a gr aph with p ossible p ar al lel e dges. Then t ( G ) = ( − 1) i + j det ( L ( i,j ) ( G )) for every i, j ∈ V ( G ) , i.e. t ( G ) e quals every c ofactor of L ( G ) . Here is y et another v ersion of the Matrix T ree Theorem. Let, as b efore, V = V ( G ) = { v 1 , . . . , v n } and let ¯ L ( G ) b e the ( n + 1) × ( n + 1)-matrix obtained from L ( G ) = ( l i,j ) b y adding the elements l i,n +1 = 1 if i ∈ { 1 , . . . , n } , l n +1 ,j = 1 if j ∈ { 1 , . . . , n } , and l n +1 ,n +1 = 0. 3.4 [11] L et G b e a gr aph with p ossible p ar al lel e dges. Then t ( G ) = n − 2 det ( ¯ L ( G )) . Let L ( λ, G ) = det ( λI − L ( G ) and S pctr ( G ) b e the m ulti-set of the eigenv alues of L ( G ). It is easy to see the follo wing. 3.5 [19, 34] L et G ∈ ¯ G n . Then ( a 1) L ( G ) is a p ositive semi-deﬁnite matrix and det ( L ( G )) = 0 , and so ( a 2) al l eigenvalues of L ( G ) ar e non-ne gative r e al numb ers : S pctr ( G ) = (0 = λ 0 ( G ) ≤ λ 1 ( G ) . . . ≤ λ n − 1 ( G )) , and ( a 3) L ( λ, G ) = λP ( λ, G ) , wher e P ( λ, G ) is a p olynomial of de gr e e n − 1 with the r o ot se quenc e ( λ 1 ( G ) ≤ . . . ≤ λ n − 1 ( G )) (we denote it b y S p ( G )): P ( λ, G ) = P { ( − 1) s c s ( G ) λ n − 1 − s : s ∈ { 0 , . . . , n − 1 }} , wher e c s ( G ) = σ s ( S p ( G )) for 0 ≤ s ≤ n − 1 . Let, as ab o ve, ∆( G n ) and δ ( G n ) denote the maximum and the minimum vertex degree of G n , resp ectively , and λ ( G n ) = λ n − 1 ( G n ). 3.6 [19, 29, 34] G ∈ ¯ G n . Then ( a 1) λ ( G ) ≤ max { d ( x, G ) + d ( y , G ) : x, y ∈ V ( G ) , x 6 = y } , ( a 2) λ i ( G ) ≥ λ i ( G − e ) for every e ∈ E ( G ) and i ∈ { 1 , . . . , n − 1 }} , and so ( a 3) λ ( G ) ≥ ∆( G ) + 1 and λ 1 ( G ) ≤ δ ( G ) . W e call P ( λ, G ) the L aplacian p olynomial of G . F or a graph F , let γ ( F ) = Q { v ( C ) : C ∈ C mp ( F ) } if F is a forest and γ ( F ) = 0, otherwise. Recall that F ( G ) is the set of spanning forests of G . Using 3.2 , w e obtained the following combinatorial in terpretation of the co eﬃcien ts of P ( λ, G ). 11 3.7 [19, 21, 29, 34] L et G b e a gr aph with n vertic es, V = V ( G ) , s an inte ger, and 0 ≤ s ≤ n − k , wher e k is the numb er of c omp onents of G . Then c s ( G ) = P { t ( G V − S ) : S ⊆ V , | S | = s } = P { f ( G, V − S ) : S ⊆ V , | S | = s } = P { γ ( F ) : F ∈ F ( G ) , e ( F ) = s } . Ob viously , c s ( G ) = 0 for s ≥ min { e ( G ) , v ( G ) − k } . Let ∇ ( G ) denote the num b er of triangles of G and δ i [ G ] = P { d ( v , G ) i : v ∈ v ( G ) } . F rom 3.7 w e ha v e, in particular: 3.8 [19, 29] L et G b e a gr aph with n , and m e dges. Then ( a 0) c 0 ( G ) = 1 , ( a 1) c 1 ( G ) = δ 1 [ G ] = 2 m , ( a 2) c 2 ( G ) = 2 m 2 − m − 1 2 δ 2 [ G ] , and ( a 3) c 3 ( G ) = 4 3 m 3 − 2 m 2 − ( m − 1) δ 2 [ G ] + 1 3 δ 3 [ G ] − 2 ∇ ( G ) , and ( a 4) c n − 1 ( G ) = nt ( G ) = ( − 1) n − 1 P (0 , G ) . The co eﬃcients of the p olynomial P ( λ, G ) satisfy the follo wing recursion. 3.9 [19, 29] L et G b e a gr aph with m e dges. Then ( m − s ) c s ( G ) = P { c s ( G − e ) : e ∈ E ( G ) } for s ∈ { 0 , . . . , m } . Giv en G ∈ G m n , let Φ( λ, G ) = λ m − n +1 P ( λ, G ) = λ m − n L ( λ, G ). This mo diﬁcation of the Laplacian p olynomial of a graph has the follo wing useful prop ert y . 3.10 [19, 29] L et G b e a gr aph and G 0 obtaine d fr om G by adding some isolate d vertic es. Then Φ( λ, G ) = Φ( λ, G 0 ) . Using 3.9 , we obtained the following recursion for Φ( λ, G ). 3.11 [19, 29] Φ( λ, G ) = Φ( a, G ) + P { R λ a Φ( t, G − u ) dt : u ∈ E ( G ) } . In [38] w e pro v ed the following imp ortant and frequently used Recipro cit y Theorem (that was later redisco vered and published several times). Let, as ab o ve, [ G ] c = K n − E ( G ), where G ∈ G n . 3.12 [19, 38] L et G b e a simple gr aph with n vertic es. Then ( a 1) λ i ( G ) + λ n − i ([ G ] c ) = n for every i ∈ { 1 , . . . , n − 1 } or, e quivalently, ( a 2) P ( λ, [ G ] c ) = ( − 1) n − 1 P ( n − λ, G ) . Recipro cit y Theorem 3.12 is a particular case of the following Recipro city Theorem for so called directed balanced graphs [33]. A simple dir e cte d gr aph ( or digr aph ) D is a pair ( V , E ), where V is a non-empt y set 12 and E ⊆ [ V ] 2 , where [ V ] 2 = { V × V \ { ( x, x ) : x ∈ V } (and so D has no parallel edges and no lo ops). A digraph ~ K ◦ = ( V , [ V ] 2 ) is called a simple c omplete digr aph with the v ertex set V . A digraph D c = ( V , [ V ] 2 \ E ) = ~ K ◦ \ E is called the c omplement of D = ( V , E ). Let V = V ( D ) = { v 1 , . . . , v n } . Let I n ( D ) b e the (diagonal) ( V × V )-matrix ( d ij ) suc h that d ii = d in ( v i ) and d ij = 0 for i 6 = j . Let A ( D ) b e the ( V × V )-matrix ( a ij ) such that a ij = 1 if ( v i , v j ) ∈ E ( D ) and a ij = 0 if ( v i , v j ) 6∈ E ( D ). Let L in ( D ) = I n ( D ) − A ( D ) and L out ( D ) = O ut ( D ) − A ( D ). Clearly , [ L in ( D )] > = L out ( D − 1 ). Let L in ( λ, D ) = det( λI − L in ( D )) and L out ( λ, D ) = det( λI − L out ( D )). W e put L ( D ) = L in ( D ) and L ( λ, D ) = L in ( λ, D ). Obviously , L ( D ) has an eigenv alue 0. Let L ( λ, D ) = λP ( λ, D ), and so P ( λ, D ) is a p olynomial. Let S p ( D ) denote the set of all n − 1 ro ots of P ( λ, D ). A digraph D is called b alanc e d if d in ( v , D ) = d out ( v , D ) 6 = 0 for ev ery v ∈ V ( D ). 3.13 [33] L et D b e a simple b alanc e d digr aph with n vertic es. Then ( a 1) ther e exists a bije ction σ : S p ( D ) → S p ( D c ) such that x + σ ( x ) = n for every x ∈ S p ( D ) or, e quivalently, ( a 2) P ( λ, D c ) = ( − 1) n − 1 P ( n − λ, D ) . Mor e over, the matric es L ( D ) and L ( D c ) ar e simultane ously diagonalizable. F urthermore, the follo wing Recipro cit y Theorem is true for all simple digraphs. 3.14 [33, 38] L et D b e a simple digr aph with n vertic es. Then ( a 1) P ( λ, D c ) = ( − 1) n − 1 P ( n − λ, D ) or, e quivalently, ( a 2) ther e exists a bije ction ε : S p ( D ) → S p ( D c ) such that e + ε ( e ) = n for every e ∈ S p ( D ) . The Recipro city Theorem 3.12 can also b e generalized as follo ws. 3.15 [19] L et D = ( V , E ) , D 1 = ( V , E 1 ) , and D 2 = ( V , E 2 ) b e a simple digr aphs such that E 1 ∪ E 2 = E and E 1 ∩ E 2 = ∅ (and so D = D 1 ∪ D 2 is de c omp ose d in two p arts D 1 and D 2 ) . Supp ose that L ( D 1 ) L ( D 2 ) = L ( D 2 ) L ( D 1 ) . Then ther e exist bije ctions α j : S p ( D ) → S p ( D j ) , j ∈ { 1 , 2 } , such that α 1 ( e ) + α 2 ( e ) = e for every e ∈ S p ( D ) . Theorem 3.15 can b e further generalized for the case when G is decomp osed in to p parts D 1 , . . . , D p , p ≥ 2. Here is an extension the Recipro city Theorem 3.12 to the class of bipartite graphs. 3.16 [13] L et X and Y b e ﬁnite disjoint sets, | X | = | Y | = s . L et B b e the c omplete ( X , Y ) -bip artite gr aph (and so v ( B ) = 2 s = n ) . Supp ose that B 1 and B 2 b e ( X , Y ) - bip artite gr aphs such that ( c 1) B 1 and B 2 ar e e dge disjoint and B 1 ∪ B 2 = B and ( c 2) B 1 is an r -r e gular gr aph, and so B 2 is an ( s − r ) -r e gular gr aph, and so 13 V ( B 1 ) = V ( B 2 ) = V ( B ) = X ∪ Y . L et S p ( B j ) = ( λ 1 ( B j ) ≤ . . . ≤ λ n − 1 ( B j )) b e the L aplacian sp e ctr a of B j , j = 1 , 2 . Then ( a 1) λ 2 s − 1 ( B 1 ) + λ 2 s − 1 ( B 2 ) = 2 s and λ i ( B 1 ) + λ 2 s − 1 − i ( B 2 ) = s for every i ∈ { 1 , . . . , 2 s − 2 } and, e quivalently, ( a 2) P ( λ, B 2 ) = ( − 1) 2 s − 1 ( λ − 2 s + λ 2 s − 1 )(( λ − s + λ 2 s − 1 ) − 1 P ( s − λ, B 1 ) . F or example, let B = K 3 , 3 , B 1 b e a 6-vertex cycle in B , B 2 = B \ E ( B 1 ), and so B 2 is a 3-edge matching, B = B 1 ∪ B 2 , B 1 is 2-regular, B 2 is 1-regular, and s = 3. Then S p ( B 1 ) = (1 , 1 , 3 , 3 , 4) and S p ( B 2 ) = { 0 , 0 , 2 , 2 , 2 } . Therefore λ 5 ( B 1 ) + λ 5 ( B 2 ) = 4 + 2 = 6 = 2 s and ( λ 1 ( B 1 ) , λ 2 ( B 1 ) , λ 3 ( B 1 ) , ( λ 4 ( B 1 )) + ( λ 4 ( B 2 ) , λ 3 ( B 2 ) , λ 2 ( B 2 ) , ( λ 1 ( B 2 )) = (1 , 1 , 3 , 3) + (2 , 2 , 0 , 0) = (3 , 3 , 3 , 3) = ( s, s, s, s ). F rom 3.12 w e ha ve: 3.17 [19, 38] L et A , B , and G b e simple gr aphs, v ( G ) = n , v ( A ) = a , and v ( B ) = b . Then ( a 1) 0 ≤ λ 1 ( G ) ≤ . . . ≤ λ n − 1 ( G ) ≤ n , ( a 2) cmp ( G ) = µ (0) and cmp ( G c ) = µ ( n ) + 1 , wher e µ ( z ) is the multiplicity of the eigenvalue z of L ( G ) , ( a 3) P ( λ, A + B ) = λP ( λ, A ) P ( λ, B ) , i.e. Φ( λ, A + B ) = Φ( λ, A )Φ( λ, B ) , and ( a 4) P ( λ, A × B ) = ( λ − a − b ) P ( λ − b, A ) P ( λ − a, B ) . Since nt ( G ) = c n − 1 ( G ) = ( − 1) n − 1 P (0 , G ), w e ha v e from 3.12 ( a 2): 3.18 [19, 38] L et G, F ∈ G n . Then ( a 1) t ( K n + r − E ( G )) = ( n + r ) r − 1 P ( n + r, G ) , and so ( a 2) t ( K n − E ( G )) = n − 1 P ( n, G ) = P { ( − 1) s c s ( G ) n n − 2 − s : s ∈ { 0 , . . . , n − 1 }} . 3.19 [19, 29] Equality ( a 2) in 3.18 is the inclusion-exclusion formula for the numb er t ( K n − E ( G )) of sp anning tr e es of K n avoiding the e dges of its sub gr aph G . Pro of. Let F b e a forest in K n and t ( K n , F ) denote the n um b er of spanning trees in K n con taining F . Let Σ s ( K n ) = P { t ( K n , F ) : F ∈ F ( K n ) , e ( F ) = s } . As we hav e shown in [28], t ( K n , F s ) = γ ( F ) n n − 2 − s . Therefore Σ s ( K n ) = c s ( G ) n n − 2 − s . By 3.18 ( a 2), t ( K n − E ( G )) = Σ 0 − Σ 1 + · · · + ( − 1) s Σ s + · · · + ( − 1) m Σ m , where m = e ( G ).  14 Let G b e a simple graph. Let G l denote the line gr aph of G , i.e. V ( G l ) = E ( G ) and ( a, b ) ∈ E ( G l ) if and only if edges a and b in G are adjacen t. Let ˙ G denote the graph obtained from G b y sub dividing every edge e of G b y exactly one v ertex s ( e ). Let ˇ G denote the graph obtained from ˙ G b y adding edge s ( a ) s ( b ) if and only if edges a and b in G are adjacent. 3.20 [34] L et G b e an r -r e gular gr aph with n vertic es and mr e dges ( and so m = 1 2 nr ) . Then ( a 1) P ( λ, G l ) = ( λ − 2 r ) m − n P ( λ, G ) , ( a 2) P ( λ, ˙ G ) = ( − 1) n ( λ − 2) m − n P ( λ ( r + 2 − λ )) , G ) , and ( a 3) P ( λ, ˇ G ) = ( λ − r − 1) n ( λ − 2 r − 2) m − n P ( λ 2 − ( r +2) λ λ − r − 1 , G ) . F rom 3.8 ( a 4) and 3.20 we ha ve: 3.21 [34] L et G b e an r -r e gular gr aph with n vertic es and mr e dges (and so m = 1 2 nr ). Then t ( ˇ G ) = n m + n 2 m − n ( r + 1) m − 1 ( r + 2) t ( G ) , t ( ˙ G ) = n m + n 2 m − n ( r + 2) t ( G ) , and t ( G l ) = n m 2 m − n r m − n t ( G ) . W e will see b elo w that threshold graphs (see deﬁnition 2.7 ) pla y a sp ecial role in problems T min and R min as well as some other optimization problems. It is known that a threshold graph is uniquely deﬁned by its degree sequence. F or a connected threshold graph G with n vertices, there exists a partition of V ( G ) = S ∪ K into tw o disjoin t sets S and K (with | S | = s , | K | = k , and so s + k = n ) such that K 6 = ∅ , G [ S ] has no edges, G [ K ] is a maximum complete subgraph of G , and there is an ordering K = ( x 1 , . . . , x k ) of the vertices in K and an ordering S = ( x k +1 , . . . , x n ) of vertices in S suc h that i > j ⇒ N ( x i , A ) ⊆ N ( x j , A ). Let d i = d ( x i , G ). Then d 1 ≥ . . . ≥ d n . The recursive description of a threshold graph G allo ws to giv e the following explicit form ulas for the Laplacian sp ectrum S p ( G ) and the n um b er of spanning trees t ( G ) in terms of the degree sequence of G . 3.22 [9, 19] L et G b e a c onne cte d thr eshold gr aph. Then ( a 1) S p ( G ) = ( d 1 + 1 ≥ . . . ≥ d k − 1 + 1 ≥ d k +1 ≥ . . . ≥ d n ) , wher e d 1 + 1 = n , ( a 2) t ( G ) = n − 1 Q { d i + 1 : i = 1 , . . . k − 1 } × Q { d j : i = k + 1 , . . . , n } , and ( a 3) t ( G ) = d n ( n n − 1 ) d n − 1 t ( G − x n ) = n − 1 k k − 1 Q { d k + i ( k + i k + i − 1 ) d k + i : i = 1 , . . . , s } . It is also easy to pro v e the following. 15 3.23 [9, 19] Every thr eshold gr aph is uniquely deﬁne d by its L aplacian sp e ctrum as wel l as by its de gr e e function. 3.24 [39], see also [34] L et G, F ∈ G m n . Supp ose that ( h 1) e ach c omp onent of G is a c omplete gr aph and ( h 2) isl ( G ) ≤ isl ( F ) . Then G and F ar e isomorphic if and only if P ( λ, G ) = P ( λ, F ) . The following example sho ws that condition ( h 2) in 3.24 is essen tial. Let G = K 6 + K 10 and F a graph obtained from the line graph of K 6 b y adding an isolated v ertex. Then P ( λ, G ) = P ( λ, F ) = λ ( λ − 6)( λ − 10) and, obviously , G and F are not isomorphic. Notice that the description of our theorem 3.24 in [5], page 163, is incorrect, namely , condition ( h 2) is missing. It turns out that among totally decomp osable graphs there are inﬁnitely man y non- isomorphic L -cosp ectral graphs. 3.25 [39] (see also [34]) F or every n ≥ 16 ther e exist total ly de c omp osable gr aphs G and F with n vertic es such that G and F ar e not isomorphic and P ( λ, G ) = P ( λ, F ) . Pro of. Let A = g s − 1 + (2 g ) g s − 1 and B = 2 g s , where s ≥ 2. Let G = A × B c and F = A c × B . Then G and F are totally decomp osable, v ( G ) = v ( F ) = 8 s , G and F are not isomorphic, and P ( λ, G ) = P ( λ, F ).  3.26 [19, 38] Supp ose that a gr aph G is obtaine d fr om disjoint simple gr aphs F 1 , . . . , F k by a series of op er ations + and × . If every F i has an inte ger L aplacian sp e ctrum, then G has also an inte ger L aplacian sp e ctrum. In p articular, every total ly de c omp osable gr aph has an inte ger sp e ctrum. In [31] we pro ved a more general result of this type for w eighted graphs. 3.27 [19] L et G b e a simple gr aph having n vertic es and the L aplacian sp e ctrum { 1 , . . . , n − 1 } . Then n ≥ 16 . Our pro of of theorem 3.27 does not use an y results obtained by computer except for the table in [43] of some sp ecial graphs on 7 vertices. Let T b e a tree and d ( x, y , T ) = d ( x, y ) b e the num b er of edges in the path of T with the ends x and y . Let W ( T ) = P { d ( x, y ) : { x, y } ⊆ V ( T ) , x 6 = y } . The parameter W ( T ) is called the Weiner index of a tr e e T . The W einer index turns out to b e a useful notion in organic c hemistry [46]. Let T ( x, y ) b e the graph obtained from T by identifying vertices x and y , x 6 = y . Then obviously , d ( x, y , T ) = t ( T ( x, y )). Therefore from 3.5 and 3.7 we ha ve: 16 3.28 [19, 20, 46] L et T b e a tr e e with n vertic es. Then W ( T ) = c n − 2 ( T ) = P { t ( T V − S ) : S ⊆ V , | S | = n − 2 } = P { f ( T , V − S ) : S ⊆ V , | S | = n − 2 } = P { γ ( F ) : F ∈ F ( T ) , e ( F ) = n − 2 } = σ n − 2 ( λ 1 ( G ) , . . . , λ n − 1 ( T )) . 4 Some op erations on graphs Let  b e a partial order relation on a subset A of G m n and Q : A → A a function on A . W e say that Q is an op er ation on A and that Q is  -incr e asing on A (  -de cr e asing on A ) if Q ( G )  G (resp ectiv ely , G  Q ( G )) for ev ery G ∈ A . W e sa y that Q is a  -monotone op er ation on A if either Q is  -increasing or  -decreasing on A . A function f : A → R induces the following quasy-linear order  f : for G, G 0 ∈ A , let G  f F if f ( G ) ≥ f ( G 0 ). In this case instead of  f -incr e asing ,  f -monotone , etc. w e sa y simply f -incr e asing , f -monotone , etc. A set Q of op erations on A induces a partial order relation as follows: given G, F ∈ A , w e deﬁne G  Q F if F can b e obtained from G by a series of op erations from Q . Ob viously , a graph A in A is  Q -minimal if and only if no op eration from Q can b e applied to A . W e will use the following simple observ ation. 4.1 If Q is  -incr e asing (  -de cr e asing ) op er ation on A , then for every G ∈ A ther e exists A ∈ A such that A  G ( r esp e ctively, G  A ) and op er ation Q c annot b e applie d to A ( i.e. A is  Q -minimal ) . In [17, 18, 25 – 27] w e in tro duced v arious op erations on graphs that preserve the num ber of v ertices and edges of a graph and that are monotone with resp ect to some graph parameters. Here are some of these op erations [25, 27]. Let H b e a graph, x, y ∈ V ( H ), and x 6 = y . W e call xH y a two-p ole with p oles x and y . Let uAv b e another t w o-p ole. Let G b e obtained from disjoint xH y and uAv b y identifying x with u and y with v . Let H xy ( G ) = ( G − [ x, X ]) ∪ [ y , X ], where X = N x ( A ) \ ( N y ( A ) ∪ y ) and Y = N y ( A ) \ ( N x ( A ) ∪ x ). W e call the t w o-p ole xH y an ( x, y )- hammo ck in G and call this op eration the H xy - op er ation or just a hammo ck op er ation (see Figure 5). A tw o-p ole xH y is called symmetric if H has an automorphism α : V ( G ) → V ( G ) suc h that α ( x ) = y and α ( y ) = x . W e call the H xy -op eration symmetric if xH y is a symmetric tw o-p ole. Obviously , if u or v is an isolated vertex in A , then H xy ( G ) is isomorphic to G . Therefore, when applying this op eration, w e will alwa ys assume that b oth u and v are not isolated vertices in A . W e call H xy -op eration an ( x, y ) -p ath op er ation or xP y -op er ation if xH y is an ( x, y )- path. Obviously , an ( x, y )-path op eration is symmetric. If d ( x, G ) = d ( x, H ) + 1, then we call the H xy -op eration a close-do or H xy -op er ation . The reverse of a close-do or H xy -op eration is called an op en-do or H xy -op er ation . 17 Figure 5: H xy -op eration A hammo c k op eration is a particular case of more general op erations in [25, 27] (see also 9.8 and 9.9 b elo w). One of p ossible sp eciﬁcations of the H xy -op eration is when V ( H ) = { x, y } (see Figure 6). W e call this simpler op eration the ♦ xy -op er ation [25, 27]. Obviously , the ♦ xy -op er ation is symmetric. W e will also use the following particular case of the close-do or ♦ xy -op eration [25, 27]. Let G b e a graph with three vertices x , y , z suc h that xz ∈ E ( G ) and y z 6∈ E ( G ). Let G 0 = G − xz + y z . W e put G 0 = D xy z ( G ) if xz is the only edge in G incident to x and sa y that G 0 is obtaine d fr om G by the D xy z -op er ation . W e also call a D xy z -op eration a close-do or op er ation (it ‘closes’ the ‘do or’ xz ). Accordingly , the reverse of a close-do or op eration is called an op en-do or op er ation . The following is a natural generalization of the H xy -op eration [19, 20]. Let G b e a graph, x, y ∈ V ( G ), and x 6 = y . Let K b e an induced subgraph of G con taining x and y , and so x K y is a t w o-p ole. Let X = N x ( G ) − ( V ( K ) ∪ N y ( G )) and Y = N y ( G ) − ( V ( K ) ∪ N x ( G )). Let [ x, X ] = { xv : v ∈ X } and [ y , X ] = { y v : v ∈ X } , and so [ x, X ] ⊆ E ( G ) and [ y , X ] ∩ E ( G ) = ∅ . Let G 0 = K xy ( G ) = ( G − [ x, X ]) ∪ [ y , X ], and so [ y , X ] ⊆ E ( G 0 ) and [ x, X ] ∩ E ( G ) = ∅ . W e sa y that K xy ( G ) is obtaine d fr om G by the K xy -op er ation (see Figure 7). Ob viously , if X or Y is empty , then K xy ( G ) = G . Therefore, when applying this op eration, we will alwa ys assume that b oth X and Y are not empty sets. It turns out that under certain conditions on ( G, K , x, y ) some graph parameters are 18 Figure 6: ♦ xy -op eration Figure 7: K xy -op eration 19 ‘monotone’ with resp ect to this op eration. A K xy -op eration on G is called α -symmetric or, simply , symmetric if G − ([ x, X ]) ∪ [ y , Y ]) has an automorphism α : V ( G ) → V ( G ) such that α ( x ) = y and α ( y ) = x , α ( z ) ∈ V ( K ) for ev ery z ∈ V ( K ) and α ( v ) = v for ev ery v ∈ X ∪ Y , and so α ( u ) ∈ V ( G − K ) for every u ∈ V ( G − K ). Clearly , H xy ( G ) is obtained from G b y the K xy -op eration, where x K y = xH y . The ab o v e described op erations prov ed to b e v ery useful for “impro ving” some graph c haracteristic and ﬁnding graphs with some extremal prop erties. It turns out that many results on the H xy -op eration are also true for the K xy -op eration. Here are some useful prop erties of the K xy -op eration. 4.2 L et ( G, K , x, y ) b e as describ e d ab ove and let G xy = K xy ( G ) . Then ( a 1) V ( G xy ) = V ( G y x ) , E ( G ) = E ( G xy ) , and ther e exists an isomorphism ω fr om G xy to G y x such that ω ( x ) = y and ω ( y ) = x , ( a 2) for a simple gr aph G , [ K xy ( G )] c = ( K c y x )([ G ] c ) and if the K xy -op er ation on G is α -symmitric, then the [ K ] c xy -op er ation on [ G ] c is also α -symmitric; in p articular, [ ♦ xy ( G )] c = ♦ y x ([ G ] c ) , and ( a 3) If K ∗ is the sub gr aph of G induc e d by V ( K ) ∪ ( N x ( G ) ∩ N y ( G ) , then K xy ( G ) = K ∗ xy ( G ) and the K xy -op er ation on G is symmetric if and only if the K ∗ xy -op er ation on G is sym- metric. Ob viously , the ♦ -op eration cannot b e applied to a graph G (i.e. G is  ♦ -minimal) if and only if for every tw o distinct vertices x , y of G either N ( x, G ) − { x, y } ⊆ N ( y , G ) or N ( y , G ) − { x, y } ⊆ N ( x, G ), i.e. if and only if G is a vertex comparable graph (or, the same, a threshold graph). Therefore w e ha ve: 4.3 [19, 20] If G is a non-thr eshold gr aph, then ther e exists a thr eshold gr aph F obtaine d fr om G by a series of ♦ -op er ations. It is easy to pro ve the following strengthening of 4.5 . 4.4 [19, 20] If G is a c onne cte d non-thr eshold gr aph, then ther e exists a c onne cte d thr eshold gr aph F obtaine d fr om G by a series of ♦ -op er ations. Th us, from 4.1 and 4.4 w e ha ve: 4.5 [19, 20] Supp ose that the ♦ -op er ation is  -de cr e asing. Then for every G in G m n ( in C m n ) and every clique K in G ther e exists F in G m n ( r esp e ctively, in C m n ) such that F is a thr eshold gr aph in G m n ( r esp e ctively, in C m n ) , F c ontains K , and G  F . 4.6 [19, 20, 27] Supp ose that the close-do or op er ation is  -de cr e asing. L et G ∈ C m n and let ¨ G b e the gr aph obtaine d fr om G by adding m − n + 1 isolate d vertic es. Then for every sp anning tr e e T of G ther e exists a tr e e D with m e dges such that T is a sub gr aph of D and D  ¨ G . 20 Theorem 4.6 follows from the fact that there exists a series of op en-do or op erations that transforms ¨ G to a tree D with e ( G ) edges and with T ⊆ D . 4.7 [19, 20, 27] Supp ose that the ♦ -op er ation is  -de cr e asing. L et F b e a simple gr aph with no isolate d vertic es, with r e dges, and with at most n vertic es. Then ( a 1) K n − E ( r P 1 )  K n − E ( P 2 + ( r − 2) r P 1 )  K n − E ( F ) for every F not isomorphic to r P 1 or P 2 + ( r − 2) P 1 and for r ≤ n/ 2 and ( a 2) K n − E ( F )  K n − E ( S r ) for every F not isomorphic to S r and for r ≤ n − 1 . The similar claims are true for a  -increasing ♦ -op eration. 5 Deﬁnitions of some p osets of graphs W e will ﬁrst describ e some Laplacian p osets, namely some partial order relations on G m and G m n related with the Laplacian p olynomials of graphs [17, 18, 26, 29]. Let ν ( G, F ) = max { v ( G ) , v ( F ) } . W e remind that P ( λ, G ) = P { ( − 1) s c s ( G ) λ n − 1 − s : s ∈ { 0 , . . . , n − 1 }} and Φ( λ, G ) = λ m − n +1 P ( λ, G ). where n = v ( G ). Let s and r b e non-negative integers and x b e a real num b er. W e write: ( τ ) G  τ s F if t ( K s + r − E ( G )) ≥ t ( K s + r − E ( F )) for ev ery integer r ≥ 0 and s ≥ ν ( G, F ) } , G  τ s F if G  τ s F and t ( K s + r − E ( G )) > t ( K s + r − E ( F )) for some r ≥ 0, and G  τ s F if G  τ s F and t ( K s + r − E ( G )) > t ( K s + r − E ( F )) for ev ery r ≥ 0. ( p ) G  p x F if v ( G ) = v ( F ) and P ( λ, G ) ≥ P ( λ, F ) for ev ery λ ≥ x , G  p x F if G  p x F and P ( λ, G ) > P ( λ, F ) for some λ ≥ x , and G  p x F if G  p x F and P ( λ, G ) > P ( λ, F ) for every λ ≥ x , ( φ ) G  φ x F if Φ( λ, G ) ≥ Φ( λ, F ) for every λ ≥ x , G  φ x F if G  φ x F and Φ( λ, G ) > Φ( λ, F ) for some λ ≥ x , and G  φ x F if G  φ x F and Φ( λ, G ) > Φ( λ, F ) for every λ ≥ x , ( c ) G  c F if v ( G ) = v ( F ) = n , c s ( G ) ≥ c s ( F ) for every s ∈ { 2 , . . . , n − 1) } and G  c F if G  c F and c s ( G ) > c s ( F ) for some s ∈ { 2 , . . . , n − 1 } , ( t ) G  t F if t ( G ) ≥ t ( F ) and G  t F if t ( G ) > t ( F ), ( λ ) G  λ F if λ ( G ) ≥ λ ( F ) and G  λ F if λ ( G ) > λ ( F ), ( ∞ ) G  ∞ F if there exists a num ber N such that P ( λ, G ) ≥ P ( λ, F ) for λ ≥ N . If v ( G ) = v ( F ) = n , we write  p instead of  p n and  τ instead of  τ n . If G  φ x F 21 and x = max { λ ( G ) , λ ( F ) } , we write  φ instead of  φ x , and so in this case λ ( G ) ≤ λ ( F ). Notice that if v ( G ) = v ( F ), then relations  p x and  φ x are the same and  φ ⇒  p . No w we will describ e some other p osets on graphs. Let R k ( p, G ) denote the prob- abilit y that the random graph ( G, p ) has at most k components. Let a k s ( G ) is the n um b er of spanning subgraphs of G with s edges and at most k comp onen ts (and so f k ( G ) = a k n − k ( G ) is the num b er of spanning forests of G with exactly k comp onen ts). Then obviously , R k ( p, G ) = P { a k s ( G ) p s q m − s : s ∈ { n − k , . . . , m }} . Let A ( λ, G ) denote the characteristic p olynomial of the adjacency matrix A ( G ) of a graph G . Let h 0 ( G ) and h 1 ( G ) denote the num ber of Hamiltonian cycles and Hamilto- nian paths in G , resp ectiv ely . Let G, F ∈ G m n . W e write: ( a ) G  a ( k ) F if a k s ( G ) ≥ a k s ( F ) for every s ∈ { n − k , . . . , m } , ( r ) G  r ( k ) F if R k ( p, G ) ≥ R k ( p, F ) for every p ∈ [0 , 1], G  r ( k ) F if R k ( p, G ) > R k ( p, F ) for every p ∈ (0 , 1), ( α ) G  α F if α ( G ) ≤ α ( F ) and A ( λ, G ) ≥ A ( λ, F ) for λ ≥ α ( G ), G  α F if G  α F and A ( λ, G ) > A ( λ, F ) for some λ ≥ α ( F ), and G  α F if G  α F and A ( λ, G ) > A ( λ, F ) for all λ ≥ α ( F ), and ( h ) G  h i F if h i ( G ) ≥ h i ( F ) for i ∈ { 1 , 2 } . Put a s ( G ) = a 1 s ( G ), R ( p, G ) = R 1 ( p, G ), and let  r b e  r (1) . Notice that a n − 1 ( G ) = t ( G ) = n − 1 c n − 1 ( G ), where ( − 1) n − 1 c n − 1 ( G ) is the last co eﬃcien t of P ( λ, G ). F or G, F ∈ G m n and z ∈ { c, p, λ, t, a, r, ∞} , w e write G  z F if G  z F if G  z F and F  z G . W e also write G  α,β F instead of G  α F and G  β F . In order to deﬁne relation  c on G m n w e need the follo wing simple observ ation. F or G, F ∈ G m n , G and F are called C mp -c osp e ctr al if b oth G and F are forests and there exists a bijection σ : C mp ( G ) → C mp ( F ) suc h that v ( C ) = v ( σ ( C )) for ev ery C ∈ C mp ( G ). It is easy to see that if G and F are C mp -cosp ectral, then c m ( G ) = c m ( F ) = γ ( F ). No w in view of 3.8 we deﬁne  c on G m n as follows: G  c F if c s ( G ) > c s ( F ) for ev ery s ∈ { 2 , . . . , m − 1 } in case G and F are C mp -cosp ectral, and c s ( G ) > c s ( F ) for ev ery s ∈ { 2 , . . . , n − 1 } in case G and F are not C mp -cosp ectral. 5.1 L et G, F ∈ G m n . Then ( a 1) G  φ F ⇒ G  λ F and G  φ F ⇒ G ≺ λ F , ( a 2) G  φ F ⇒ G  p F ⇒ G  τ F ⇒ G  ∞ F , 22 ( a 3) G  φ F ⇔ G  c F ⇔ G  p F ⇔ G  τ F ⇔ G  ∞ F ⇔ P ( λ, G ) ≡ P ( λ, F ) , ( a 4)  φ ,  c ,  p ,  τ ,  a ( k ) ,  r ar e p artial quasi-or der r elations and  t ,  λ ,  ∞ ,  h i ar e line ar quasi-or der r elations on G m n , and ( a 5) G  a ( k ) F ⇒ G  r ( k ) F , G  r ( k ) F ⇒ a k n − k ( G ) ≥ a k n − k ( F ) , and G  r ( k ) F ⇔ R k ( p, G ) ≡ R k ( p, F ) . Pro of. By 3.12 ( a 2), G  φ F ⇒ G  p F . By 3.18 , G  p F ⇒ G  τ F . All other claims ab ov e are ob vious.  It is interesting to compare relations  c and  p . Let P + ( λ, G ) = P { ( − 1) 2 r c 2 r ( G ) λ n − 1 − 2 r : r ∈ { 0 , . . . , d 1 2 n e} and P − ( λ, G ) = P { ( − 1) 2 r +1 c 2 r +1 ( G ) λ n − 2 r : r ∈ { 0 , . . . , d 1 2 n e − 1 } . W e call P + ( λ, G ) and P − ( λ, G ) the p ositive and the ne gative p art of P ( λ, G ), resp ec- tiv ely . If G, F ∈ G m n and G  c F , then b oth p ositive and negative parts of P ( λ, G ) are greater or equal to the p ositive and negative parts of P ( λ, F ), resp ectiv ely , for λ ≥ 0. Therefore it is v ery p ossible and not a surprise at all that there are pairs G, F ∈ G m n suc h that G  p F but G 6 c F or G  c F but G 6 p F or, moreov er, G  p F but F  c G . It is more surprising when b oth G  c F and G  p F . Here are tw o examples illustrating the ab ov e observ ation. Example 1. Let F b e a star with the center vertex c and with at least three edges and let G b e obtained from F − u b y b y ad ding a new edge betw een tw o v ertices adjacen t to c . Then G, F ∈ G m n for some n = m + 1 ≥ 4, and G exactly one triangle, an isolated v ertex, exactly t wo comp onents. It is easy to sho w that G  p F . It is also easy to show that c 2 ( G ) − c 2 ( F ) = 2 m − 6 > 0, c n − 1 ( G ) = 0, and c n − 1 ( F ) = n . Therefore G 6 c F . Example 2. Let G b e the graph ha ving three comp onen ts, namely , a triangle and t w o one edge comp onents. Let F b e the graph tw o comp onen ts, namely , P 3 and P 4 . Then P ( λ, G ) = λ 2 ( λ − 2) 2 ( λ − 3) 2 = λ 2 ( λ 4 − 10 λ + 37 λ 2 − 60 λ + 36) and P ( λ, F ) = λ ( λ − 1)( λ − 2)( λ − 3)(( λ − 2) 2 − 2) = λ ( λ 5 − 10 λ 4 + 37 λ 3 − 62 λ 2 + 46 λ − 12). Therefore G  p F and F  c G . As we will see b elo w, the symmetric op erations describ ed in Section 4 turn out to b e monotone with resp ect to almost all relations on graphs men tioned in this Section and in Section 8. Let ∈ { a ( k ) ,  c ,  p ,  τ ,  r ( k ) ,  t ,  α ,  h i ,  ∞ } . In particular, w e will hav e: 5.2 L et G 0 b e a gr aph obtaine d fr om a gr aph G by a ♦ -op er ation. Then G  G 0 . 6 On Laplacian p osets of graphs In this section we will describ e some results on the monotonicity of the op erations 23 deﬁned in Section 4 with resp ect to some Laplacian p osets in Section 5 and on the problems T max and T min of ﬁnding graphs with the maximum and minimum num b er of spanning trees among the graphs of the same size, resp ectively . Since t ( G ) is prop or- tional to R ( p, G ) for p close to zero, some results in Section 7 on R ( p, G ) provide the corresp onding results on t ( G ) (see, for example, 7.9 ). Using 3.5 ( a 3) we can prov e the following inequalities. 6.1 [19, 34] L et G ∈ ¯ G m n . Then ( a 1) c s ( G ) ≤ (2 m/n ) s  n − 1 s  , and in p articular, ( a 2) c s ( G ) ≤ c s ( K n ) for every s ∈ { 0 , . . . , n − 1 } , and so ( a 3) K n  c G , and mor e over, K n  c G for G not isomorphic to K n . Giv en a symmetric function δ : R n → R and a graph G , let δ [ G ] = δ ( { d ( v , G ) : v ∈ V ( G ) } . In particular, let δ s [ G ] = P { d ( v , G ) s : v ∈ V ( G ) } . 6.2 [19, 20] L et G, F ∈ G m n and F b e obtaine d fr om G by a symmetric hammo ck- op er ation. L et δ : R n → R b e a c onvex symmetric function. Then δ [ G ] ≤ δ [ F ] , and so, in p articular, δ s [ G ] ≤ δ s [ F ] for every p ositive inte ger s . 6.1 Some results on relations  t ,  τ , and  p 6.3 [19, 27] L et G, G 0 ∈ G m n and F b e obtaine d fr om G by a symmetric H xy -op er ation. Then ( a 1) t ( G ) ≥ t ( G 0 ) and ( a 2) t ( G ) = t ( G 0 ) if and only if G is isomorphic to G 0 or x is a cut vertex in G . F rom 4.5 , 6.2 , and 6.3 we ha ve: 6.4 [19, 20] F or every G ∈ G m n and a clique K in G ther e exist thr eshold gr aphs F 0 and F 00 in G m n c ontaining K and such that t ( G ) ≥ t ( F 0 ) and δ [ G ] ≤ δ [ F 00 ] , wher e δ : R n → R is a c onvex symmetric function. Since a ♦ -op eration is a particular case of an H -op eration, the claim of 6.3 is also true for a ♦ -op eration. Therefore from 4.2 ( a 2) and 6.3 we hav e: 6.5 [19, 27] L et G, F ∈ G m n and F b e obtaine d fr om G by a ♦ -op er ation. Then t ([ G ] c ) ≥ t ([ F ] c ) . F rom 3.11 and 6.5 we ha ve: 6.6 [19, 20] L et G b e a gr aph and G 0 b e obtaine d fr om G by a ♦ -op er ation. Then G  τ G 0 . The following is a generalization of 6.6 for an H -op eration and  p . 24 6.7 [19] L et G b e a gr aph and G 0 b e obtaine d fr om G by a symmetric H -op er ation. Then G  p G 0 . F rom 4.7 and 6.5 we hav e, in particular, the following c haracterization of graphs with n v ertices and e ( K n ) − r edges ha ving the maximum n umber of spanning trees pro vided r ≤ n/ 2. W e recall that r P 1 is a graph-matching with r edges and P 2 + k P 1 is a disjoint union of the 2-edge path P 2 and the k -edge matc hing. 6.8 [19, 27, 29] L et F b e a simple gr aph with no isolate d vertic es and with r e dges. Supp ose that F is not isomorphic to r P 1 or P 2 + ( r − 2) P 1 . Then t ( K n − E ( rP 1 )) > t ( K n − E ( P 2 + ( r − 2) P 1 ) > t ( K n − E ( F )) for every n and r such that r ≤ n/ 2 . In [23] we were able to obtain more general result by giving a complete characteriza- tion of graphs with n v ertices and  n 2  − r edges having the maxim um n um b er of spanning trees, where r ≤ n . The pro of of this result uses essentially some of the ab o v e op erations and some more delicate  p -monotone op erations for sp ecial classes of graphs [17]. Here is the description of all graphs with n vertices,  n 2  − n edges and the maximum n um b er of spanning trees. 6.9 [23] L et Q ∈ G n n . Supp ose that t ( K n − E ( Q )) ≥ t ( K n − E ( F )) for every F ∈ G n n . Then ( a 0) if n = 0 mo d 3 , say n = 3 k and k ≥ 1 , then Q = k O 3 , ( a 1) if n = 1 mo d 3 , say n = 3 k + 1 and k ≥ 1 , then Q = O 4 + ( k − 1) O 3 , and ( a 2) if n = 0 mo d 3 , say n = 3 k + 2 and k ≥ 1 , then Q = O 5 + ( k − 1) O 3 . Let, as ab ov e, S r b e the graph-star with r edges. F rom 4.7 and 6.5 we ha v e, in particular: 6.10 [19, 29] L et F b e a simple gr aph with no isolate d vertic es, with r e dges, and with at most n vertic es. Supp ose that F is not isomorphic to S r . Then t ( K n − E ( F )) > t ( K n − E ( S r )) for every n and r such that r ≤ n − 1 . F rom 4.6 and 6.5 we ha ve, in particular: 6.11 [19, 27] L et G b e a c onne cte d gr aph. Then for every sp anning tr e e T of G ther e exists a tr e e D with e ( G ) e dges such that D c ontains T and D  τ n G , wher e n = e ( G ) + 1 . As ab o v e, P is a path, O is a cycle, S is a star, and K − 4 is obtained from K 4 b y deleting one edge. Also let Z b e obtained from a star S by adding a new vertex x and a new edge b et ween x and a leaf of S and let W b e a windmill, i.e. W is obtained from a star S with at least t wo edges by adding an edge b et ween tw o leav es of S . 25 6.12 [23] L et G m b e a gr aph with m e dges. Supp ose that in e ach claim b elow G m is not isomorphic to any of the sp e cial gr aphs liste d in this claim. We write  p inste ad of  p n , wher e n = 2 m . ( a 1) If m ∈ { 2 , 3 , 4 } , then 2 P 1  p P 2 , 3 P 1  p P 1 + P 2  p P 3  p O 3  p S 3 , and 4 P 1  p 2 P 1 + P 2  p P 1 + P 3  p P 1 + O 3  p P 4  p P 1 + S 3  p O 4  p Z 4  p W 4  p S 4 . ( a 2) F or m = 5 , Z 5  ∞ K − 4 and ( Z 5 , K − 4 ) is the only  p -inc omp ar able p air of gr aphs. ( a 3) If m ≥ 6 , then mP 1  p ( m − 2) P 1 + P 2  p ( m − 4) P 1 + 2 P 2  p ( m − 3) P 1 + P 3  p ( m − 3) P 1 + O 3  p ( m − 6) P 1 + 3 P 2  p ( m − 5) P 1 + P 2 + P 3  p ( m − 4) P 1 + P 4  p G m  p W m  p S m . ( a 4) If m ≥ 7 , then G m  p Z m  p W m  p S m . The following is a generalization of 6.6 for ( G m n ,  p ). 6.13 [19, 20] L et G b e a simple gr aph and G 0 b e obtaine d fr om G by a ♦ -op er ation. Then G  p G 0 . Ob viously , G − e  τ G . It turns out that this inequalit y remains true for  p . 6.14 [19, 20] L et G b e a simple gr aph and e an e dge of G . Then ( λ − 1) P ( λ, G − e ) > λP ( λ, G ) for λ ≥ n , and so G − e  p G . Here are some results on  p -comparison of n -v ertex trees that we were able to pro v e using the tree op erations from [26]. Notice that if G and F are trees with the same num b er of v ertices, then P ( λ, G ) = Φ( λ, G ). Let T ( a, b, c ) denote the tree T from S ( r ) with w ( T ) = ( a, b, c ), where a ≤ b ≤ c . 6.15 [26] L et T n b e a tr e e with n vertic es. The tr e es ar e numb er e d ac c or ding to the Har ary list of T n with n ≤ 10 (see [10]) . ( a 1) If 1 ≤ n ≤ 9 , then  p is a line ar or der on T n . In p articular, we have: ( a 1 . 1) the  p -or der of the tr e es in T 7 is 1 , 2 , 3 , 4 , 9 , 10 , 5 , 6 , 11 , 7 , 8 , ( a 1 . 2) the  p -or der of the tr e es in T 8 is 1 , 2 , 3 , 5 , 4 , 13 , 14 , 17 , 16 , 15 , 23 , 6 , 7 , 8 , 18 , 20 , 19 , 22 , 9 , 10 , 21 , 11 , 12 , and ( a 1 . 3) the  p -or der of the tr e es in T 8 is  p -or der is 26 1 , 2 , 3 , 4 , 5 , 6 , 19 , 20 , 23 , 22 , 21 , 24 , 26 , 25 , 27 , 44 , 42 , 45 , 7 , 8 , 9 , 10 , 11 , 28 , 30 29 , 40 , 39 , 31 , 38 , 32 , 46 , 47 , 33 , 41 , 12 , 13 , 14 , 37 , 35 , 36 , 42 , 15 , 16 , 34 , 17 , 18 . ( a 2) Ther e ar e  p -non-c omp ar able tr e es in T 10 . ( a 3) If n ≥ 10 , then in T n ther e exist four suc c essively  p -b est and six suc c essively  φ -worst tr e es ( se e Figur e 8 ) , and in p articular, P n  p T (1 , 1 , n − 3)  p T (1 , 2 , n − 4)  p T (1 , 3 , n − 5)  p T n  p Z n  p S n , wher e T n is not isomorphic to any of the sp e cial tr e es liste d ab ove. Here are some more details on  p -comparison of n -vertex trees with n ≤ 10. 6.16 L et T n b e a tr e e with n vertic es. The tr e es ar e numb er e d ac c or ding to the Har ary list of T n with n ≤ 10 (see [10]) . ( a 1) If 1 ≤ n ≤ 7 , then  p =  φ =  c . ( a 2) In T 8  p =  p x for x = 5 , 6639 but  p 6 =  φ , namely, ther e ar e two  φ -non-c omp ar able p airs (4 , 13) and (8 , 18) of tr e es in T 8 , namely, λ (8) = 5 . 236 , λ (18) = 5 . 125 , and 0 ≤ L ( λ, 8) < L ( λ, 18) for λ (8) ≤ λ < x , L ( x, 8) = L ( x, [18]) , and L ( λ, 8) > L ( λ, 18) for λ > x . Ther e is one  c -non-c omp ar able p air (16 , 6) in T 8 , namely, c 0 (16) = c 0 (6) = 1 , c 1 (16) = c 1 (6) = 14 , c 5 (16) = c 5 (6) = 204 , c 7 (16) = c 7 (6) = 8 , c i (16) < c i (6) for 2 ≤ i ≤ 4 , and c 6 (16) > c 6 (6) . ( a 3) In T 9  p =  p x for x = 6 . 842 but  p 6 =  φ . Ther e ar e 26  φ -non-c omp ar able p airs of tr e es in T 9 , for example, p air (14 , 37) is one of them, namely, x = 6 . 84129 is the “cr ossing p oint” for p air (14 , 37) , λ (14) = 6 . 147 , λ (37) = 6 . 062 , 0 ≤ L ( λ, 14) < L ( λ, 37) for λ (14) ≤ λ < x , L ( x, 14) = L ( x, 37) , and L ( λ, 14) > L ( λ, 37) for λ > x . ( a 4) (71 , 82) , (70 , 82) , and (69 , 82) ar e  p -non-c omp ar able p airs of tr e es in T 10 . In p articular, x = 11 . 4772 > 10 is the “cr ossing p oint” for p air (71 , 82) , namely, λ (71) = 7 . 119 , λ (82) = 6 . 702 , 0 ≤ L ( λ, 71) < L ( λ, 82) for λ (71) ≤ λ < x , L ( x, 71) = L ( x, 82) , and L ( λ, 71) > L ( λ, 82) for λ > x . 6.2 Some results on relation  φ Using the recursion 3.11 w e pro ved by induction the follo wing inequalities. 6.17 [19, 29] L et G m b e a gr aph with m e dges not isomorphic to any of the sp e cial gr aphs liste d b elow. Then mP 1  φ P 2 + ( m − 2) P 1  φ G m  φ S m . 6.18 [19, 26] L et G m b e a c onne cte d gr aph with m ≥ 4 e dges. If G m is not a p ath, then P m  φ G m . 27 Figure 8: F our  p -“b est” and six  p -“w orst” trees in T n , n ≥ 9. In [26] w e were able to ﬁnd an inductive pro of of 6.18 using 3.11 . On the other hand, we found some  φ -increasing op erations on trees that allo w us to giv e another pro of of 6.18 and some other results on the p oset ( T n ,  φ ). It turns out that the revers of an ( x, y )-path op eration is one of  φ -increasing op erations on trees and that every tree T n whic h is not a path can b e transformed to the path P n b y a series of such op erations. Later we used these metho ds to pro v e similar results for the p oset ( G m n ,  c ). In particular, we found that some of the ab o v e mentioned  φ -increasing op erations on trees are also  c -increasing op erations (see 6.52 b elo w). Using 3.11 we obtain from 6.18 : 6.19 [26] L et G m b e an e dge 2-c onne cte d gr aph with m ≥ 5 e dges. If G m is not a cycle, then O m  φ G m . 6.20 [26] L et i and j b e p ositive inte gers such that 2 ≤ j + 1 ≤ i − 1 . Then Φ( λ, P i − 1 + P j +1 ) > Φ( λ, P i + P j ) for λ ≥ 4 , i.e. P i − 1 + P j +1  φ 4 P i + P j . In [30] (whic h is a contin uation of [26]) we obtained similar  φ x -comparison results for some other classes of graphs. In particular, we extended 6.20 to the class of graphs whose each comp onen t is either a path or a cycle. Here are some of these results. W e remind that if G  φ x F for x = max { λ ( G ) , λ ( F ) } , we write  φ instead of  φ x , and so in this case λ ( G ) ≤ λ ( F ). Also notice that 3 ≤ λ ( O 2 s +1 ) < λ ( O 2 k ) = 4. 28 6.21 [30] L et n and k b e inte gers such that n ≥ 2 and k ≥ 0 . Then ( a 0) O 2 n − 1 + O 2 n +1  φ O 4 n  φ 2 O 2 n , ( a 1) O 2 n − 2 k − 3 + O 2 n +2 k +3  φ x O 2 n − 2 k − 1 + O 2 n +2 k +1  φ x O 2 n − 1 + O 2 n +1 for n ≥ k + 3 and x = 4 . 05 , and ( a 2) 2 O 2 n  φ O 2 n − 2 k + O 2 n +2 k  φ O 2 n − 2 k − 2 + O 2 n +2 k +2 for n ≥ k + 3 . 6.22 [30] L et n and k b e inte gers such that n ≥ 1 and k ≥ 0 . Then ( a 0) 2 O 2 n +1  φ O 4 n +2  φ O 2 n + O 2 n +2 , ( a 1) O 2 n − 2 k − 1 + O 2 n +2 k +3  φ x O 2 n − 2 k +1 + O 2 n +2 k +1  φ x 2 O 2 n +1 for n ≥ k + 1 and x = 4 . 05 , and ( a 2) O 2 n + O 2 n +2  φ O 2 n − 2 k + O 2 n +2 k  φ O 2 n − 2 k − 2 + O 2 n +2 k +2 for n ≥ k + 3 . 6.23 [30] L et n and k b e inte gers such that n ≥ 2 and k ≥ 0 . Then ( a 0) O 2 n − 1 + O 2 n +2  φ x O 4 n +1  φ O 2 n + O 2 n +1 for x = 4 . 133 , ( a 1) O 2 n − 2 k − 1 + O 2 n +2 k +2  φ O 2 n − 2 k +1 + O 2 n +2 k  φ O 2 n − 1 + O 2 n +2 for n ≥ k + 2 , and ( a 2) O 2 n − 1 + O 2 n +2  φ O 2 n + O 2 n +1  φ O 2 n − 2 k + O 2 n +2 k +1  φ O 2 n − 2 k − 2 + O 2 n +2 k +3 for n ≥ k + 2 . 6.24 [30] L et n and k b e inte gers such that n ≥ 2 and k ≥ 0 . Then ( a 0) O 2 n − 1 + O 2 n  φ x O 4 n − 1  φ O 2 n − 2 + O 2 n +1 for x = 4 . 325 , ( a 1) O 2 n − 2 k − 1 + O 2 n +2 k  φ O 2 n − 2 k +1 + O 2 n +2 k − 2  φ O 2 n − 1 + O 2 n for n ≥ k + 2 , and ( a 2) O 2 n − 1 + O 2 n  φ O 2 n − 2 + O 2 n +1  φ O 2 n − 2 k + O 2 n +2 k − 1  φ O 2 n − 2 k − 2 + O 2 n +2 k +1 for n ≥ k + 2 . W e also obtain similar results for graphs O n + P k . Here are some of them. 6.25 [30] L et n and k b e inte gers such that n ≥ 2 and k ≥ 2 . Then ( a 1) if n ≥ 2 k + 1 , then P 2 k + O 2 n +1  φ x P 2 k +1 + O 2 n +1 for x ≥ 4 . 74 , ( a 2) if k ≤ n < 2 k + 1 , then P 2 k +1 + O 2 n +1  φ x P 2 k + O 2 n +1 for x > 4 and ( a 3) if n < k , then P 2 k + O 2 n +1  φ x P 2 k +1 + O 2 n +1 for x ≥ 4 . 6.26 [30] L et n and k b e inte gers such that n ≥ 2 and k ≥ 2 . Then ( a 1) if n < k or n ≥ 2 k , then P 2 k +1 + O 2 n  φ x P 2 k − 1 + O 2 n +2 for x > 4 and ( a 2) if k ≤ n < 2 k , then P 2 k − 1 + O 2 n +2  φ x P 2 k +1 + O 2 n for x > 4 . 29 6.27 [30] L et n and k b e inte gers such that n ≥ 2 and k ≥ 1 . Then P 2 k +2 + O 2 n  φ x P 2 k + O 2 n +1 for x > 3 . 5 . 6.28 [30] L et n and k b e inte gers such that n ≥ 2 . Then P 2 n − 3 + O 4  φ x P 1 + O 2 n for x > 4 . F rom 6.21 - 6.24 we ha ve the following corollaries. 6.29 [30] Al l ine qualities in 6.21 - 6.24 r emain true if the r elations  φ and  φ x ar e r eplac e d by  p . 6.30 [30] L et M n b e a  φ x -maximal 2-r e gular gr aph in G n n for x = 4 . 05 . Then ( a 0) if n = 0 mo d 3 , say n = 3 k and k ≥ 1 , then M n = k O 3 , ( a 1) if n = 1 mo d 3 , say n = 3 k + 1 and k ≥ 1 , then M n = O 4 + ( k − 1) O 3 , and ( a 2) if n = 2 mo d 3 , say n = 3 k + 2 and k ≥ 1 , then M n = O 5 + ( k − 1) O 3 . 6.31 [30] L et W n b e a  φ -minimal 2-r e gular gr aph in G n . Then ( a 0) if n = 0 mo d 4 , say n = 4 k and k ≥ 1 , then W n = k O 4 , ( a 1) if n = 1 mo d 4 , say n = 4 k + 1 and k ≥ 1 , then W n = O 5 + ( k − 1) O 4 , ( a 2) if n = 2 mo d 4 , say n = 4 k + 2 and k ≥ 1 , then W n = O 6 + ( k − 1) O 4 , and ( a 3) if n = 3 mo d 4 , say n = 4 k + 3 and k ≥ 1 , then W n = O 7 + ( k − 1) O 4 . 6.32 [30] L et B M 2 n b e a  φ -maximal bip artite 2-r e gular gr aph in G 2 n . Then B M 2 n = O 2 n . 6.33 [30] L et B W 2 n b e a  φ -minimal bip artite gr aph in G 2 n . Then ( a 0) if n = 0 mo d 2 , say 2 n = 4 k and k ≥ 1 , then B W 2 n = k O 4 and ( a 1) if n = 1 mo d 2 , say 2 n = 4 k + 2 and k ≥ 1 , then B W 2 n = O 6 + ( k − 1) O 4 . No w using 3.8 ( a 4) and Recipro cit y Theorem 3.12 we ha v e from 6.29 - 6.31 : 6.34 [30] L et F b e a 2-r e gular gr aph with 2 n vertic es (and so ev ery comp onent of F is an ev en cycle) . Supp ose that F is not isomorphic to M n in 6.30 or to W n in 6.31 . Then t ( K n + s \ E ( M n )) > t ( K n + s \ E ( F )) > t ( K n + s \ E ( W n )) for every non-ne gative inte ger s . Also using 3.8 ( a 4) and Recipro cit y Theorem 3.16 for regular bipartite graphs we obtain from 6.29 , 6.32 , and 6.33 the description of regular bipartite graphs with 2 n v ertices and n 2 − 2 n edges having the maximum and the minim um num b er of spanning trees: 30 6.35 [19] L et B b e a 2-r e gular bip artite gr aph with 2 n vertic es (and so every comp onent of B is an ev en cycle) . Supp ose that B is not a cycle and not isomorphic to B W 2 n in 6.33 . Then t ( K n,n \ E ( B M 2 n )) > t ( K n,n \ E ( B )) > t ( K n,n \ E ( B W 2 n )) . W e were also able to prov e the follo wing inequalities in addition to 6.16 ( a 3). 6.36 [19] If m ≥ 4 , then P m  φ O m  φ T (1 , 1 , m − 2) . Using some  φ -monotone op erations on n -v ertex trees from [26], we ha ve obtained the following tw o theorems on the p osets ( D n ( r ) ,  φ ) and ( L n ( r ) ,  φ ) (see the deﬁnitions in 2.15 and 2.16 ). It turns out that similar results hold for  c as well (see 6.70 and 6.71 ). Let δ 0 ( G ) b e the minim um non-leaf vertex degree in G . 6.37 [19, 20] L et r ≥ 3 and n ≥ r + 2 . ( a 1) for every D ∈ K n ( r ) \ K n ( r ) ther e exists Y ∈ K n ( r ) such that D  φ Y , ( a 2) D  φ K n ( r ) for every D ∈ K n ( r ) \ { K n ( r ) } , and ther efor e, (from ( a 1) and ( a 2)) , ( a 3) D  φ K n ( r ) for every D ∈ D n ( r ) \ { K n ( r ) } , ( a 4) ( D n (3) ,  φ ) is a line ar or der p oset, namely, for T , T 0 ∈ D n (3) we have: T  φ T 0 ⇔ δ 0 ( T ) > δ 0 ( T 0 ) , and ( a 5) ( D n (4) ,  φ ) is a line ar p oset, namely, for T , T 0 ∈ D n (4) we have: T  φ T 0 ⇔ T > u T 0 . 6.38 [19, 20] L et r ≥ 3 and n ≥ r + 2 . Then ( a 0) L n ( r )  φ L n ( r + 1) for every r ∈ { 2 , . . . , n − 2 } , ( a 1) ( S n ( r ) ,  φ ) is a line ar p oset, namely, for T , T 0 ∈ S n ( r ) we have: T  φ T 0 ⇔ T > w T 0 , ( a 2) M n ( r )  φ L for every L ∈ S n ( r ) \ { M n ( r ) } , ( a 3) for every L ∈ L n ( r ) \ S n ( r ) ther e exists Z ∈ S n ( r ) such that L  φ Z , and ( a 4) L  φ L n ( r ) for every L ∈ S n ( r ) \ { L n ( r ) } , ther efor e (from ( a 3) and ( a 4)) ( a 5) L  φ L n ( r ) and, in p articular, λ ( L ) > λ ( L n ( r )) for every L ∈ L n ( r ) \ { L n ( r ) } . Let T b e an n -v ertex tree of maximum degree r . Then T can b e transformed b y some  φ -increasing op erations from from [26], to an n -vertex star-tree S with r leav es. Therefore by 6.38 ( a 2), w e hav e: 6.39 [19] L et T b e an n -vertex tr e e of maximum de gr e e r and T is not isomorphic to M n ( r ) . Then M n ( r )  φ T . 31 As we mentioned in 5.1 ( a 1), G  φ F ⇒ G  λ F and G  φ F ⇒ G ≺ λ F . Therefore, the replacement of  φ b y  λ and  φ b y ≺ λ in an y theorem results in another correct theorem. F or example, from 6.17 and 6.18 , we hav e, in particular: 6.40 [19, 26] L et G m b e a gr aph with m e dges and with no isolate d vertic es. ( a 1) if G m is not isomorphic to S m , then λ ( G m ) < λ ( S m ) , ( a 2) if m ≥ 4 and G m has a vertex of de gr e e at le ast thr e e, then λ ( P m ) < λ ( G m ) , and so ( a 3) if G m is a tr e e not isomorphic to P m or S m , then λ ( P m ) < λ ( T m ) < λ ( S m ) . Here is an alternative pro of of 6.40 . Pro of. W e prov e ( a 1). Obviously , λ ( S m ) = m + 1. Let Q b e a comp onen t of G m with the maxim um n um b er of v ertices. Let v ( Q ) = n 0 . Then n 0 ≤ m + 1. By 3.17 ( a 1), λ ( G m ) ≤ n 0 . If n 0 < m + 1, then w e are done. Therefore let n 0 = m + 1. Then Q is a tree with m edges. Since G m has no isolated v ertices, clearly Q = G m . Hence Q is a tree not isomorphic to S m . Then the complemen t of Q is connected. Th us, b y 3.12 , λ ( Q ) < n 0 = m + 1. No w w e prov e ( a 2). Since G m has a v ertex of degree at least three, λ ( G m ) ≥ 4, by 3.6 ( a 3). By 3.6 ( a 1), λ ( P m ) ≤ 4, and so λ ( P m ) ≤ λ ( G m ). It is kno wn [5] that λ ( P m ) = 4 sin 2 ( m 2 m +2 π ). Thus, λ ( P m ) < 4 ≤ λ ( G m ).  Claim ( a 2) in 6.40 also follo ws from 6.41 b elo w. There are inﬁnitely man y examples showing that theorem 6.14 on the  p -monotonicit y of the ♦ -op eration is not true if  p is replaced b y  φ . How ev er, it may b e true under some additional condition. 6.41 [19] L et G ∈ G and G 0 b e obtaine d fr om G by the ♦ x,y -op er ation. Supp ose that G is a c onne cte d gr aph and G 0 is a bip artite gr aph. Then ( a 1) λ ( G ) ≤ λ ( G 0 ) , and mor e over, ( a 2) G  φ G 0 . Pro of (uses 3.6 , 3.11 , 9.11 , and 9.12 ). Let L + ( G ) = D ( G ) + A ( G ) and λ + ( G ) the maxim um eigen v alue of L + ( G ). Since G is a connected graph, we hav e from Theorem 8.4.5 in [12]: Claim 1. λ ( G ) ≤ λ + ( G ) . ( p1 ) W e prov e claim ( a 1) in our theorem. By Claim 1 , λ ( G ) ≤ λ + ( G ). By 9.11 below, λ + ( G ) ≤ λ + ( G 0 ). Since G 0 is a bipartite graph, b y 9.12 , λ + ( G 0 ) = λ ( G 0 ). Thus, λ ( G ) ≤ λ + ( G ) ≤ λ + ( G 0 ) = λ ( G 0 ). ( p2 ) Now we pro v e claim ( a 2) in our theorem. Let X = N ( x, G ) \ ( N ( y , G ) ∪ y ) and Y = N ( y , G ) \ ( N ( x, G ) ∪ x ). Ob viously , 32 e ( G ) = e ( G 0 ). F or e ∈ E ( G ) let α ( e ) = e if e 6∈ [ x, X ] and ε ( e ) = e 0 if e = xs ∈ [ x, X ] and e 0 = y s for some s ∈ X . Since G 0 is obtained from G by the ♦ x,y -op eration, ε is a bijection from E ( G ) to E ( G 0 ). By deﬁnition of  φ , G  φ G 0 if and only if λ ( G ) ≤ λ ( G 0 ) and Φ( λ, G ) ≥ Φ( λ, G 0 ) for λ ≥ λ ( G 0 ). W e prov e claim ( a 2) by induction on e ( G ). Our claim is obviously true for e ( G ) = 0. No w w e supp ose that our claim is true for e ( G ) = m − 1 and we hav e to prov e that it is also true for e ( G ) = m , where m ≥ 1. Let E = E ( G ), λ 0 = λ ( G 0 ), and ∆( λ, G ) = Φ( λ, G ) − Φ( λ, G 0 ). Obviously , ♦ x,y ( G − e ) = G 0 − ε ( e ). Therefore ∆( t, G − e ) = Φ( t, G − e ) − Φ( t, G 0 − ε ( e )). By 3.11 , Φ( λ, G ) = Φ( s, G ) + P { R λ s Φ( t, G − e ) dt : e ∈ E } . Therefore ∆( λ, G ) = ∆( s, G ) + P { R λ s ∆( t, G − e ) dt : e ∈ E } . Since G 0 is bipartite, G 0 − ε ( e ) is also bipartite. Therefore by the induction hypothesis, ∆( t, G − e ) ≥ 0 for t ≥ λ ( G 0 − ε ( e )). By 3.6 , λ ( G 0 − ε ( e )) ≤ λ ( G 0 ) = λ 0 . Therefore from the ab o v e inequalit y w e ha v e: ∆( t, G − e ) ≥ 0 for t ≥ λ ( G 0 ) = λ 0 . By our claim ( a 1) (that w e already prov ed in ( p1 )), λ ( G ) ≤ λ ( G 0 ) = λ 0 . Therefore ∆( λ 0 , G ) = Φ(( λ 0 , G ) ≥ 0. Thus, if λ ≥ λ 0 , then ∆( λ, G ) = ∆( λ 0 , G ) + P { R λ λ 0 ∆( t, G − e ) dt : e ∈ E } ≥ 0.  W e can also prov e the following generalization of 6.41 . 6.42 [19] L et G ∈ G and G 0 b e obtaine d fr om G by a symmetric K xy -op er ation. If G 0 is a bip artite gr aph, then G  φ G 0 . It turns out that a symmetric K xy -op eration (and in particular, the ♦ -op eration) is also  α -monotone. 6.43 [19] L et G b e a c onne cte d gr aph and G 0 b e the gr aph obtaine d fr om G by a symmetric K xy -op er ation. Then G  α G 0 . Here is another useful  α -inequalit y . 6.44 [19] L et G b e a c onne cte d gr aph. Then G − e  α G for every e ∈ E ( G ) . 6.3 Some results on relation  c 6.45 [19, 20] L et G b e a simple gr aph, x, y ∈ V ( G ) , x 6 = y , and C x , C y diﬀer ent c omp onents of G such that x ∈ V ( C x ) , y ∈ V ( C y ) , v ( C x ) ≥ 2 , and v ( C y ) ≥ 2 . Supp ose that G 0 is obtaine d fr om G by the ♦ xy -op er ation. Then G  c G 0 . 6.46 [19, 20] L et G b e a simple c onne cte d gr aph and G 0 b e obtaine d fr om G by a symmetric H xy -op er ation, wher e H is c onne cte d. Then ( a 1) c s ( G ) > c s ( G 0 ) for every s ∈ { 2 , . . . , n − 2 } and ( a 2) c n − 1 ( G ) = c n − 1 ( G 0 ) if and only if G is isomorphic to G 0 or x is a cut vertex in G . 33 Alternativ e pro ofs of 6.46 as w ell as more general results and some other  c - monotone op erations on graphs are given b elo w (see 6.61 and 6.67 - 6.69 ). F rom 6.46 w e ha ve, in particular: 6.47 [19] L et A , B , and H b e disjoint c onne cte d gr aphs, a ∈ V ( A ) , b ∈ V ( B ) , and x, y ∈ V ( H ) , x 6 = y . L et G b e obtaine d fr om A , B , and H by identifying a with x and b with y and let G 0 b e obtaine d fr om A , B , and H by identifying a and b with x . Supp ose that the two-p ole xH y is symmetric. Then c s ( G ) > c s ( G 0 ) for every s ∈ { 2 , . . . , n − 2 } and c n − 1 ( G ) = c n − 1 ( G 0 ) . Since a close-do or op eration is a particular case of a symmetric hammo ck-operation, w e ha v e from 6.46 : 6.48 [19] L et G b e a c onne cte d gr aph and G 0 b e obtaine d fr om G by a close-do or op er ation. L et s ∈ { 2 , . . . v ( G ) − 1 } . Then c s ( G ) > c s ( G 0 ) and c s ( G ) = c s ( G 0 ) if and only if G and G 0 ar e isomorphic. F rom 4.6 and 6.48 we ha ve, in particular: 6.49 [19, 20] L et G b e a c onne cte d gr aph and let ¨ G b e the gr aph obtaine d fr om G by adding e ( G ) − v ( G ) + 1 isolate d vertic es. Then for every sp anning tr e e T of G ther e exists a tr e e D with e ( G ) e dges such that D c ontains T and D  c ¨ G . Mor e over, if e ( G ) ≥ v ( G ) , then D  c ¨ G . F rom 4.5 and 6.46 we ha ve: 6.50 [19, 20] F or every gr aph G in C m n ther e exists a thr eshold gr aph F in C m n such that G  c F . In the next theorem we will use the notions of an extreme threshold graph F m n and the corresp onding set of graphs H m n deﬁned in 2.10 and 2.12 . 6.51 [19, 20] L et G ∈ C m n and G 6 = F m n . ( a 0) If m = n − 1 ≥ 3 , then c s ( G ) > c s ( F n − 1 n ) for every s ∈ { 2 , . . . , n − 2 } and c n − 1 ( G ) = c n − 1 ( F n − 1 n ) = n . ( a 1) If m = n ≥ 3 , then c s ( G ) > c s ( F n n ) for every s ∈ { 2 , . . . , n − 2 } and c n − 1 ( G ) = c n − 1 ( H ) = 3 n for every H ∈ H n n , and so c n − 1 ( G ) = c n − 1 ( F n n ) . ( a 2) If n ≥ 4 and m = n + 1 , then c s ( G ) > c s ( F n +1 n ) for every s ∈ { 2 , . . . , n − 2 } and c n − 1 ( G ) = c n − 1 ( H ) = 8 n for every H ∈ H n +1 n , and so c n − 1 ( G ) = c n − 1 ( F n +1 n ) . ( a 3) If n ≥ 5 and m = n + 2 , then c s ( G ) > c s ( F n +2 n ) for every s ∈ { 2 , . . . , n − 2 } and c n − 1 ( G ) = c n − 1 ( H ) = 16 n for every H ∈ H n +2 n , and so c n − 1 ( G ) = c n − 1 ( F n +2 n ) . ( a 4) If n ≥ 6 and n + 2 ≤ m ≤ 2 n − 4 , then c s ( G ) > c s ( F m n ) for every s ∈ { 2 , . . . , n − 2 } and c n − 1 ( G ) = c n − 1 ( H ) for every H ∈ H m n , and so c n − 1 ( G ) = c n − 1 ( F m n ) . ( a 5) If m = 2 n − 3 , then for every n ≥ 6 ther e exists G ∈ C m n such that G 6 c F m n . 34 Claims ( a 0) - ( a 4) in 6.51 follow basically from 6.50 . Recall that an P xy -op eration is an H xy -op eration, where xH y is an xy -path. This op erations was one of the  φ -increasing op eration on trees in [26]. It is easy to see that any tree T n can b e transformed to a path P n b y a series of rev erse P xy -op erations. Therefore we ha ve from 6.46 and 6.49 : 6.52 [19, 20] Supp ose that G ∈ G m n , G is c onne cte d, and gr aph ¨ G is obtaine d fr om G by adding m − n + 1 isolate d vertic es. If G is not a p ath, then P m  c ¨ G . As w e ha v e mentioned b efore, theorem 6.52 on the p oset ( G m n ,  c ) is similar to theorem 6.18 on the p oset ( G m n ,  φ ). No w we wan t to demonstrate another pro of of 6.52 that uses 3.9 and that is similar to the pro of of 6.18 in [26] using 3.11 . W e need the following claim that can b e easily pro v ed b y induction using 3.9 . 6.53 [19] L et i ∈ { 1 , 2 } and ( A i , B i ) b e a p air of disjoint p aths. Supp ose that ( h 1) v ( A 1 ∪ B 1 ) = v ( A 2 ∪ B 2 ) = n , ( h 2) v ( A i ) ≤ v ( B i ) for every i ∈ { 1 , 2 } , and ( h 3) v ( A 1 ) > v ( A 2 ) . Then c s (( A 1 ∪ B 1 ) > c s ( A 2 ∪ B 2 ) for every s ∈ { 2 , . . . , n − 1 } . Notice that 6.53 is also a simple particular case of 6.68 b elo w. W e also need the follo wing result interesting in itself. Let mx ( G ) denote the num b er edges of a comp onen t with the maximum n um b er of edges. 6.54 [19, 26] L et P b e a p ath, T a tr e e, and e ( P ) = e ( T ) . Then ther e exists a bije ction ε : E ( T ) → E (( P ) such that mx ( T − u ) ≥ mx ( P − ε ( u )) . No w w e are ready to prov e 6.52 . Obviously , b ecause of 6.49 theorem 6.52 follows from the theorem for trees b elo w. 6.55 [19, 20] Supp ose that G is a tr e e with n vertic es, G is not a p ath, and n ≥ 4 . Then P n  c G . Pro of (uses 3.9 , 6.53 , and 6.54 ). W e pro ve our claim by induction on m = n − 1. Recall that for t wo trees Q and R of m edges, Q  c R if c s ( Q ) > c s ( R ) for 2 ≤ s ≤ m − 1. Supp ose that m = 3. Then T 3 = { P 3 , S 3 } and c 2 ( P 3 ) = 10 > 9 = c 3 ( S 3 ). Therefore our claim is true for m = 3. W e assume that our claim is true for e ( P ) < m and prov e that it is also true for e ( P ) = m ≥ 5. Let s ∈ { 2 , . . . , m − 1 } . By 6.54 , there exists a bijection ε : E ( T ) → E (( P ) suc h that mx ( T − u ) ≥ mx ( P − ε ( u )). By 3.9 , ( m − s ) c s ( T ) = X { c s ( T − u ) : u ∈ E ( T ) (6.1) 35 and ( m − s ) c s ( P ) = X { c s ( P − ε ( u )) : u ∈ E ( T ) . (6.2) Let A u and B u b e the tw o comp onen ts of T − u and e ( A u ) ≤ e ( B u ). Similarly , let A 0 u and B 0 u b e the tw o comp onen ts of P − ε ( u ) and e ( A 0 u ) ≤ e ( B 0 u ). Then e ( B u ) ≥ e ( B 0 u ). If A and B are disjoint graphs, then c s ( A ∪ B ) = X { c i ( A ) c j ( B ) : i + j = s } . (6.3) Therefore c s ( T − u ) = c s ( A u ∪ B u ) = P { c i ( A u ) c j ( B u ) : i + j = s } . Let A 00 u and B 00 u b e t wo disjoin t paths such that e ( A 00 u ) = e ( A u ) and e ( B 00 u ) = e ( B u ). Since e ( A u ) < m and e ( B u ) < m and since A u and B u are trees, we hav e b y the induction h yp othesis: c i ( A 00 u ) ≥ c i ( A u ) and c j B 00 u ) ≥ c j ( B u ). Therefore by (6.3), c s ( A 00 u ∪ B 00 u ) ≥ c s ( A u ∪ B u ) = c s ( T − u ) . (6.4) Since e ( A 00 u ) = e ( A u ) and e ( B u ) ≥ e ( B 0 u ), we hav e by 6.4 and 6.53 : c ( s ( P − ε ( u )) = c s ( A 0 u ∪ B 0 u ) ≥ c s ( A 00 u ∪ B 00 u ) = c s ( T − u ). Since T is not a path, there is u ∈ E ( T ) such that either A u or B u is not a path. Let D u b e one of A u , B u whic h is not a path. Then b y the induction h yp othesis, c i ( D 00 u ) > c i ( D u ) for some i ≤ s . No w b y (6.1) and (6.2) we ha v e: c s ( P ) > c s ( T ).  The next theorem on  c is similar to theorem 6.19 on  φ in [26]. 6.56 [19, 20] Supp ose that G ∈ G m n , G is e dge 2-c onne cte d, and gr aph ¨ G is obtaine d fr om G by adding m − n + 1 isolate d vertic es. If G is not a cycle, then O m  c G . Pro of (uses 3.9 and 6.52 ). W e pro v e that O n  c G . Since e ( O n ) = e ( G ), we can assume that E ( O n ) = E ( G ). Let s ∈ { 2 , . . . , n − 1 } . By 3.9 , ( n − s ) c s ( O n ) = P { c s ( O n − u ) : u ∈ E ( C n ) } and ( n − s ) c s ( G ) = P { c s ( G − u ) : u ∈ E ( G ) } . Ob viously , O n − u is a path with n vertices for every u ∈ E ( O n ). Since G is edge 2- connected, G − u is connected for ev ery u ∈ E ( G ). Therefore b y 6.52 , c s ( O n − u ) ≥ c s ( G − u ) for every u ∈ E ( G ). Since G is not a cycle, G − t is not a path for some t ∈ E ( G ). Hence by 6.52 , c s ( O n − t ) > c s ( G − t ). Thus from the ab o ve recursions w e ha v e: c s ( O n ) > c s ( G ).  6.57 [19, 20] L et O , G ∈ C n n , wher e O is a cycle. If G is not a cycle, then O  c G . 36 6.4 More on Laplacian p osets of graphs It turns out that a matching M with m edges is not only the  φ -maxim um but also the  c -maxim um in G m . 6.58 L et M b e a gr aph-matching, G a simple gr aph not isomorphic to M , and e ( G ) = e ( M ) . Then M  c,φ G . Pro of (uses 3.7 , 3.9 , 3.10 , and 6.17 ). Let e ( G ) = m . By 6.17 , M  φ G . Thus, it suﬃces to pro v e that c s ( M ) > c s ( G ) for ev ery s ∈ { 2 , . . . , m } . W e prov e our claim by induction on m . By 3.10 , Φ( λ, H ) do es not dep end on the num b er of isolated vertices of a graph H . Supp ose that m = 2. Then G is a 3-vertex path plus an isolated vertex. No w Φ( λ, M ) = ( λ − 2) 2 = λ 2 − 4 λ + 4 and Φ( λ, G ) = ( λ − 3)( λ − 1) = λ 2 − 4 λ + 3. It follows that for m = 2 our claim is true. Now w e assume that our claim is true for e ( G ) = m − 1 and we pro v e that it is also true for e ( G ) = m ≥ 3. Since e ( M ) = e ( G ), there is a bijection from E ( M ) to E ( G ). W e can assume that this bijection is the identit y , i.e. that E ( M ) = E ( G ). By 3.9 , ( m − s ) c s ( M ) = P { c s ( M − u ) : u ∈ E ( G ) } and ( m − s ) c s ( G ) = P { c s ( G − u ) : u ∈ E ( G ) } for 2 ≤ s ≤ m − 1. Supp ose that 2 ≤ s ≤ m − 1. Obviously , e ( M − u ) = e ( G − u ) = m − 1 and M − u consists of a matching M u plus tw o isolated vertices, and so e ( M u ) = e ( M − u ) = m − 1. Since Φ( λ, M − u ) do es not dep end on the num b er of isolated vertices of a graph M − u , eac h c s ( M u ) = c s ( M − u ). By the induction h yp othesis, c s ( M − u ) ≥ c s ( G − u ) for ev ery u ∈ E . Since G is not a matc hing, G has an edge t such that the graph obtained from G − t b y removing the isolated v ertices is not a matching. Then by the induction h yp othesis, c s ( M − t ) > c s ( G − t ). Thus, our claim for m follows from the ab ov e recursions. Finally , supp ose that s = m . Then b y 3.7 , c m ( M ) = γ ( M ) = 2 m . If G is not a forest, then c m ( G ) = 0. So we assume that G is a forest, and so b y 3.7 , c m ( G ) = γ ( G ). Since G is not a matching, it has a comp onent C with r = e ( C ) ≥ 2. Consider in M the the subgraph N induced by the edge subset E ( C ). Then N is a matc hing with r edges and γ ( N ) = 2 r . If r = m , then γ ( G ) = m + 1, and so γ ( M ) = 2 m > m + 1 = γ ( G ). So we assume that r < m . Then γ ( M ) = γ ( N ) γ ( M − N ) and γ ( G ) = γ ( C ) γ ( G − C ). Since r ≥ 2, γ ( N ) = 2 r > r + 1 = γ ( C ) and by the induction h yp othesis, γ ( M − N ) ≥ γ ( G − C ). Therefore c m ( M ) = γ ( M ) > γ ( G ).  Using 3.9 , 6.17 , and 6.58 we can also pro v e the following stronger result. 6.59 L et G b e a simple gr aph with m e dges not isomorphic to mP 1 , P 2 + ( m − 2) P 1 , and S m . Then mP 1  c,φ P 2 + ( m − 2) P 1  c,φ G  c,φ S m . No w we need to recall the deﬁnition of a K xy -op eration on a graph G . Let x, y ∈ V ( G ) and K b e an induced subgraph of G con taining x and y . Let X = N x ( G ) \ ( V ( K ) ∪ N y ( G )), 37 Y = N y ( G ) \ ( V ( K ) ∪ N x ( G )), [ x, X ] = { xv : v ∈ X } , and [ y , X ] = { y v : v ∈ X } , and so [ x, X ] ⊆ E ( G ) and [ y , X ] ∩ E ( G ) = ∅ . Let G 0 = K xy ( G ) = ( G − [ x, X ]) ∪ [ y , X ], and so [ y , X ] ⊆ E ( G 0 ) and [ x, X ] ∩ E ( G 0 ) = ∅ . W e say that K xy ( G ) is obtaine d fr om G by the K xy -op er ation . W e call K xy -op eration on G symmetric if G − ([ x, X ]) ∪ [ y , Y ]) has an automorphism α : V ( G ) → V ( G ) such that α ( x ) = y , α ( y ) = x , α [ K ] = K , α ( v ) = v for every v ∈ X ∪ Y , and so α [ G − ( K ∪ X ∪ Y )] = G − ( K ∪ X ∪ Y ). Let, as ab o v e, δ s [ G ] = P { d ( v , G ) s : v ∈ V ( G ) } . It is easy to prov e the following generalization of 6.2 . 6.60 [19, 20] L et G, G 0 ∈ G m n and G 0 b e obtaine d fr om G by a symmetric K xy -op er ation. L et f : R n → R b e a c onvex symmetric function and f [ G ] = f ( { d ( v , G ) : v ∈ V ( G ) } ) . Then f [ G ] ≤ f [ G 0 ] , and so, in p articular, δ s [ G ] ≤ δ s [ G 0 ] for every p ositive inte ger s . 6.61 L et G b e a gr aph and G 0 b e obtaine d fr om G by a symmetric K xy -op er ation. Then c s ( G ) ≥ c s ( G 0 ) for every s ∈ { 0 , . . . , v ( G ) − 1 } . Pro of (uses 3.8 , 3.9 , and 4.2 ( a 3)). If α : V ( G ) → V ( G ) is an automorphism of a graph G and F is a subgraph of G , then let α [ F ] denote the image of F under the automorphism α . In particular, if e = pq ∈ E ( G ), then α [ e ] = α ( p ) α ( q ) and if A ⊆ E ( G ), then let α [ A ] = { α [ a ] : a ∈ A } . By 4.2 ( a 3), we can assume that N x ( G ) ∩ N y ( G ) ⊆ V ( K ). No w, since G 0 is obtained from G b y the K xy -op eration, we ha v e: G 0 = K xy ( G ) = ( G − [ x, X ]) ∪ [ y , X ], where X = N x ( G ) \ V ( K ), Y = N y ( G ) \ V ( K ), and so [ x, X ] ⊆ E ( G ) and [ y , X ] ∩ E ( G ) = ∅ . Since the K xy -op eration in G is symmetric, G \ ([ x, X ] ∪ [ y , Y ]) has an automorphism α : V ( G ) → V ( G ) such that α ( x ) = y , α ( y ) = x , α [ K ] = K , and α ( v ) = v for every v ∈ X ∪ Y , and so α [ G − ( K ∪ X ∪ Y )] = G − ( K ∪ X ∪ Y ) and the K xy -op eration is α -symmetric. Let E − ( G ) = E ( G \ ([ x, X ] ∪ [ y , Y ])). Let A ( G ) = { e ∈ E − ( G ) : α [ e ] = e } and B ( G ) = { e ∈ E − ( G ) : α [ e ] 6 = e } , and so A ( G ) ∩ B ( G ) = ∅ and E ( G ) = A ( G ) ∪ B ( G ) ∪ [ x, X ] ∪ [ y , Y ]. By 3.8 , we can assume that s ∈ { 2 , . . . , v ( G ) − 1 } . W e prov e our claim b y induction on e ( G ) = m . If m = 0, then our claim is ob viously true. Supp ose that our claim is true for every graph G with e ( G ) = m − 1. W e prov e that it is also true for ev ery graph G with e ( G ) = m ≥ 1. Let e v = xv and e 0 v = y v for v ∈ X and let E [ X ] = [ x, X ] = { e v : v ∈ X } and E 0 [ X ] = [ y , X ] = { e 0 v : v ∈ X } . Then G 0 = ( G − E [ X ]) ∪ E 0 [ X ]. Obviously , E ( G ) − E [ X ] = E ( G 0 ) − E 0 [ X ]. Let ε ( u ) = u if u ∈ E ( G ) − E [ X ] and ε ( e v ) = e 0 v if e v ∈ E [ X ] (and so v ∈ X ). Then ε : E ( G ) → E ( G 0 ) is a bijection. By 3.9 , we ha v e the follo wing recursions for s ≤ m − 1: ( m − s ) c s ( G ) = X { c s ( G − u ) : u ∈ E ( G ) } (6.5) 38 and ( m − s ) c s ( G 0 ) = X { c s ( G 0 − ε ( u ) : u ∈ E ( G ) } (6.6) Supp ose ﬁrst that s = m . If G is not a forest, then G 0 is also not a forest. Therefore w e ha ve: c m ( G ) = c m ( G 0 ) = 0. So let G b e a forest. Then c m ( G ) = γ ( G ) and c m ( G 0 ) = γ ( G 0 ). In this case it is easy to show that c m ( G ) ≥ c m ( G 0 ). No w supp ose that s ≤ m − 1. Claim 1. If u ∈ A ( G ) , ( i.e. α [ u ] = u ) , then c s ( G − u ) ≥ c s ( G 0 − u ) . Pr o of. If u ∈ E ( G ) − E ( K ), then G 0 − ε ( u ) = ( G − u ) − E [ X ]) ∪ E 0 [ X ] = K xy ( G − u ). If u ∈ E ( K ), then G 0 − u = ( G − u ) − E [ X ]) ∪ E 0 [ X ] = K 0 xy ( G − u ), where K 0 = K − u . Since α [ u ] = u , clearly K 0 xy -op eration in G − u is α -symmetric. In b oth cases, since e ( G − u ) < e ( G ), w e ha ve by the inductiv e h yp othesis: c s ( G − u ) ≥ c s ( G 0 − u ). ♦ Claim 2. If u ∈ [ x, X ] ∪ [ y , Y ] , then c s ( G − u ) ≥ c s ( G 0 − ε ( u )) . Pr o of. Supp ose that u = [ x, X ]). Then u = xv for some v ∈ X . Ob viously , G 0 − ε [ u ] = ( G − u ) − E [ X − v ]) ∪ E 0 [ X − v ] = K 0 xy ( G − u ), where K 0 is the subgraph of G induced by K ∪ v . Since α ( z ) = z for every z ∈ X ∪ Y and K xy -op eration in G is α -symmetric, clearly K 0 xy -op eration in G − u is also α -symmetric. Since e ( G − u ) < e ( G ), w e ha v e b y the inductive hypothesis: c s ( G − u ) ≥ c s ( G 0 − u ). Similar arguments show that our claim is also true for u ∈ [ y , Y ]. ♦ By the recursions (6.5) and (6.6) and by Claims 1 and 2, it is suﬃcient to prov e the follo wing inequalit y for every u ∈ B ( G ) (and so α [ u ] 6 = u ): c s ( G − u ) + c s ( G − α [ u ]) ≥ c s ( G 0 − u ) + c s ( G 0 − α [ u ]) . (6.7) This inequality is a particular case of the follo wing claim. Claim 3. If Z ⊆ B ( G ), then c s ( G − Z ) + c s ( G − α [ Z ]) ≥ c s ( G 0 − Z ) + c 0 s ( G − α [ Z ]) . Pr o of. Supp ose, on the contrary , that our claim is not true. Let ( G, Z ) b e an ( | E | , | B − Z | )-lexicographically smallest (or lg-smallest) counterexample to our claim, where E = E ( G ) and B = B ( G ). If Z = B , then our claim is obviously , true. Therefore suc h a counterexample exists. Let σ ( G, Z ) = c s ( G − Z ) + c s ( G − α [ Z ]) and σ ( G 0 , Z ) = c s ( G 0 − Z ) + c s ( G 0 − α [ Z ]). By the recursions (6.5) and (6.6) for c s ( G ) and c s ( G 0 ), we hav e: ( m − s ) σ ( G, Z ) = P { σ ( G − u, Z ) : u ∈ [ x, X ] ∪ [ y , Y ] } + P { σ ( G − u, Z ) : u ∈ A ( G ) } + P { σ ( G − u, Z ) : u ∈ B − Z } . F or σ ( G 0 , Z ) we hav e a similar form ula obtained from the ab o ve formula by replacing G 39 b y G 0 . Let ∆( G, Z ) = σ ( G, Z ) − σ ( G 0 , Z ). Since ( G, Z ) is a coun terexample, we hav e: ∆( G, Z ) < 0. Our goal is to get a contradiction by sho wing that ∆( G, Z ) ≥ 0. F rom the ab o v e relations we hav e: ( m − s )∆( G, Z ) = P { ∆( G − u, Z ) : u ∈ [ x, X ] ∪ [ y , Y } + P { ∆( G − u, Z ) : u ∈ A ( G ) } + P { ∆( G − u, Z ) : u ∈ B − Z } . Supp ose that u = z v ∈ [ x, X ] ∪ [ y , Y ]. Then argumen ts similar to those in the pro of of Claim 2 , sho w that G 0 − ε ( u ) = K 0 xy ( G − u ), where K 0 is the subgraph of G induced b y K ∪ v and K 0 xy -op eration in G − u is α -symmetric. Since ( | E − u | , | B − Z | ) is lg-smaller than ( | E | , | B − Z | ), clearly ( G − u, Z ) is not a coun terexample. Therefore P { ∆( G − u, Z ) : u ∈ [ x, X ] ∪ [ y , Y ] } ≥ 0. Supp ose that u ∈ A ( G ). If u 6∈ E ( K ), then G 0 − ε ( u ) = K xy ( G − u ). Since ( | E − u | , | B − Z | ) is lg-smaller than ( | E | , | B − Z | ), ob viously , ( G − u, Z ) is not a coun terexample. Therefore ∆( G − u, Z ) ≥ 0. If u ∈ E ( K ), then G 0 − ε ( u ) = K 0 xy ( G − u ), where K 0 = K − u , and the K 0 xy -op eration is α -symmetric in G − u . Since ( | E − u | , | Z | ) is lg-smaller than ( | E | , | B − Z | ), again ( G − u, Z ) is not a counterexample. Therefore ∆( G − u, Z ) ≥ 0. Th us, P { ∆( G − u, Z ) : u ∈ A ( G ) } ≥ 0. Finally , supp ose that u ∈ B − Z . Since ( | E | , | B − u − Z | ) is lg-smaller than ( | E | , | B − Z | ), again ( G, Z ∪ u ) is not a counterexample. Therefore ∆( G − u, Z ) = ∆( G, Z ∪ u ) ≥ 0, and so P { ∆( G − u, Z ) : u ∈ B − Z } ≥ 0. Th us, from the ab o ve recursion for ∆( G, Z ) we ha ve: ∆( G, Z ) ≥ 0. ♦ Ob viously , inequality (6.7) is a particular case of Claim 3 , when | B | = 1.  Using 4.2 , we ha ve from 6.61 : 6.62 L et G, F ∈ G m n and F b e obtaine d fr om G by a symmetric K xy -op er ation. Then t ( G ) ≥ t ( F ) and t ([ G ] c ) ≥ t ([ F ] c ) . No w from 6.62 we ha ve the following strengthening of 6.5 : 6.63 L et G, F ∈ G m n and F b e obtaine d fr om G by a symmetric K xy -op er ation. Then G  τ F and [ G ] c  τ [ F ] c . The following is a generalization of 6.13 for a K xy -op eration. 6.64 L et G b e a gr aph and G 0 b e obtaine d fr om G by a symmetric K xy -op er ation. Then G  p G 0 . Pro of (uses 3.11 , 6.61 , and 6.62 ). Let v ( G ) = n . Our goal is to pro v e that Φ( λ, G ) ≥ Φ( λ, G 0 ) for λ ≥ n . W e prov e b y induction on e ( G ) = m . Our claim is 40 ob viously true for e ( G ) = 1. Supp ose that our claim is true for e ( G ) = m − 1. W e will pro v e that it is also true for e ( G ) = m ≥ 2. By 4.2 ( a 3), we can assume that N x ( G ) ∩ N y ( G ) ⊆ V ( K ). No w, since G 0 is obtained from G b y the K xy -op eration, we ha v e: G 0 = G − [ x, X ] ∪ [ y , X ], where X = N x ( G ) \ V ( K ), Y = N y ( G ) \ V ( K ), [ x, X ] = { xv : v ∈ X } and [ y , X ] = { y v : v ∈ X } , and so [ x, X ] ⊆ E ( G ) and [ y , X ] ∩ E ( G ) = ∅ . Let e v = xv and e 0 v = y v for v ∈ X and let E [ X ] = [ x, X ] = { e v : v ∈ X } and E 0 [ X ] = [ y , X ] = { e 0 v : v ∈ X } . Then G 0 = ( G − E [ X ]) ∪ E 0 [ X ]. Obviously , E ( G ) − E [ X ] = E ( G 0 ) − E 0 [ X ]. Let ε ( u ) = u if u ∈ E ( G ) − E [ X ] and ε ( e v ) = e 0 v if e v ∈ E [ X ] (and so v ∈ X ). Then ε : E ( G ) → E ( G 0 ) is a bijection. By 3.11 , Φ( λ, G ) = Φ( n, G ) + P { R λ n Φ( t, G − u ) dt : u ∈ E ( G ) } . Let ∆( λ, G ) = Φ( λ, G ) − Φ( λ, G 0 ) and ∆( λ, G − u ) = Φ( λ, G − u ) − Φ( λ, G 0 − ε ( u )) . Then ∆( λ, G ) = ∆( n, G ) + X { Z λ n (∆( t, G − u )) dt : u ∈ E ( G ) } . (6.8) By 3.18 , Φ( n, G ) = n m − n +2 t ( K n − E ( G )). Therefore ∆( n, G ) = Φ( n, G ) − Φ( n, G 0 ) = n m − n +2 ( t ( K n − E ( G )) − t ( K n − E ( G 0 )). By 6.62 , t ( K n − E ( G )) − t ( K n − E ( G 0 )) ≥ 0. Therefore ∆( n, G ) = Φ( n, G ) − Φ( n, G 0 ) ≥ 0 . (6.9) No w, using the induction h yp othesis and the arguments similar to those in the pro of of 6.61 , it can b e shown that X { Z λ n (∆( t, G − u )) dt : u ∈ E ( G ) } ≥ 0 . (6.10) Th us, our claim follows from (6.8), (6.9), and (6.10).  F rom 4.2 ( a 2) and 6.64 , we ha v e the follo wing strengthening of 6.63 : 6.65 L et G, F ∈ G m n and F b e obtaine d fr om G by a symmetric K xy -op er ation. Then t ( G ) ≥ t ( F ) and [ G ] c  p [ F ] c . F rom 6.64 w e ha ve, in particular, the follo wing strengthening of 6.11 . 6.66 L et G b e a simple c onne cte d gr aph and ¨ G b e the gr aph obtaine d fr om G by adding e ( G ) − v ( G ) + 1 isolate d vertic es. Then for every sp anning tr e e T of G ther e exists a tr e e D with e ( G ) e dges such that T is a sub gr aph of D and D  p ¨ G . 41 W e can also prov e that under the assumption in 6.66 , if e ( G ) − v ( G ) + 1 > 0 (i.e. if ¨ G 6 = G ), then D  p G . Another wa y to pro v e 6.61 is by ﬁxing a spanning forest in G and analyzing how it is transformed b y the op eration that brings G to G 0 . W e demonstrate this approach b y giving another pro of of a particular case of 6.61 when G 0 is obtained from G by a ♦ -op eration. 6.67 L et G b e a c onne cte d gr aph with n vertic es, x and y two distinct vertic es in G . L et G 0 b e obtaine d fr om G by the ♦ xy -op er ation. Then ( a 1) c s ( G ) > c s ( G 0 ) for every s ∈ { 2 , . . . , n − 2 } and ( a 2) c n − 1 ( G ) = c n − 1 ( G 0 ) if and only if x is a cut vertex in G . Pro of (uses 3.7 and 3.8 ). By 3.8 , we can assume that s ∈ { 2 , . . . , n − 1 } . Let Z = N x ( G ) ∩ N y ( G ), X = N x ( G ) \ ( Z ∪ { y } ), and Y = N y ( G ) \ ( Z ∪ { x } ). Let e v = xv and e 0 v = y v for v ∈ X and let E x = [ x, X ] = { e v : v ∈ X } and E y = [ y , X ] = { e 0 v : v ∈ X } . Since G 0 is obtained from G b y the ♦ xy -op eration, G 0 = ( G − E x ) ∪ E y and [ { x, y } , Z ] ⊆ E ( G ). Obviously , E ( G ) − E x = E ( G 0 ) − E y . Let ε ( u ) = u if u ∈ E ( G ) − E x and ε ( e v ) = e 0 v if e v ∈ E x (and so v ∈ X ). Then ε : E ( G ) → E ( G 0 ) is a bijection. F or U ⊆ E ( G ), let ε [ U ] = { ε ( u ) : u ∈ U } . F or a subgraph S of G , let ϑ ( S ) b e the subgraph of G such that V ( ϑ ( S )) = V ( S ) and E ( ϑ ( S )) = ( E ( S ) \ E x ) ∪ ε [ E ( S ) ∩ E x ]. Let S ( H ) denote the set of subgraphs of a graph H . Ob viously , ϑ is a bijection from S ( G ) to S ( G 0 ). F or A ⊆ S ( G ), let ϑ [ A ] = { ϑ ( A ) : A ∈ A} . Let P 0 b e a forest in G 0 with at most tw o comp onen ts eac h meeting { x, y } . Let P 0 x and P 0 y b e the comp onen ts of P 0 con taining x and y , resp ectiv ely , and so if P 0 has one comp onen t, then P 0 x = P 0 y . Let P b e the subgraph in G suc h that P 0 = ϑ ( P ). Clearly , { x, y } ⊆ V ( P 0 ) = V ( P ) and e ( P 0 ) = e ( P ). Let σ ( xz ) = y z and σ ( y z ) = xz for every z ∈ Z . Obviously , σ : [ { x, y } , Z ] → [ { x, y } , Z ] is a bijection. F or S ⊆ [ { x, y } , Z ], let σ [ S ] = { σ ( s ) : s ∈ S } . Let A = A ( P ) = [ { x, y } , Z ] ∩ E ( P ). Obviously , A ( P ) = A ( P 0 ). Let ¯ P = ( P − A ) ∪ σ [ A ] and ¯ P 0 = ( P 0 − A ) ∪ σ [ A ]. Obviously , ¯ P = P and ¯ P 0 = P 0 if and only if σ [ A ] = A . W e need the following facts. Claim 1. Supp ose that P 0 is a tr e e and P is not a tr e e. Then ¯ P is a tr e e and ¯ P 0 is not a tr e e. Pr o of. Since v ( P ) = v ( P 0 ), e ( P ) = e ( P 0 ), P 0 is a tree, and P is not a tree, w e hav e: P has a cycle C . If C do es not con tain v ertex x or C contains t wo edges from E x then P 0 has a cycle, a contradiction. Therefore C is the only cycle in P and C contains exactly one edge xc with c ∈ X and exactly one edge xz with z ∈ Z . Th us, P has exactly tw o comp onen ts. No w our claim follows. ♦ 42 It is easy to see that the conv erse of Claim 1 may b e not true. Namely , we hav e: Claim 1’. The fol lowing ar e e quivalent: ( a 1) x is not a cut vertex in G and ( a 2) ther e exists a sp anning tr e e P of G such that ¯ P is a sp anning tr e e and b oth P 0 and ¯ P 0 ar e not tr e es. Claim 2. If P 0 is a for est with exactly two c omp onents, then P is also a for est with exactly two c omp onents. Pr o of. Supp ose, on the con trary , P 0 is a forest with exactly tw o comp onen ts but P is not a forest with exactly t w o comp onen ts. Then P has at least three comp onen ts. Then there is a comp onen t Q of P that av oids { x, y } . Then Q is also a comp onen t of P 0 . How ev er, P 0 has exactly tw o comp onen ts eac h con taining exactly one vertex from { x, y } , a con tradiction. ♦ Actually , the con v erse of Claim 2 is also true. Claim 2’. P 0 is a for est with exactly two c omp onents if and only if P is also a for est with exactly two c omp onents. Supp ose that b oth P 0 and P are forest with exactly tw o comp onen ts. Let X ∗ = { v ∈ X : xv ∈ E ( P ) } and Y ∗ = { v ∈ Y : y v ∈ E ( P ) } . Let, as ab o v e, P 0 x and P 0 y b e the t w o comp onen ts of P 0 suc h that x ∈ V ( P 0 x ) and y ∈ V ( P 0 y ). Let us remov e from P 0 y the edges from y to X ∗ ∪ Y ∗ , denote by Q 0 y the comp onen t of the resulting forest con taining v ertex y , and put R = P 0 y − Q 0 y . Then each comp onent of the forest R has exactly one v ertex in X ∗ ∪ Y ∗ . Let R x = R x ( P ) = R x ( P 0 ) b e the union of the comp onen ts meeting X ∗ and R y = R y ( P ) = R y ( P 0 ) b e the union of the comp onen ts meeting Y ∗ . Let ∆( P , P 0 ) = ( c r ( P ) + c r ( ¯ P )) − ( c r ( P 0 ) + c r ( ¯ P 0 )), where r = e ( P ). Claim 3. Supp ose that b oth P and P 0 ar e for est with exactly two c omp onents. Then ∆( P , P 0 ) = 2 v ( R x ) v ( R y ) ≥ 0 , and so ∆( P , P 0 ) > 0 ⇔ v ( R x ) > 0 and v ( R y ) > 0 . Pr o of. Ob viously , c r ( P 0 ) = v ( P 0 x )( v ( Q 0 y ) + v ( R x ) + v ( R y )), c r ( ¯ P 0 ) = v ( Q 0 y )( v ( P 0 x ) + v ( R x ) + v ( R y )), c r ( P ) = ( v ( P 0 x ) + v ( R x ))( v ( Q 0 y ) + v ( R y )), and c r ( ¯ P ) = ( v ( Q 0 y ) + v ( R x ))( v ( P 0 x ) + v ( R y )). F rom the ab o v e formulas we ha v e: ∆( P , P 0 ) = ([( v ( P 0 x ) + v ( R x ))( v ( Q 0 y ) + v ( R y ))] + [( v ( Q 0 y ) + v ( R x ))( v ( P 0 x ) + v ( R y ))]) − ([ v ( P 0 x )( v ( Q 0 y ) + v ( R x ) + v ( R y ))] + [ v ( Q 0 y )( v ( P 0 x ) + v ( R x ) + v ( R y ))] = [ v ( P 0 x )( v ( Q 0 y ) + v ( R y ) + v ( R x )( v ( Q 0 y ) + v ( R y ))] − [ v ( P 0 x )( v ( Q 0 y ) + v ( R y )) + v ( P 0 x ) v ( R x )] + [( v ( Q 0 y )( v ( P 0 x ) + v ( R y )) + v ( R x )( v ( P 0 x ) + v ( R y ))] − [ v ( Q 0 y )( v ( P 0 x ) + v ( R y ))] + v ( Q 0 y ) v ( R x )] = v ( R x ) v ( Q 0 y ) + v ( R x ) v ( R y ) − v ( P 0 x ) v ( R x ) + v ( R x ) v ( P 0 x ) + v ( R x ) v ( R y ) − v ( Q 0 y ) v ( R x ) = 43 2 v ( R x ) v ( R y ) . Our claim follows. ♦ Giv en a spanning forest F of G 0 , let F xy b e the minimal subforest of F containing x and y and suc h that eac h comp onen t of F xy is a comp onent of F . Obviously , F xy has at most t w o comp onen ts eac h meeting { x, y } . Let F s ( G 0 ) denote the set of spanning forests F with s edges in G 0 . Let P ( G 0 ) denote the set of all forests P 0 in G 0 ha ving at most t w o comp onen ts eac h meeting { x, y } and let P r ( G 0 ) denote the set of all forests in P ( G 0 ) having r edges. Now from 3.7 we ha ve: c s ( G 0 ) = { P { P { γ ( F ) : F ∈ F s ( G 0 ) , F xy = P 0 } : P 0 ∈ P ( G 0 ) } . Therefore c s ( G 0 ) = X { X { c r ( P 0 ) c s − r ( G − V ( P 0 )) : P 0 ∈ P r ( G 0 ) } : 0 ≤ r = e ( P 0 ) ≤ s } . (6.11) Similarly , c s ( G ) = P { P { γ ( F ) : F ∈ F s ( G ) , F xy = Q } : Q ∈ P ( G ) } = { P { P { c r ( Q ) c s − r ( G − V ( Q )) : Q ∈ P r ( G ) } : 0 ≤ r = e ( Q ) ≤ s } . Recall that P is the subgraph in G such that P 0 is obtained from P b y the ♦ xy -op eration. Let e P 0 = P if P is a forest and e P 0 = ¯ P if P is not a forest. By Claim 1 and Claim 2, if P is not a forest, then b oth P 0 and ¯ P are trees. Let c 0 s ( G ) = X { X { c r ( e P 0 ) c s − r ( G − V ( P 0 ) : P 0 ∈ P r ( G 0 ) } : 0 ≤ r = e ( P 0 ) ≤ s } . (6.12) Ob viously , c s ( G ) ≥ c 0 s ( G ). If P 0 is a tree with r edges , then e P 0 is also a tree with r edges and c r ( P 0 ) = c r ( e P 0 ) = r + 1. Therefore by Claim 3, we hav e from (6.11) and (6.12): c 0 s ( G ) ≥ c s ( G 0 ). Th us, c s ( G ) ≥ c 0 s ( G ) ≥ c s ( G 0 ), and so claim ( a 1) of our theorem is true. F or 2 ≤ s ≤ n − 2, there exists a forest F in G with the prop erties: b oth F xy and F 0 xy are forests with exactly t w o comp onen ts, R x ( F xy ) > 0, and R y ( F xy ) > 0. Therefore by Claim 3 , c s ( G ) > c s ( G 0 ). Obviously , if x is a cut v ertex of G , then c n − 1 ( G ) = c n − 1 ( G 0 ). If x is not a cut v ertex of G , then b y Claim 1’ , c n − 1 ( G ) > c 0 n − 1 ( G ), and so c n − 1 ( G ) > c 0 n − 1 ( G ) ≥ c n − 1 ( G 0 ). Thus, claim ( a 2) is true.  The arguments similar to those in the pro of of 6.67 provide one of p ossible pro ofs of 6.46 . 6.68 L et A , B , D , and H b e disjoint gr aphs, x, y ∈ V ( H ) and x 6 = y , and d is a vertex in V ( D ) incident to an e dge. L et A b e a p ath with an end-vertex a and B a p ath with an end-vertex b . L et gr aph R b e obtaine d fr om A , B , and H by identifying x with a and y with b . L et G a and G b b e obtaine d fr om R and D by identifying d with a and b , r esp e ctively. Supp ose that ( h 1) ther e exist an automorphism η : V ( H ) → V ( H ) such that η ( a ) = b and η ( b ) = a 44 (and so tw o-p ole aH b is symmetric) and ( h 2) v ( A ) ≤ v ( B ) . Then ( a 1) G a  c G b and, mor e over, ( a 2) G a  c G b ⇔ v ( A ) < v ( B ) . Pro of. W e pro v e our claim b y induction on v ( A ∪ B ) = n . If n = 2, then our claim is true b y 6.47 . So w e assume that n ≥ 3 and that our claim is true if v ( A ∪ B ) < n . Our goal is to pro ve that the claim is also true for v ( A ∪ B ) = n . Let s ∈ { 2 . . . v ( G ) − 1 } . Let F s ( G ) denote the set of spanning forests of G with s edges. Let T b e a tree in D such that d ∈ V ( T ). Put { v , z } = { a, b } and let σ s ( G z ) = P { γ ( F ) : F ∈ F s ( G z ) , T ⊆ F } . It is suﬃcien t to sho w that σ s ( G a ) ≥ σ s ( G b ) . Let e ( T ) = t and P 1 denote the set of trees P in R containing a and b and suc h that e ( P ) ≤ s − t . Let P 2 denote the set of pairs ( P a , P b ) suc h that V ( P a ) ∩ V ( P b ) = ∅ , P a and P b are trees, a ∈ V ( P a ), b ∈ V ( P b ), and e ( P a ∪ P b ) ≤ s − t . Giv en a spanning forest F of G and z ∈ V ( G ), let F z denote the comp onen t of F containing z . F or P ∈ P 1 , let F s ( G, P ) = { F ∈ F s ( G ) : F a = F b = P } . F or ( P a , P b ) ∈ P 2 , let F s ( G, ( P a , P b )) = { F ∈ F s ( G ) : F a = P a , F b = P b } . F or z ∈ { a, b } , let σ s ( G z , ( P a , P b )) = P { γ ( F ) : F ∈ F s ( G z , ( P a , P b ) } and σ s ( G z , P ) = P { γ ( F ) : F ∈ F s ( G z , ( P ). Then σ s ( G z , ( P a , P b )) = ( v ( P z ) + t ) v ( P v ) P { γ ( F − ( P a ∪ P b )) : F ∈ F s ( G z , ( P a , P b )) } and σ s ( G z , P ) = ( v ( P ) + t ) P { γ ( F − P ) : F ∈ F s ( G z , P ) } . Let σ 1 s ( G z ) = P { σ s ( G z , P ) : P ∈ P 1 } and σ 2 s ( G z ) = P { σ s ( G z , ( P a , P b )) : ( P a , P b ) ∈ P 2 } . Then σ s ( G z ) = σ 1 s ( G z ) + σ 2 s ( G z ). Let ∆ s ( R ) = σ s ( G a ) − σ s ( G b ). W e need to pro v e that ∆ s ( R ) ≥ 0 . Clearly , σ s ( G a , P ) = σ s ( G b , P ). Therefore σ 1 s ( G a ) = σ 1 s ( G b ), and so ∆ s ( R ) = σ 2 s ( G a ) − σ 2 s ( G b ). Given ( P a , P b ) ∈ P 2 , let ∆ s ( R, ( P a , P b )) = σ s ( G a , ( P a , P b )) − σ s ( G b , ( P a , P b )). Let σ ( R , ( P a , P b )) = P { γ ( F − ( P a ∪ P b )) : F ∈ F s ( G, ( P a , P b )) } . Then ∆ s ( R, ( P a , P b )) = [( v ( P a ) + t ) v ( P b ) − ( v ( P b ) + t ) v ( P a )] σ ( R, ( P a , P b )) = t [ v ( P b ) − v ( P a )] σ ( R, ( P a , P b )). Let P 0 = { ( P a , P b ) ∈ P } : v ( P b ) < v ( A ∪ H − b ) , v ( P a ) 6 = v ( P b ) } and P 00 = { ( P a , P b ) ∈ P : v ( P b ) ≥ v ( A ∪ H − b ) } . 45 Let, accordingly , ∆ 0 s ( R ) = P { ∆ s ( R, ( P a , P b )) : ( P a , P b ) ∈ P 0 } and ∆ 00 s ( R ) = P { ∆ s ( R, ( P a , P b )) : ( P a , P b ) ∈ P 00 } . Then ∆ s ( R ) = ∆ 0 s ( R ) + ∆ 00 s ( R ). Since P a ⊆ A ∪ H − b , we hav e: v ( P b ) ≥ v ( A ∪ H − b ) } ⇒ v ( P b ) ≥ v ( P a ). Since t ≥ 0, clearly v ( P b ) ≥ v ( P a ) ⇒ ∆ s ( R, ( P a , P b )) ≥ 0. No w b y the previous inequality , v ( P b ) ≥ v ( A ∪ H − b ) } ⇒ ∆ s ( R, ( P a , P b )) ≥ 0. Therefore ∆ 00 s ( R ) ≥ 0. Moreov er, if v ( B ) > v ( A ), then there exist P a A and P b suc h that v ( P b ) > v ( P a ). Therefore, if v ( B ) > v ( A ) and t > 0, then ∆ 00 s ( R > 0. Since v ertex d in D is not an isolated v ertex, there exists a tree T in D containing d with t = e ( T ) > 0. Th us, it is suﬃcien t to sho w that ∆ 0 s ( R ) ≥ 0. Let T denote the set of pairs ( T a , T b ) such that T a and T b are trees in H , V ( T a ) ∩ V ( T b ) = ∅ , a ∈ V ( T a ), and b ∈ V ( T b ). Let T 0 a = η [ T a ] and T 0 b = η [ T b ] and put π ( T a , T b ) = ( T 0 a , T 0 b ). Then π : T → T is a bijection. Let T 0 = { ( T a , T b ) ∈ T : π ( T a , T b ) = ( T a , T b ) } and T 00 = { ( T a , T b ) ∈ T : π ( T a , T b ) 6 = ( T a , T b ) } . Let L denote the set of pairs ( L a , L b ) such that L a is a path in A containing a , L b is a path in B containing b , and v ( L a ) ≤ v ( L b ). Let µ ( L a , L b ) = ( L 0 a , L 0 b ), where L 0 a is the path in A suc h that a ∈ V ( L 0 a ) and v ( L 0 a ) = v ( L b ) and L 0 b is the path in B such that b ∈ V ( L 0 b ) and v ( L 0 b ) = v ( L a ). Let L 0 = { ( L a , L b ) ∈ L : µ ( L a , L b ) = ( L a , L b ) } and L 00 = { ( L a , L b ) ∈ L : µ ( L a , L b ) 6 = ( L a , L b ) } . F or ( T a , T b ) ∈ T and ( L a , L b ) ∈ L , let σ s ( G z , ( T a , T b ) , ( L a , L b )) = σ s ( G z , ( P a , P b )), where P a = T a ∪ L a and P b = T b ∪ L b . Let r = r ( T a , T b ) , ( L a , L b )) = s − t − e ( T a ∪ T b ∪ L a ∪ L b ) } . Since e ( T z ) = e ( T 0 z ) and e ( L z ) = e ( L 0 z ) for z ∈ { a, b } , w e ha v e: r ( T a , T b ) , ( L a , L b )) = r ( T 0 a , T 0 b ) , ( L a , L b )) = r ( T a , T b ) , ( L 0 a , L 0 b )) = r ( T 0 a , T 0 b ) , ( L 0 a , L 0 b )). Therefore c r ( R − T a − T b − L a − L b ) = c r ( R − T 0 a − T 0 b − L a − L b ) and c r ( R − T a − T b − L 0 a − L 0 b ) = c r ( R − T 0 a − T 0 b − L 0 a − L 0 b ). Then σ s ( G a , ( T a , T b ) , ( L a , L b )) = ( v ( T a ) + e ( L a ) + t )( v ( T b ) + e ( L b )) c r ( R − T a − T b − L a − L b ), σ s ( G a , ( T 0 a , T 0 b ) , ( L a , L b )) = ( v ( T 0 a ) + e ( L a ) + t )( v ( T 0 b ) + e ( L b )) c r ( R − T 0 a − T 0 b − L a − L b ), 46 σ s ( G a , ( T a , T b ) , ( L 0 a , L 0 b )) = ( v ( T a ) + e ( L 0 a ) + t )( v ( T b ) + e ( L 0 b )) c r ( R − T a − T b − L 0 a − L 0 b ), σ s ( G a , ( T 0 a , T 0 b ) , ( L 0 a , L 0 b )) = ( v ( T 0 a ) + e ( L 0 a ) + t )( v ( T 0 b ) + e ( L 0 b )) c r ( R − T 0 a − T 0 b − L 0 a − L 0 b ). The similar form ulas for σ s ( G b , ... )’s are obtained from the ab o v e form ulas for σ s ( G a , ... )’s b y mo ving t from the right brack et to the left one. Let ω s ( G z , ( T a , T b ) , ( L a , L b )) = σ s ( G z , ( T a , T b ) , ( L a , L b )) + σ s ( G z , ( T 0 a , T 0 b ) , ( L a , L b )) + σ s ( G z , ( T a , T b ) , ( L 0 a , L 0 b )) + σ s ( G z , ( T 0 a , T 0 b ) , ( L 0 a , L 0 b )) and ∆ 0 s ( R, ( T a , T b ) , ( L a , L b )) = ω s ( G a , ( T a , T b ) , ( L a , L b )) − ω s ( G b , ( T a , T b ) , ( L a , L b )). Then ∆ 0 s ( R ) = P { ∆ 0 s ( R, ( T a , T b ) , ( L a , L b ) : ( T a , T b ) ∈ T 00 , ( L a , L b ) ∈ L 00 } + 1 2 P { ∆ 0 s ( R, ( T a , T b ) , ( L a , L b ) : ( T a , T b ) ∈ T 0 , ( L a , L b ) ∈ L 00 } + 1 2 P { ∆ 0 s ( R, ( T a , T b ) , ( L a , L b ) : ( T a , T b ) ∈ T 00 , ( L a , L b ) ∈ L 0 } + 1 4 P { ∆ 0 s ( R, ( T a , T b ) , ( L a , L b ) : ( T a , T b ) ∈ T 0 , ( L a , L b ) ∈ L 0 } . Finally , it is suﬃcien t to sho w that eac h ∆ 0 s ( R, ( T a , T b ) , ( L a , L b )) ≥ 0 . F rom the ab o v e formula we ha v e: ∆ 0 s ( R, ( T a , T b ) , ( L a , L b )) = t ( v ( L b ) − v ( L a ))[ c r ( R − T a − T b − L a − L b ) − c r ( R − T a − T b − L 0 a − L 0 b )] (6.13) Let R ab = R − T a − T b . Then R ab is the disjoint union of three graphs: R ab = ( A − a ) ∪ ( B − b ) ∪ ( H − T a − T b ), and so R − T a − T b − L a − L b = ( A − L a ) ∪ ( B − L b ) ∪ ( H − T a − T b ). Therefore c r ( R − T a − T b − L a − L b ) = P { c p (( A − L a ) ∪ ( B − L b )) c q ( H − T a − T b ) : p + q = r } and c r ( R − T a − T b − L 0 a − L 0 b ) = P { ( c p ( A − L 0 a ) ∪ ( B − L 0 b )) c q ( H − T a − T b ) : p + q = r } . Th us [ c r ( R − T a − T b − L a − L b ) − c r ( R − T a − T b − L 0 a − L 0 b )] = P { [ c p (( A − L a ) ∪ ( B − L b )) − c p (( A − L 0 a ) ∪ ( B − L 0 b ))] c q ( H − T a − T b ) : p + q = r } . Let A − L a = A 1 , B − L b = B 1 , A − L 0 a = A 2 , and B − L 0 b = B 2 . Then v ( A 1 ∪ B 1 ) = v ( A 2 ∪ B 2 ). Since v ( L 0 a ) = v ( L b ), v ( L 0 b ) = v ( L a ), and v ( L a ) ≤ v ( L b ), clearly v ( A 1 ) ≥ v ( A 2 ). Let ¯ A = A 2 , ¯ B = B 2 , and ¯ D b e the path with v ( A 1 ) − v ( A 2 ) + 1 vertices disjoint from ¯ A ∪ ¯ B . Let ¯ a , ¯ b , and ¯ d b e end-v ertices of paths ¯ A , ¯ B , and ¯ D , resp ectively . Let ¯ H 47 b e the graph consisting of exactly tw o isolated v ertices ¯ x and ¯ y . Ob viously , v ( A ) ≤ v ( B ) ⇒ v ( ¯ A ) ≤ v ( ¯ B ). Let us put in the claim (we are pro ving) ( A, a ) := ( ¯ A, ¯ a ), ( B , b ) := ( ¯ B , ¯ b ), ( D , d ) := ( ¯ D , ¯ d ), ( H , x, y ) := ( ¯ H , ¯ x, ¯ y ), G a := A 1 ∪ B 1 , and G b := A 2 ∪ B 2 . Clearly , after this replacement the assumptions ( h 1) and ( h 2) are satisﬁed and v ( ¯ A ∪ ¯ B ) < v ( A ∪ B ) = n . Therefore by the induction hypothesis, c p ( A 1 ∪ B 1 ) ≥ c p ( A 2 ∪ B 2 ). Obviously , c p ( A 1 ∪ B 1 ) − c p ( A 2 ∪ B 2 ) = c p (( A − L a ) ∪ ( B − L b )) − c p (( A − L 0 a ) ∪ ( B − L 0 b )). No w since v ( L b ) ≥ v ( L a ), we hav e from (6.13): ∆ 0 s ( R, ( T a , T b ) , ( L a , L b )) ≥ 0.  Using the argumen ts similar to those in the pro of of 6.68 , it is not hard to pro ve the follo wing generalization of 6.68 . 6.69 L et A , D , F , and H b e disjoint gr aphs, d b e a non-solate d vertex of D , b ∈ V ( F ) , and x, y ∈ V ( H ) , wher e x 6 = y . L et A b e a p ath with an end-vertex a . L et R b e obtaine d fr om A , F , and H by identifying x with a and y with b . L et gr aphs G a and G b b e obtaine d fr om R and D by identifying d with a and b , r esp e ctively. Supp ose that ( h 1) two-p ole xH y is symmetric and ( h 2) F has a p ath bB t such that v ( A ) ≤ v ( B ) . Then ( a 1) G a  c G b and ( a 2) v ( A ) < v ( B ) ⇒ G a  c G b . The op eration describ ed in 6.69 is shown in Figure 9. No w, using the  c -monotonicit y of the xP y -operation and the op eration describ ed in 6.69 , we can obtain the results on the p oset ( T n ,  c ) similar to 6.37 and 6.38 on the p oset ( T n ,  φ ). W e remind that for n -v ertex trees T and T 0 , T  c T 0 if and only if c s ( T ) > c s ( T 0 ) for ev ery s ∈ { 2 , . . . , n − 2 } . Let, as ab ov e, δ 0 ( G ) b e the minim um non-leaf vertex degree in G . The following theorem is an analog of 6.37 with  φ replaced b y  c (see the corresp onding deﬁnitions in 2.15 ). 6.70 [19] L et r ≥ 3 and n ≥ r + 2 . Then ( a 1) for every D ∈ D n ( r ) \ K n ( r ) ther e exists Y ∈ K n ( r ) such that D  c Y , ( a 2) D  c K n ( r ) for every D ∈ K n ( r ) \ { K n ( r ) } , and ther efor e, (from ( a 1) and ( a 2)) ( a 3) D  c K n ( r ) for every D ∈ D n ( r ) \ { K n ( r ) } (see Figure 10) , ( a 4) ( D n (3) ,  c ) is a line ar or der p oset, namely, for T , T 0 ∈ D n (3) we have: T  c T 0 ⇔ δ 0 ( T ) > δ 0 ( T 0 ) , and ( a 5) ( D n (4 , r ) ,  c ) is a line ar p oset, namely, for T , T 0 ∈ D n (4) we have: 48 Figure 9: The op eration in 6.69 T  c T 0 ⇔ T > u T 0 . Ob viously , claim ( a 3) in 6.70 follo ws from claims ( a 1) and ( a 2). Claim ( a 1) in 6.70 follo ws from 6.46 and the fact that every tree in D ∈ D n ( r ) \ K n ( r ) can b e transformed in to a tree in K n ( r ) b y a series of xP y -operations (so that ev ery intermediate tree is also in D n ( r )). Claim ( a 2) in 6.70 follows from 6.69 and the fact that ev ery tree in D ∈ K n ( r ) \ { K n ( r ) } can b e transformed in to K n ( r ) b y a series of op erations describ ed in 6.69 (so that every intermediate tree is also in K n ( r )). Claims ( a 4) and ( a 5) can b e pro v en in the same wa y using 6.69 . Similarly , w e can prov e the follo wing analog of 6.38 for  c (see the corresp onding deﬁnitions in 2.16 ): 6.71 [19] L et r ≥ 3 , n ≥ r + 2 , and L ∈ L n ( r ) . Then ( a 0) L n ( r )  c L n ( r + 1) for every r ∈ { 2 , . . . , n − 2 } , ( a 1) ( S n ( r ) ,  c ) is a line ar p oset, namely, for T , T 0 ∈ S n ( r ) we have: T  c T 0 ⇔ T > w T 0 , ( a 2) M n ( r )  c L for every L ∈ S n ( r ) \ { M n ( r ) } , ( a 3) for every L ∈ L n ( r ) \ S n ( r ) ther e exists Z ∈ S n ( r ) such that L  c Z , ( a 4) L  c L n ( r ) for every L ∈ S n ( r ) \ { L n ( r ) } , and ther efor e (from ( a 3) and ( a 4)) 49 Figure 10: T  c,φ K ( a 5) L  c L n ( r ) for every L ∈ L n ( r ) \ { L n ( r ) } , i.e. L n ( r ) is the  c -minimum gr aph in L n ( r ) . Figure 11 illustrates claims ( a 2) and ( a 5) in 6.71 . Let L 1 n ( r ) denote the set of graphs with n vertices, r leav es, and exactly one cycle (and so v ( G ) = e ( G ) = n ). It can b e sho wn that unlike L n ( r ) the set L 1 n ( r ) do es not hav e in general the  c -minim um graph. Here are some results illustrating this situation. Let Y = Y s ( r ) b e the tree obtained from r ≥ 1 disjoint paths P i , 1 ≤ i ≤ r , of s ≥ 1 edges b y sp ecifying one end-v ertex of eac h path and iden tifying these sp eciﬁed end-v ertices of all paths. Then Y has n = r s + 1 vertices, exactly one vertex (sa y , y ) of degree r , exactly r v ertices of degree 1, and the other vertices of degree t w o, and so Y ∈ S n ( r ). Let Z b e a path with s ≥ 2 edges, z a leav e of Z , and z 0 the vertex in Z adjacen t to z . Let 4 b e the triangle and F b e the graph obtained from disjoint 4 and Z b y iden tifying a vertex in 4 with vertex z in Z . No w let A = A s ( r ) and B = B s ( r ) b e the graphs obtained from disjoin t F and Y = Y s ( r − 1) b y identifying vertex y in Y with z and z 0 , resp ectiv ely . Then A s ( r ) is isomorphic to Y s ( r ). Both A and B hav e n = r s + 3 v ertices and edges, exactly r leav es, and exactly one cycle whic h is the triangle 4 , and so A, B ∈ L 1 n ( r ). As ab ov e, c i ( G ) is the i -th co eﬃcien t of the Laplacian p olynomial of G , and so c 0 ( G ) = 1, c 1 ( G ) = 2 e ( G ), and c n − 1 ( G ) = nt ( G ). 6.72 [19] L et n and r b e inte gers such that n ≥ r + 3 and r ≥ 2 . Then ( a 0) c 1 ( A ) = c 1 ( B ) = 2 n and c n − 1 ( A ) = c n − 1 ( B ) = 3 n , ( a 1) c n − 2 ( A ) > c n − 2 ( B ) , and ( a 2) c 2 ( B ) > c 2 ( A ) . 50 Figure 11: M  c,φ T  c,φ L Here are more details ab out graphs A and B for r = 2. 6.73 [19] L et A and B b e gr aphs describ e d ab ove with r = 2 , and so v ( A ) = v ( B ) = n = 2 s + 3 . Then ( a 1) if s = 4 ( i.e. n = 11) , then c 9 ( A ) > c 9 ( B ) and c j ( B ) > c j ( A ) for 8 ≥ j ≥ 2 , ( a 2) if s = 5 or 6 , then c i ( A ) > c i ( B ) for 2 s + 1 ≥ i ≥ 2 s and c j ( B ) > c j ( A ) for 2 s − 1 ≥ j ≥ 2 , ( a 3) if s = 7 , then c i ( A ) > c i ( B ) for 2 s + 1 ≥ i ≥ 2 s − 1 and c j ( B ) > c j ( A ) for 2 s − 2 ≥ j ≥ 2 , and ( a 4) if s ≥ 7 , then c i ( A ) > c i ( B ) for 2 s + 1 ≥ i ≥ 2 s − 1 and c 2 ( B ) > c 2 ( A ) . Let T b e an n -vertex tree of maximum degree r . Then T can b e transformed to an n -v ertex star-tree S with r leav es b y a series of the in v erse P xy -op erations. Then b y 6.46 , S  c T . Therefore b y 6.71 ( a 2), we hav e: 6.74 [19] L et T b e an n -vertex tr e e of maximum de gr e e r and T is not isomorphic to M n ( r ) . Then M n ( r )  c T . Using the  c -monotonicit y of the op eration describ ed in 6.68 , we can pro v e the follo wing results on trees with exactly three leav es (i.e. on trees in S (3)). 6.75 [19] F or every tr e e F non-isomorphic to a p ath ther e exists T ∈ S (3) with v ( T ) = v ( F ) such that T  c F . F rom 6.16 , 6.49 , and 6.75 we ha ve: 51 6.76 [19] L et G b e c onne cte d gr aph and n = e ( G ) + 1 (and so v ( G ) ≤ n ) . Supp ose that G is not a p ath and not in S n (3) . Then ther e exists T ∈ S n (3) such that P n  c,φ M n (3)  c,φ T  c,φ G . F rom 4.5 , 6.49 , 6.64 , and 6.67 we hav e: 6.77 [19] L et G b e a c onne cte d gr aph in G m . Then for every sp anning tr e e T of G and every clique K in G ther e exist a tr e e D and a c onne cte d thr eshold gr aph H in G m such that D c ontains T , H c ontains K , and D  c,p G  c,p H . Giv en a symmetric function g on k v ariables and a graph F with k comp onen ts, let g [ F ] = g { v ( C ) : C ∈ C mp ( F ) } . 6.78 Remark. L et G b e a gr aph with n vertic es and let q s ( G ) = P { g [ F ] : F ∈ F ( G ) , e ( F ) = s } , wher e g is a symmetric c onc ave function on n − s variables. Then the or ems 6.61 and 6.69 r emain true if c s ( G ) is r eplac e d by q s ( G ) . 7 On reliabilit y p osets of graphs In this section we will describ e some results on the monotonicity of the op erations deﬁned in Section 4 with resp ect to some reliability p osets and on the problems R max and R min of ﬁnding maximum and minimum reliable graphs among the graphs of the same size. As abov e, R k ( p, G ) denotes the probabilit y that the random graph ( G, p ) has at most k comp onen ts and R 1 ( p, G ) = R ( p, G ). Let f k ( G ) denote the n umber of spanning forests of G with k comp onen ts. Ob viously , if G ∈ G m n and k ≥ n − 2, then R k ( p, G ) dep ends only on p and m (and is easy to ﬁnd). Therefore we will alwa ys assume that if G ∈ G m n , then k ≤ n − 3. F or a graph G ∈ ¯ G m n , we hav e: R k ( p, G ) = P { a k s ( G ) p s q m − s : s ∈ { n − k , . . . , m }} , where q = 1 − p and a k s ( G ) is the n um b er of spanning subgraphs of G with s edges and at most k comp onents, and so a k n − k ( G ) = f k ( G ). Let Q k 1 ( x, G ) = P { a k s ( G ) x m − s : s ∈ { n − k , . . . , m }} and Q k 0 ( x, G ) = P { a k s ( G ) x s : s ∈ { n − k , . . . , m }} . Then R k ( p, G ) = p m Q k 1 ( q /p, G ) = q m Q k 0 ( p/q , G ). It turns out that the co eﬃcients of R k ( x, G ) (and accordingly , of Q k 0 ( x, G ) and Q k 1 ( x, G )) satisfy the follo wing recursions similar to those for Φ( λ, G ) (see 3.9 and 3.11 ). 7.1 [19, 32] L et G ∈ ¯ G m n . Then 52 ( a 1) ( m − s ) a k s ( G ) = P { a k s ( G − u ); u ∈ E ( G ) }} for s ∈ { n − k , . . . , m } , and ( a 2) sa k s ( G ) = P { a k s ( G/u ); u ∈ E ( G ) }} for s ∈ { n − k , . . . , m } , and so ( a 3) Q k 0 ( x, G ) = P { R x t =0 Q k 0 ( t, G − u ) dt : u ∈ E ( G ) } and ( a 4) Q k 1 ( x, G ) = P { R x t =0 Q k 1 ( t, G/u ) dt : u ∈ E ( G ) } . F unction R k ( p, G ) and its co eﬃcien ts a k s ( G ) satisfy the follo wing useful “ deletion- c ontr action ” formulas: 7.2 [19, 25, 35] L et G ∈ ¯ G m n b e a gr aph with at le ast one e dge and u ∈ E ( G ) . Then ( a ) a k s ( G ) = a k s ( G − u ) + a k s ( G/u ) for every s ∈ { n − k , . . . , m } , and so ( r ) R k ( p, G ) = R k ( p, G − u ) + R k ( p, G/u ) . The relations in 7.1 and 7.2 are prett y useful for pro ving some claims on a k s ( G ) and R k ( p, G ) by induction. F or example, it is very easy to pro ve 7.9 ( a 0) b elo w b y induction using 7.1 ( a 4). The following result is a generalization of 6.3 . 7.3 [19, 25] L et G ∈ G m n and let gr aph H b e obtaine d fr om G by a symmetric hammo ck- op er ation. Then a k s ( G ) ≥ a k s ( H ) for every s ∈ { n − k , . . . , m } , i.e. G  a ( k ) H , and so G  r ( k ) H . In [25] this theorem (and in [27] theorem 6.3 ) w as pro ved using the deletion- con traction form ula in 7.2 . Theorem 7.3 can also b e prov ed using the approach in the pro of of 6.67 and in Section 8. F rom 4.5 and 7.3 we ha ve: 7.4 [19] F or every gr aph G in C m n ther e exists a thr eshold gr aph F in C m n such that a k s ( G ) ≥ a k s ( F ) for every s ∈ { n − k , . . . , m } , i.e. G  a ( k ) F , and so G  r ( k ) F . F rom 4.2 ( a 2) and 7.3 we ha ve: 7.5 [19, 27] L et G ∈ G m n and let gr aph H b e obtaine d fr om G by a symmetric hammo ck- op er ation. Then ( c 1) a k s ([ G ] c ) ≥ a k s ([ H ] c ) for every s ∈ { n − k , . . . , m } , i.e. [ G ] c  a ( k ) [ H ] c , and so ( c 2) [ G ] c  r ( k ) [ H ] c . The arguments in the pro ofs of 6.61 can b e used to prov e the following. 7.6 [19] L et G ∈ G m n , F b e the gr aph obtaine d fr om G by a symmetric K xy -op er ation, and k b e a p ositive inte ger at most n . Then G  r ( k ) F . F rom 4.2 ( a 2) and 7.6 we ha ve: 53 7.7 [19] L et G ∈ G m n , F b e the gr aph obtaine d fr om G by a symmetric K xy -op er ation, and k b e a p ositive inte ger at most n . Then [ G ] c  r ( k ) [ F ] c . Ob viously , 7.6 and 7.7 are generalizations of 7.3 and 7.5 , resp ectiv ely . F rom 4.6 and 7.5 we ha ve the following generalization of 6.11 . 7.8 [19, 27] Supp ose that H is a c onne cte d gr aph and n ≥ e ( H ) + 1 . Then for every sp anning tr e e T of F ther e exists a tr e e D with e ( H ) e dges such that D c ontains T and a k s ( K n − E ( D )) ≥ a k s ( K n − E ( H )) for every s ∈ { n − k , . . . , m } , i.e. K n − E ( D )  a ( k ) K n − E ( H ) , and so G  r ( k ) H . W e remind that the n um b ers r ( G ) = v ( G ) − cmp ( G ) and r ∗ ( G ) = e ( G ) − v ( G ) + cmp ( G ) are called the r ank and the c or ank (or the cyclomatic numb er ) of a graph G . 7.1 On the  a ( k ) -maximization problem for graphs with “small” corank Let us consider the following generalization of problem of problem T max in Section 1: ﬁnd a graph M ∈ C m n suc h that a k s ( M ) = max { a k s ( G ) : G ∈ ¯ G m n } , where 1 ≤ k < n and s ∈ { n − k , . . . , m } . Do es a solution of this problem dep ends on s or on k ? It turns out that a solution ma y dep end on s (see 7.14 ). F rom claim 7.9 it follows that if m ≤ n + 2, then a solution do es not dep end on s , and therefore it is a solution for the  a ( k ) - and  r ( k ) - maximization problems as well. Let F b e a graph and σ a p ositiv e integer. Let F σ denote the set of graphs H that are obtained from F by sub dividing each edge u by s ( u ) vertices, where P { s ( u ) : u ∈ E ( F ) } = σ , and so v ( H ) = v ( F ) + σ and e ( H ) = e ( F ) + σ ). W e call a graph H from F σ σ - uniform if | s ( u ) − s ( u 0 ) | ≤ 1 for ev ery u, u 0 ∈ E ( G . Let z = b σ /e ( F ) c . If H is a σ -uniform graph in F σ , then s ( u ) ∈ { z , z + 1 } for ev ery u ∈ E ( F ). Let E 0 = { u ∈ E ( F ) : s ( u ) = z } and E 1 = { u ∈ E ( F ) : s ( u ) = z + 1 } . Let F i b e the subgraphs induced by E i in F , i ∈ { 0 , 1 } . Obviously , if σ = 0 mo d e ( F ), then s ( u ) = z for every u ∈ E ( F ), and therefore there is exactly one σ -uniform graph in F σ up to isomorphism; we denote his graph F ∗ . Let B denote the graph with t w o vertices and three parallel edges. Then all σ - uniform graph in B σ are isomorphic to the same graph which we denote by Θ ∗ n , where n = σ + 2 = v (Θ ∗ n ). Let Q = K 4 and n = σ + 4. If σ mo d 6 ∈ { 0 , 1 , 5 } , then all σ -uniform graph in Q σ are isomorphic to the same graph which we denote by Q ∗ n . No w w e will deﬁne the σ -uniform graph Q ∗ n in case when σ mo d 6 6∈ { 0 , 1 , 5 } , i.e. when σ mo d 6 = r ∈ { 2 , 3 , 4 } , and so e ( Q 1 ) = r . Let H b e a σ -uniform graph in Q σ . 54 Put Q ∗ n = H if one of the following holds: ( h 1) r = 2 and Q 1 is a matching, ( h 2) r = 4 and Q 0 is a matching, and ( h 3) r = 3 and Q 1 (as well as Q 0 ) is a 3-edge path. Let, as ab o ve, O n denote the cycle with n v ertices. Using 7.1 and 7.2 , w e w ere able to prov e the follo wing. 7.9 [19, 32] L et G ∈ ¯ G m n and k ∈ { 1 , . . . , n − 3 } . Then the fol lowing holds. ( a 0) Supp ose that e ( G ) = n . If G is not a cycle, then O n  a ( k ) G , and so O n  r ( k ) G . ( a 1) Supp ose that e ( G ) = n + 1 . If G is not Θ ∗ n , then Θ ∗ n  a ( k ) G , and so Θ ∗ n  r ( k ) G . ( a 2) Supp ose that e ( G ) = n + 2 . If G is not Q ∗ n , then Q ∗ n  a ( k ) G , and so Q ∗ n  r ( k ) G . 7.2 On the  a ( k ) -maximization problem for graphs with “large” corank Let us reform ulate the problem in the previous part 7.1 as follo ws: ﬁnd in a complete graph K n a set Z of z edges (and the corresp onding subgraph of K n induced by Z ) such that a k s ( K n − Z ) = max { a k s ( K n − A ) : A ⊂ E ( K n ) , and | A | = z } . The follo wing result gives a solution to this problem for the graphs with relativ ely “large” corank, i.e. with relativ ely “small” | A | . F rom 7.5 w e ha ve, in particular, the follo wing generalization of 6.9 . 7.10 [19, 25] L et H b e a sub gr aph of K n and with r e dges with no isolate d vertic es. Supp ose that H is not isomorphic to r P 1 or P 2 + ( r − 2) P 1 . ( a 1) If n ≥ 2 r , then K n − E ( rP 1 )  a ( k ) K n − E ( P 2 + ( r − 2) P 1 )  a ( k ) K n − E ( H ) , and so K n − E ( rP 1 )  r ( k ) K n − E ( P 2 + ( r − 2) P 1 )  r ( k ) K n − E ( H ) . ( a 2) If n = 2 r − 1 , then K n − E ( P 2 + ( r − 2) P 1 )  a ( k ) K n − E ( H ) , and so K n − E ( P 2 + ( r − 2) P 1 )  r ( k ) K n − E ( H ) . 55 7.3 On the  a ( k ) -minimization problem for graphs with “small” corank Let us consider the following generalization of problem of problem T min in Section 1: ﬁnd a graph L ∈ C m n suc h that a k s ( L ) = min { a k s ( G ) : G ∈ C m n } , where 1 ≤ k < n and s ∈ { n − k , . . . , m } . Do es a solution of this problem dep ends on s or on k ? The following result gives a solution to this problem for the graphs of relatively “small” corank. It turns out that in this case the solution do es not dep end on s and is the solution for the  a ( k ) - and  r ( k ) -minimization problems as well. 7.11 [19, 20] L et G ∈ C m n and G 6∈ H m n . ( a 0) If m = n ≥ 3 , then G  a ( k ) L for every L ∈ H n n , and so G  a ( k ) F n n and G  r ( k ) F n n = W . ( a 1) If n ≥ 4 and m = n + 1 , then G  a ( k ) L for every L ∈ H n +1 n , and so G  a ( k ) F n +1 n and G  r ( k ) F n +1 n . ( a 2) If n ≥ 5 and m = n + 2 , then G  a ( k ) L for every L ∈ H n +2 n , and so G  a ( k ) F n +2 n and G  r ( k ) F n +2 n . ( a 3) If n ≥ 6 and n + 2 ≤ m ≤ 2 n − 2 , then G  a ( k ) L for every L ∈ H m n , and so G  a ( k ) F m n and G  r ( k ) F m n . 7.4 On the  a ( k ) -minimization problem for graphs with “large” corank Let us reform ulate the problem in the previous part 7.3 as follo ws: ﬁnd in a complete graph K n a set Z of z edges (and the corresp onding subgraph of K n induced by Z ) such that a k s ( K n − Z ) = min { a k s ( K n − A ) : A ⊂ E ( K n ) , K n − A is connected, and | A | = z } . A similar question is whether a solution of this problem dep ends on s or on k . The follo wing result gives a solution to this problem for the graphs with relatively “large” corank, i.e. with relatively “small” | A | . 7.12 [19, 20] L et H b e a sub gr aph of K n with no isolate d vertic es and with z ≥ 1 e dges. ( a 0) If z ≤ n − 2 and H is not isomorphic to F z z +1 = S z , then K n − E ( H )  a ( k ) K n − E ( F z z +1 ) , and so K n − E ( H )  r ( k ) K n − E ( F z z +1 ) . ( a 1) If z = n − 1 , then 56 K n − E ( H )  a ( k ) K n − E ( F z z ) , and so K n − E ( H )  r ( k ) K n − E ( F z z ) . ( a 2) If z = n , then K n − E ( H )  a ( k ) K n − E ( F z z − 1 ) , and so K n − E ( H )  r ( k ) K n − E ( F z z − 1 ) . Notice that claim ( a 0) in 7.12 is a generalization of 6.10 and follows from 4.5 and 7.5 . In 2.10 we deﬁned the so-called extreme threshold graphs F m n and the set H m n . Ob viously , F m n ∈ H m n ⊆ C m n and if G, G 0 ∈ H m n , then G  r G 0 . 7.13 [19, 20] L et n and z b e p ositive inte gers, n ≥ 3 , and n ≥ z . L et m =  n 2  − z . Supp ose that m ≥ n − 1 ( i.e. C m n 6 = ∅ ) and 1 ≤ z ≤ n . Then G ∈ C m n \ H m n ⇒ G  a ( k ) L for every L ∈ H m n , and in p articular, G  a ( k ) F m n . Pro of The follo wing claim is obviously true. Claim. ( c 1) If 1 ≤ z ≤ n − 2 , then F m n = K n − E ( F z z +1 ) . ( c 2) If z = n − 1 , then F m n = K n − E ( F z z ) . ( c 3) If z = n , then F m n = K n − E ( F z z − 1 ) . No w the claim of the theorem follo ws from the ab o v e Claim and 7.12 .  7.5 Some problems on the reliabilit y p oset of graphs Using 6.9 , we w ere able to pro v e the follo wing ab out the existence of  r -maxim um graphs in G m n . 7.14 [19, 22, 24, 32] L et, as ab ove, G  r F if R ( p, G ) ≥ R ( p, F ) for every p ∈ [0 , 1] . Then ( a 1) for every n ≤ 5 and n − 1 ≤ m ≤ e ( K n ) , G m n has an  r -maximum gr aph and ( a 2) for every n ≥ 6 ther e exists m = m ( n ) such that G m n do es not have an  r -maximum gr aph. F rom 7.14 w e ha ve, in particular: 7.15 [19, 22, 24, 32] F or every n ≥ 6 ther e exists m = m ( n ) such that G m n do es not have an  a -maximum gr aph. P ap er [24] con taining theorem 7.14 was also mentioned in a survey pap er [15] pub- lished in Journal of Graph Theory in 1982. Acciden tally , in 1986 the author of [2] put forw ard a conjecture con tradicting 7.14 and 7.15 , namely , saying that G m n has a  r - maxim um graph for every ( n, m ) with n − 1 ≤ m ≤ e ( K n ) and claiming in addition that 57 his conjecture is true for ev ery n ≤ 6. Here are some in teresting problems related with 7.14 . As ab o v e, let M ( p ) be a graph in ¯ G m n , namely , R ( p, M ( p )) = max { R ( p, G ) : G ∈ ¯ G m n } , i.e. M ( p ) is a most reliable p -random graph in ¯ G m n . Let M m n = { M ( p ) ∈ G m n : p ∈ [0 , 1] } . Ob viously , M m n is a ﬁnite set. By 7.14 , there are inﬁnitely many pairs ( n, m ) such that |M| m n ≥ 2. 7.16 Problem. [19] Is ther e a numb er N such that |M m n | ≤ N for every n ≥ 2 and m ≤ n ( n − 1) / 2 ? Giv en G, F ∈ G m n suc h that R ( p, G ) 6≡ R ( p, F ), let cr s { R ( G, F ) } denote the n umber of zeros of R ( p, G ) − R ( p, F ) in (0 , 1)(with their m ultiplicities). In [35] w e observed that there are pairs ( G, F ) of equi-size graphs such that cr s { R ( G, F ) } ≥ 1. Are there suc h pairs ( G, F ) with cr s { R ( G, F ) } ≥ 2 ? This was a long standing question un til we gav e in [16] a construction pro viding for ev ery in teger k ≥ 1 an equi-size pair ( G, F ) suc h that cr s { R ( G, F ) } = k . A similar question arises for equi-size pairs ( G, F ) of R ( p )-maxim um graphs. Let cr s { R m n } = max { cr s { R ( G, F ) } : G, F ∈ M m n , R ( p, G ) 6≡ R ( p, F ) } . As w e ha v e mentioned in 7.14 , for every n ≥ 6 there exists m = m ( n ) such that cr s { R m n } ≥ 1. 7.17 Problem. [19] Ar e ther e p airs ( n, m ) such that cr s { R m n } ≥ 2 ? F urthermor e, is it true that for every inte ger k ≥ 1 ther e exists ( n, m ) such that cr s { R m n } ≥ k ( or, mor e over, cr s { R m n } = k ) ? In Section 1 we men tioned the problem on the minimal elemen ts of the  r -p oset on C m n . Here is a more general versio n of this problem. 7.18 Problem. [19, 20] Is the fol lowing claim true ? Claim . L et m ≥ n − 1 , and so C m n 6 = ∅ . Then ther e exists L ∈ C m n such that G  r ( k ) L for every G ∈ C m n . A similar problem concerns the  a ( k ) -minimal graphs in C m n . 7.19 Problem. [19, 20] Is the fol lowing claim true ? Claim . L et m ≥ n − 1 , and so C m n 6 = ∅ . Then ther e exists L ∈ C m n such that G  a ( k ) L for every G ∈ C m n . Let, as ab o ve, F m n denote the set of connected threshold graphs with n vertices and m edges, and so F m n ⊆ C m n . F rom 7.4 it follows that Problem 7.18 is equiv alent to the follo wing problem. 7.20 Problem. [19, 20] Is the fol lowing claim true ? Claim. L et m ≥ n − 1 .Then ther e exists F ∈ F m n such that G  r ( k ) F for every G ∈ F m n . 58 Similarly , from 7.4 it follows that Problem 7.19 is equiv alent to the follo wing prob- lem. 7.21 Problem. [19, 20] Is the fol lowing claim true ? Claim. L et m ≥ n − 1 .Then ther e exists F ∈ F m n such that G  a ( k ) F for every G ∈ F m n . Ob viously , Claim in 7.19 implies Claim in 7.18 and Claim in 7.21 implies Claim in 7.20 . There are some results supp orting Claim in 7.19 and indicating that H m n is the set of all  a ( k ) -minim um graphs in C m n (see, for example, 7.11 and 7.12 ). 8 On some other graph parameters W e call a comp onen t of a graph non-trivial if it has at least one edge, and trivial otherwise. Let F ( G, v , e ) denote the set of forests F in G with v v ertices and e edges suc h that eac h comp onen t of F is non-trivial, and so F is induced by its set of edges and has v − e comp onen ts. In particular, F ( G, 2 e, e ) = M ( G, e ) is the set of matchings in G with e edges. Giv en a vertex x of G , let F x ( G, v , e ) denote the set of forests F in F ( G, v , e ) con taining v ertex x . Let |F x ( G, v , e ) | = f x ( G, v , e ) and | M ( G, f ) | = m ( G, e ). W e remind some notations and notation from the pro of of 6.67 . Let G = ( V , E ) b e a simple graph, x, y ∈ V , x 6 = y , Z = N x ( G ) ∩ N y ( G ), X = N x ( G ) \ ( Z ∪ { y } ), and Y = N y ( G ) \ ( Z ∪ { x } ). Let e v = xv and e 0 v = y v for v ∈ X and let E x = [ x, X ] = { e v : v ∈ X } and E y = [ y , X ] = { e 0 v : v ∈ X } . Let G 0 b e obtained from G by the ♦ xy -op eration, i.e. G 0 = ( G \ E x ) ∪ E y . Obviously , [ { x, y } , Z ] ⊆ E ( G ) and E ( G ) − E x = E ( G 0 ) − E y . Let ε ( u ) = u if u ∈ E ( G ) − E x and ε ( e v ) = e 0 v if e v ∈ E x (and so v ∈ X ). Then ε : E ( G ) → E ( G 0 ) is a bijection. F or U ⊆ E ( G ), let ε [ U ] = { ε ( u ) : u ∈ U } . F or a subgraph S of G , let ϑ ( S ) b e the subgraph of G such that V ( ϑ ( S )) = V ( S ) and E ( ϑ ( S )) = ( E ( S ) \ E x ) ∪ ε [ E ( S ) ∩ E x ]. Let S ( H ) denote the set of subgraphs of a graph H . Ob viously , ϑ is a bijection from S ( G ) to S ( G 0 ). F or A ⊆ S ( G ), let ϑ [ A ] = { ϑ ( A ) : A ∈ A} . 8.1 [19] L et G b e a c onne cte d gr aph with n vertic es, x and y two distinct vertic es in G . L et v and e b e inte gers such that F ( G, v , e ) 6 = ∅ . L et G 0 b e obtaine d fr om G by the ♦ xy -op er ation. Then f x ( G, v , e ) ≥ f x ( G 0 , v , e ) and m ( G, e ) ≥ m ( G 0 , e ) . 59 Pro of . The pro of of this theorem is similar to but simpler than the pro of of 6.67 . Let P 0 b e a forest in G 0 with at most t w o comp onen ts eac h b eing non-trivial and eac h meeting { x, y } . Let P be the subgraph in G suc h that P 0 = ϑ ( P ). Clearly , { x, y } ⊆ V ( P 0 ) = V ( P ) and e ( P 0 ) = e ( P ). Let σ ( xz ) = y z and σ ( y z ) = xz for every z ∈ Z . Obviously , σ : [ { x, y } , Z ] → [ { x, y } , Z ] is a bijection. F or S ⊆ [ { x, y } , Z ], let σ [ S ] = { σ ( s ) : s ∈ S } . Let A = A ( P ) = [ { x, y } , Z ] ∩ E ( P ). Obviously , A ( P ) = A ( P 0 ). Let ¯ P = ( P − A ) ∪ σ [ A ] and ¯ P 0 = ( P 0 − A ) ∪ σ [ A ]. Obviously , ¯ P = P and ¯ P 0 = P 0 if and only if σ [ A ] = A . Clearly , b oth P and P 0 = ϑ ( P ) hav e at most t w o comp onen ts. W e need the following simple facts (see similar claims in the pro of of 6.67 ). Claim 1. Supp ose that P 0 is a tr e e and P is not a tr e e. Then ¯ P is a tr e e and ¯ P 0 = ϑ ( P ) is not a tr e e. Pr o of. Since v ( P ) = v ( P 0 ), e ( P ) = e ( P 0 ), P 0 is a tree, and P is not a tree, w e hav e: P has a cycle C . If C do es not contain vertex x or C contains tw o edges from E [ X ], then P 0 has a cycle, a contradiction. Therefore C is the only cycle in P and C contains exactly one edge xc with c ∈ X and exactly one edge xz with z ∈ Z . Then ¯ P is a tree, x ∈ V ( P 0 ) = V ( ¯ P ), and ¯ P 0 = ϑ ( ¯ P ) is not a tree (namely , it has a cycle). ♦ It is also easy to pro v e the following tw o claims. Claim 2. P 0 has two c omp onents if and only if P = ϑ − 1 ( P ) has two c omp onents. Claim 3. Supp ose that b oth P 0 and P = ϑ − 1 ( P 0 ) have two c omp onents. Then ( a 1) b oth ¯ P and ¯ P 0 = ϑ ( ¯ P ) have two c omp onents and ( a 2) if P 0 has no trivial c omp onents and P has a trivial c omp onent ( namely, y ) , then ¯ P has no trivial c omp onents and ¯ P 0 has a trivial c omp onent ( namely, x ) . Giv en a forest F of G containing x and y , let F xy b e the minimal subforest of F con taining x and y and suc h that each comp onent of F xy is a component of F . Obviously , F xy has at most tw o comp onen ts eac h meeting { x, y } . Let ¯ F = F − F xy + ¯ F xy . Let F = F x ( G, v , e ), F 0 = F x ( G 0 , v , e ), and ˜ F = ϑ − 1 [ F 0 ]. Let A 0 and B 0 b e subsets of F 0 suc h that F 0 = A 0 ∪ B 0 , ϑ − 1 [ A 0 ] ⊆ F , and ϑ − 1 [ B 0 ] ∩ F = ∅ , and so A 0 ∩ B 0 = ∅ . Let ˜ A = ϑ − 1 [ A 0 ] and ˜ B = ϑ − 1 [ B 0 ]. Then ˜ F = ˜ A ∪ ˜ B , ˜ A ∩ ˜ B = ∅ , |A| = |A 0 | , and |B | = |B 0 | . Supp ose ﬁrst that B 0 = ∅ . Then ϑ − 1 is an injection from F 0 to F and we are done. No w supp ose that B 0 6 = ∅ . Let c ( H ) denote the n um b er of non-trivial comp onents of a graph H . Let B 0 ∈ B 0 , and so B = ϑ − 1 ( F 0 ) is not an elemen t of F . Obviously , e ( B ) = e ( B 0 ) = e and x ∈ V ( B 0 ). Now x ∈ V ( B 0 ) ⇒ x ∈ V ( B ). Therefore c ( B ) 6 = c ( B 0 ) or, equiv alently , c ( B xy ) 6 = c ( B 0 xy ). Then b y Claims 1 and 3, c ( ¯ B ) = c ( B 0 ) and c ( ¯ B 0 ) 6 = c ( B 0 ), and so ¯ B ∈ F and ¯ B 0 6∈ F 0 . Let ˜ ϑ ( F 0 ) = ϑ − 1 ( F 0 ) if F 0 ∈ A 0 and ˜ ϑ ( F 0 ) = ¯ F if F 0 ∈ B 0 . It is easy to see that the following holds. 60 Claim 4. L et F 0 1 , F 0 2 ∈ F 0 . Then F 0 1 6 = F 0 2 ⇔ ˜ ϑ ( F 0 1 ) 6 = ˜ ϑ ( F 0 2 ) . It follows that ˜ ϑ is an injection from F 0 to F . Thus |F | ≥ |F 0 | . Similar (but muc h simpler) arguments sho w that m ( G, e ) ≥ m ( G 0 , e ).  By the ab o v e deﬁnition, a comp onent of a graph is non-trivial if it has at least one edge. Notice that if this deﬁnition is replaced by: “a comp onen t of a graph is non-trivial if it has at least s edges with s ≥ 2”, then the claim of theorem 8.1 is no longer true. Ob viously , f x ( G, v ( G ) , v ( G ) − 1) = t ( G ). Therefore theorem 6.3 ( a 1) for the ♦ - op eration is a particular case of 8.1 . Also, it follows from 8.1 that a similar result holds for the num b er of spanning forests in G with a giv en n um b er of edges (or, the same, with a given n umber of comp onents). Let S x ( G, v , e, k ) denote the set of subgraphs of G containing vertex x and ha ving v v ertices, e edges and at most k comp onen ts. Let s x ( G, v , e, k ) = |S x ( G, v , e, k ) | . The argumen ts similar to those in the pro of of 8.1 can b e used to pro v e the following generalization of 8.1 . 8.2 [19] L et G b e a c onne cte d gr aph and x and y two distinct vertic es in G . L et v and e b e inte gers such that S x ( G, v , e, k ) 6 = ∅ . L et G 0 b e obtaine d fr om G by the ♦ xy -op er ation. Then s x ( G, v , e, k ) ≥ s x ( G 0 , v , e, k ) . Let h 0 ( G ) and h 1 ( G ) denote the num b er of Hamiltonian cycles and Hamiltonian paths, resp ectively . 8.3 [19] L et G ∈ G m n and G 0 b e the gr aph obtaine d fr om G by an ♦ xy -op er ation. Then h s ( G ) ≥ h s ( G 0 ) for s ∈ { 0 , 1 } . Pro of . The pro of is similar to but muc h simpler than the pro of of 6.67 . F or a graph F , let H 0 ( F ) b e the set of Hamiltonian cycles of F and H 1 ( F ) the set of Hamiltonian paths of F , and so h s ( F ) = |H s ( F ) | . Case 1. Consider a Hamiltonian cycle C 0 in G 0 . Let C 0 b e a Hamiltonian cycle in G 0 and C = ϑ − 1 ( C 0 ). If E ( C 0 ) ∩ [ y , X ] = ∅ , then C is also a Hamiltonian cycle in G . If x is incident to at most one edge in G 0 , then G 0 has no Hamiltonian cycle. Therefore w e assume that x is incident to at least tw o edges in G 0 and E ( C 0 ) ∩ [ y , X ] 6 = ∅ , and so E ( C 0 ) ∩ [ y , X ] has either one or tw o edges. Ob viously , C 0 has either exactly tw o edges xz 1 and xz 2 in [ x, Z ] or exactly one edge xz in [ x, Z ] and edge xy (pro vided xy ∈ E ( G 0 )). Case 1.1. Supp ose that C 0 has exactly one edge xz in [ x, Z ] and edge xy . Then C 0 has exactly one edge y x 0 in [ y , X ]. Then C = C 0 − y x 0 + xx 0 is not a Hamiltonian cycle in G , namely , y is incident to one edge in C and C − y is a cycle containing xz . Put ¯ C = C − xz + y z . Then ¯ C is a Hamiltonian cycle in G and ¯ C 0 is not a Hamiltonian cycle in G 0 . Case 1.2. Now supp ose that C 0 has exactly tw o edges xz 1 and xz 2 in [ x, Z ]. Supp ose that C 0 has exactly one edge y x 0 in [ y , X ] (and so x 0 ∈ X ). Then C = 61 C 0 − y x 0 + xx 0 is not a Hamiltonian cycle in G , namely , C has a unique cycle D and v ertex y is of degree one in C . Obviously , D has exactly one edge in { xz 1 , xz 2 } , say xz 1 . Put ¯ C = C − xz 1 + y z 1 . Then ¯ C is a Hamiltonian cycle in G and ¯ C 0 is not a Hamiltonian cycle in G 0 . No w supp ose that E ( C 0 ) ∩ [ y , X ] has exactly tw o edge y x 1 and y x 2 (and so x 1 , x 2 ∈ X ). Then C = C 0 − { y x 1 , y x 2 } + { xx 1 , xx 2 } is not a Hamiltonian cycle in G , namely , C has exactly t w o cycles D 1 and D 2 , each xx i ∈ D i , and y is an isolated v ertex in C . Put ¯ C = C − { xz 1 , xz 2 } + { yz 1 , y z 2 } . Then ¯ C is a Hamiltonian cycle in G and ¯ C 0 is not a Hamiltonian cycle in G 0 . Case 2. Now consider a Hamiltonian path P 0 in G 0 and assume that E ( P 0 ) ∩ [ y , X ] 6 = ∅ , and so E ( P 0 ) ∩ [ y , X ] has either one or tw o edges. Obviously , either P 0 con tains xy (pro vided xy ∈ E ( G 0 )) or has one or t w o edges in [ x, Z ]. Case 2.1. Supp ose that xy ∈ E ( P 0 ). If x is an end of P 0 , then P 0 has exactly one edge x 0 y in [ y , X ] and P = P 0 − x 0 y + x 0 x is a Hamiltonian path in G . Otherwise, the situation is similar to Case 1.1 on Hamiltonian cycles. Case 2.2. Now supp ose that P 0 has one or tw o edges in [ x, Z ]. It is easy to see that P is a Hamiltonian path in G if and only if P 0 has exactly one edge in [ x, Z ], exactly one edge y x 0 in [ y , X ], and in P 0 y is closer to x than x 0 . Supp ose that x is an end-vertex of P 0 , and so P 0 has exactly one edge xz in [ x, Z ]. If P 0 has exactly one edge y x 0 in [ y , X ] and in P 0 x 0 is closer to x than y , then P = P 0 − y x 0 + xx 0 is not a Hamiltonian path in G , namely , P has exactly tw o comp onen ts and one of them a cycle con taining xz . Put ¯ P = P − xz + yz . Then ¯ P is a Hamiltonian path in G and ¯ P 0 is not a Hamiltonian path in G 0 . If P 0 has t wo edges y x 1 and y x 2 in [ y , X ], then P = P 0 − { y x 1 , y x 2 } + { xx 1 , xx 2 } is not a Hamiltonian path in G , namely , P has exactly one cycle D , xz ∈ E ( D ), and y is an isolated vertex in P . Put ¯ P = P − xz + y z . Then ¯ P is a Hamiltonian path in G and ¯ P 0 is not a Hamiltonian cycle in G 0 . No w supp ose that x is not an end-vertex of P 0 , and so P 0 has exactly t wo edges in [ x, Z ]. Then the situation is similar to Case 2.1 on Hamiltonian cycles. Th us, from the ab o ve Cases it follows that the following claim is true. Claim 1. L et s ∈ { 0 , 1 } . Supp ose that Q 0 ∈ H s ( G 0 ) and Q 6∈ H s ( G ) . Then ¯ Q ∈ H s ( G ) and ¯ Q 0 6∈ H s ( G 0 ) . Let H = H s ( G ) and H 0 = H s ( G 0 ). Let A 0 and B 0 b e subsets of H 0 suc h that H 0 = A 0 ∪ B 0 , ϑ − 1 [ A 0 ] ⊆ H , and ϑ − 1 [ B 0 ] ∩ H = ∅ , and so A 0 ∩ B 0 = ∅ . Let ˜ A = ϑ − 1 [ A 0 ] and ˜ B = ϑ − 1 [ B 0 ]. Then ˜ H = ˜ A ∪ ˜ B , ˜ A ∩ ˜ B = ∅ , |A| = |A 0 | , and |B | = |B 0 | . It is easy to see that the following holds. Claim 2. L et H 0 1 , H 0 2 ∈ H 0 . Then H 0 1 6 = H 0 2 ⇔ ˜ ϑ ( H 0 1 ) 6 = ˜ ϑ ( H 0 2 ) . 62 Supp ose ﬁrst that B 0 = ∅ . Then by Claim 2, ϑ − 1 is an injection from H 0 to H and w e are done. No w suppose that B 0 6 = ∅ . Let ˜ ϑ ( H 0 ) = ϑ − 1 ( H 0 ) if H 0 ∈ A 0 and ˜ ϑ ( H 0 ) = ¯ H if H 0 ∈ B 0 . Then by Claims 1 and 2, ˜ ϑ is an injection from H 0 to H . Th us, |H | ≥ |H 0 | .  Let x ∈ V ( G ) and s a p ositiv e integer. Let h x 0 ( G, s, k ) denote the n umber of subgraphs of G containing x and having s v er- tices and at most k comp onen ts eac h b eing a cycle. Let h x 1 ( G, s, k ) denote the n umber of subgraphs of G containing x and having s v er- tices and exactly k comp onen ts eac h b eing a non-trivial path. Using the argumen ts similar to those in the ab o v e pro of of 8.3 , the follo wing gener- alization of 8.3 can b e prov ed. 8.4 L et G ∈ G m n , G 0 b e the gr aph obtaine d fr om G by an ♦ xy -op er ation. L et η ( G ) ∈ { h x 0 ( G, s, k ) , h x 1 ( G, s, k ) } . Then η ( G ) ≥ η ( G 0 ) . It can b e shown that if in the deﬁnition of h x 0 ( G, s, k ) condition “at most k comp o- nen ts” is replaced b y “exactly k comp onents” (as in the deﬁnition of h x 0 ( G, s, k )), then theorem 8.4 will b e no longer true for h x 0 ( G, s, k ). The results similar to 8.4 are also true for some other t yp es of subgraphs of a graph. 9 On w eigh ted graphs and digraphs Man y notions and results ab ov e can b e naturally generalized to weigh ted graphs and digraphs of diﬀerent t yp e. Here are some of them. Let D = ( V , E ) b e a directed graph or simply , a digraph (and so E ⊆ V × V ) and w a function from E → R . W e call the pair ( D , w ) a weighte d digr aph also denoted by D w . If ( x, y ) ∈ E ⇔ ( y , x ) ∈ E in D , then D can b e inte rpreted as an undirected graph G and ( G w ) the corresp onding w eigh ted graph. F or v ∈ V , let d ( v , G w ) = P { w ( v x ) : x ∈ V ( G − v ) } . F or V = V ( G ) = { v 1 , . . . , v n } , let A ( G w ) b e the ( V × V )-matrix ( a ij ), where each a ii = 0 and each a ij = w ( v i v j ) for i 6 = j . Let R ( D w ) b e the diagonal ( V × V )-matrix ( r ij ), where eac h r ii = d ( v i , G w ) and each r ij = 0 for i 6 = j . Let L ( D w ) = R ( G w ) − A ( G w ). A dir e cte d tr e e ( or ditr e e ) T r o ote d at z is a digraph such that its underlying graph is a tree, z ∈ V ( T ), d out ( z ) = 0, and d out ( v ) = 1 for ev ery v ∈ V ( T ) \ z . Let T z ( D ) denote the set of spanning ditrees of D ro oted at z , w ( T ) = Q { w ( e ) : e ∈ E ( T ) } for T ∈ T z ( D ), and t z ( G w ) = P { w ( T )) : T ∈ T z ( G ) } . Here is an obvious generalization of the classical Matrix T ree Theorem for w eigh ted 63 digraphs. 9.1 L et D w b e a weighte d digr aph and r ∈ V ( D ) . Then t r ( G w ) = det ( L r ( D w )) . Ob viously , det ( λI − L ( D w )) = λP ( λ, D w ), where P ( λ, D w ) is a p olynomial of degree n − 1. Let S ( G w ) = ( λ 1 ( D w ) , . . . , λ n − 1 ( D w )) b e the list of ro ot ro ots of P ( λ, D w ). Let C = ( V , E ) b e a directed graph such that E = V × V . Digraph C is called the c omplete digr aph with the vertex set V , and so every vertex of C has a lo op. Let w b e a function from V × V to R , and so C w is a w eigh ted digraph. Giv en a function a : V × V → R , let u = a − w . Then C u is another digraph with the vertex set V . W e call digraph C u a - c omplement of C w and write C u = [ C w ] a . Here is a generalization of the Recipro city Theorem 3.12 for w eighted digraphs. 9.2 [33, 38] L et C w b e a weighte d digr aph with n vertic es. L et σ : ( V × V ) → R b e a c onstant function: σ ( xy ) = q ∈ R for every xy ∈ V × V . Then ( a 1) ther e is a bije ction α : { 1 , . . . , n − 1 } → { 1 , . . . , n − 1 } such that λ i ( C w ) + λ α ( i ) ([ C w ] q ) = q n for every i ∈ { 1 , . . . , n − 1 } or, e quivalently, ( a 2) ( q n − λ ) L ( λ, [ C w ] q ) = ( − 1) n − 1 λL ( q n − λ, C w ) . A weigh ted digraph C w is called r -out-r e gular ( r -in-r e gular ) if d out ( v , C w ) = r (re- sp ectiv ely , d in ( v , C w ) = r ) for every v ∈ V ( D ). Let A ( λ, C w ) = det ( λI n − A ( D w )). Ob viously , if C w is r -out-regular or r -in-regular, then A ( r , C w ) = 0. F rom 9.2 we hav e the corresp onding Recipro cit y Theorem on A ( λ, C w ) for an out-regular (resp ectiv ely , in-regular) weigh ted digraph D w . 9.3 [33] L et C w b e a weighte d r -out-r e gular or r -in-r e gular digr aph with n vertic es. Then ( λ + r ) A ( λ, [ C w ] q ) = ( − 1) n ( λ − q n + r ) A ( − λ, C w ) . Let p ( x, y ) b e a p olynomial of t w o v ariables x and y . Assuming that xy 6 = y x , w e call p ( x, y ) a xy -p olynomial ( y x -p olynomial ) if p do es not contain “the pro duct y x ” (resp ectiv ely , “the pro duct xy ”). Therefore if xy = y x , then p ( x, y ) is b oth xy and y x -p olynomial. Using 9.3 it is also easy to prov e the following useful fact. 9.4 L et C w b e a weighte d r -out-r e gular ( r -in-r e gular) digr aph with n vertic es. L et J nn b e the n × n -matrix with al l entries e qual 1 and { α 1 , . . . , α n } b e the set of eigenvalues of A ( C w ) = A , wher e α n = r . L et p ( x, y ) b e a p olynomial with two variables x and y , and with r e al c o eﬃcients. Supp ose that p ( x, y ) is an xy -p olynomial if C w is r -out-r e gular and is an y x -p olynomial if C w is r -in-r e gular. Then matrix p ( A, J nn ) has the eigenvalues p ( r , n ) and p ( α i , 0) for i = 1 , · · · , n − 1 . 64 Let G w b e a w eigh ted graph and let T ( G ) denote the set of spanning trees of G . Giv en X ⊆ V , let F ( G, X ) denote the set of spanning forests F of G such that every comp onen t of F has exactly one vertex in X , and so T ( G ) = F ( G, x ) for every x ∈ V . F or T ∈ T ( G ) and F ∈ F ( G, X ), let w ( T ) = Q { w ( e ) : e ∈ E ( T ) } and w ( F ) = Q { w ( e ) : e ∈ E ( F ) } and let t ( G w ) = P { w ( T )) : T ∈ T ( G ) } and f ( G, X ) = P { w ( F )) : F ∈ F ( G, X ) } . Then t ( G w ) = f ( G, x ) for ev ery x ∈ V , and so v ( G ) t ( G w ) = P { f ( G, x ) : x ∈ V } . F or x, y ∈ V ( G ), x 6 = y , let G w / { x, y } denote the w eigh ted graph F u suc h that V ( F ) = V ( G ) \ { x, y } ∪ t and u ( tz ) = w ( xz ) + w ( y z ) for every z ∈ V ( G − { x, y } ) and u ( ab ) = w ( ab ) for a, b ∈ V ( G − { x, y } ). W e say that G w / { x, y } is obtaine d fr om G w by identifying its vertic es x and y . F or h = xy ∈ E ( G ), let G w /xy = G w / { x, y } and G w − h = G w 0 , where w 0 ( h ) = 0 and w 0 ( e ) = w ( e ) for every e ∈ E ( G − h ). F or a graph G w and X ⊆ V ( G ), let L X ( G w ) denote the matrix obtained from L ( G w ) by removing the rows and columns corresp onding to every vertex x ∈ X and let G w X = G w /X denote the graph obtained from G b y identifying all vertices in X . No w it is easy to see that the follo wing generalization of Theorems 3.1 and 3.2 holds. 9.5 [19] L et G w b e a weighte d gr aph, V = V ( G ) , and X ⊆ V ( G ) . Then t ( G w X ) = f ( G w , X ) = det ( L X ( G w )) , and so t ( G w ) = det ( L v ( G w )) for every v in V ( G ) . As b efore, det ( λI − L ( G w )) = λP ( λ, G w ), where P ( λ, G w ) is a p olynomial of degree n − 1 with the ro ot sequence S ( G w ) = ( λ 1 ( G w ) ≤ . . . , ≤ λ n − 1 ( K w )) and P ( λ, G w ) = P { ( − 1) s c s ( G w ) λ n − 1 − s : s ∈ { 0 , . . . , n − 1 }} . The following generalizations of 3.7 and 3.9 are true. 9.6 [19] L et G w b e a weighte d gr aph with n vertic es and m e dges, s ∈ { 0 , . . . , n − 1 } , and V = V ( G ) . Then ( a 1) c s ( G ) = P { f ( G w , V − X ) : X ⊆ V , | X | = s } = P { γ ( F ) Q { w ( e ) : e ∈ E ( F ) } : F ∈ F ( K ) , e ( F ) = s } and ( a 2) ( m − s ) c s ( K w ) = P { c s ( K w − e ) : e ∈ E ( K ) } for s ∈ { 0 , . . . , m } . Let, us ab o ve, K = ( V , E b e a complete graph, and so E =  V 2  ), and K w a w eighed graph with the vertex set V . Given a function σ :  V 2  → R , let u = σ n − w . W e call ( K u ) the σ - c omplement of K w and write K u = [ K w ] σ . 65 Here is a generalization of the Recipro city Theorem 3.12 for w eighted graphs. 9.7 [19, 31, 38] L et K w b e a weighte d gr aph with n vertic es. L et σ :  V 2  → R b e a c onstant function, namely, σ ( uv ) = q ∈ R + for every uv ∈  V 2  . Then ( a 1) λ i ( K, w ) + λ n − i ( K, w ) q ) = q n for every i ∈ { 1 , . . . , n − 1 } , and so ( a 2) ( q n − λ ) L ( λ, ( K, w ) q ) = ( − 1) n − 1 λL ( q n − λ, ( K, w )) and ( a 3) if q ≥ max { w ( e ) : e ∈ E ( K ) } and w ( e ) ≥ 0 for every e ∈ E ( K ) , then 0 ≤ λ i ( K, w ) ≤ q n and 0 ≤ λ i ( K, w ) q ) ≤ q n for every i ∈ { 1 , . . . , n − 1 } . Theorem 9.7 w as used in [31] to giv e a simple pro cedure for ﬁnding the sp ectrum of so-called totally decomp osable symmetric matrices. Let K = ( V , E ) b e a complete graph and w : E → R b e a function. Let { v 1 , v 2 } ⊆ V , S ⊆ V \ { v 1 , v 2 } , and ε : S → R b e a function. F or i ∈ { 1 , 2 } , let w i : E → R b e a function such that w i ( e ) = w ( e ) for every e ∈ E \ [ S, v i ] and w i ( sv i ) = w ( sv i ) + ε ( s ) for ev ery s ∈ S . The following result establishes an inequality b et w een t ( K , w 1 ) and t ( K , w 2 ) under certain conditions on functions w and ε . 9.8 [19, 27] L et K = ( V , E ) b e a c omplete gr aph. Supp ose that ( h 1) w : E → R and ε : S → R ar e non-ne gative functions and ( h 2) w ( sv 2 ) ≥ w ( sv 1 ) for every s ∈ S . Then t ( K , w 1 ) ≥ t ( K , w 2 ) . Ob viously , 6.3 follows from 9.8 when an H -op eration is ♦ -op eration. Here is an analog of 9.8 for graphs with randomly deleted edges. Let K = ( V , E ) b e a complete graph and q : E → [0 , 1] b e a function. Let { v 1 , v 2 } ⊆ V , S ⊆ V \ { v 1 , v 2 } , and ε : S → [0 , 1] b e a function. W e call ( K, q ) a r andom gr aph , interpret q ( e ) as the probabilit y that edge e of the complete graph K do es not exist and assume that all edge ev en ts are mutually indep enden t. Let R k ( K, q ) denote the probabilit y that the random graph ( K, q ) has at most k comp onen ts. F or i ∈ { 1 , 2 } , let q i : E → [0 , 1] b e a function suc h that q i ( e ) = q ( e ) for every e ∈ E \ [ S, v i ] and q i ( sv i ) = q ( sv i ) ε ( s ) for every s ∈ S . The follo wing result establishes an inequality b etw een R k ( K, q 1 ) and R k ( K, q 2 ) under certain conditions on function q . 9.9 [19, 25] Supp ose that q ( sv 2 ) ≥ q ( sv 1 ) for every s ∈ S . Then R k ( K, q 2 ) ≥ R k ( K, q 1 ) . Ob viously , 7.3 follows from 9.9 when an H -op eration is ♦ -op eration. No w w e will deﬁne an analogue of the K xy -op eration for w eighted graphs. Let K = ( V , E ) be a complete graph, { v 1 , v 2 } ⊆ V , S ⊆ V \{ v 1 , v 2 } and let ε b e a function from S to R . F or i ∈ { 1 , 2 } , let r i : S → R b e a function such that r i ( s ) = w ( sv i ) for s ∈ S . Let r 0 1 ( s ) = r 1 ( s ) + ε ( s ) and r 0 2 ( s ) = r 2 ( s ) − ε ( s ) for s ∈ S . Put u ( e ) = w ( e ) for 66 e ∈ E \ [ { v 1 , v 2 } , S ] and u ( sv i ) = r 0 i ( s ) for s ∈ S , and so u is a function from E to R . W e sa y that K u is obtaine d fr om K w by the ( v 1 , v 2 , ε ) -op er ation and denote it K w ( v 1 , v 2 , ε ). Let a : E → R b e the function suc h that a ( e ) = w ( e ) for every e ∈ E \ [ { v 1 , v 2 } , S ] and a ( e ) = 0, otherwise. W e call the ( v 1 , v 2 , ε )-op eration symmetric on K w if there exists an authomorphism σ : V ( K ) → V ( K ) of K a suc h that σ ( v 1 ) = v 2 and σ ( v 2 ) = v 1 . Let B ( K w ) = D 0 ( K w ) + A ( K w ), where D 0 ( K w ) is a diagonal ( V × V )-matrix. It is easy to pro ve the following claim. 9.10 L et x ∈ R n and x S = x | S . Then x t B ( K u ) x − x t B ( K w ) x = 2( x 1 − x 2 )( ε · x S ) . Since B ( K w ) is a symmetric matrix, all eigenv alues of B ( K w ) are real num b ers. Let β ( K w ) denote the maximum eigen v alue of B ( K w ). 9.11 [19] L et K = ( V , E ) b e a c omplete gr aph, { v 1 , v 2 } ⊆ V , S ⊆ V \ { v 1 , v 2 } , and ε : S → R and w : E → R b e functions. L et K u = K w ( v 1 , v 2 , ε ) b e obtaine d fr om K w by the ( v 1 , v 2 , ε ) -op er ation. Supp ose that ( h 1) w : E → R is a non-ne gative function, ( h 2) a weighte d gr aph K w is c onne cte d, ( h 3) r 1 ≥ r 2 ≥ ε ≥ ¯ 0 , i.e., w ( v 1 s ) ≥ w ( v 2 s ) ≥ ε ( s ) ≥ 0 for every s ∈ S , and ( h 4) the ( v 1 , v 2 , ε ) -op er ation is symmetric on K w . Then β ( K w ( v 1 , v 2 , ε )) ≥ β ( K w ) > 0 . Pro of. Since B ( K w ) is symmetric, all its eigen v alues are real n umbers. Since b y ( h 1), w is a non-negative function, all entries of B ( K w ) are non-negative. Since b y ( h 2), K w is connected, matrix B ( K w ) is irreducible. Let x ∈ R n b e an eigenv ector of matrix B ( K w ) corresp onding to its maximum eigenv alue β ( K w ). W e can assume that k x k = 1. Now b y the P erron-F robenius theorem [8], β ( K w ) > 0 and all co ordinates of x are p ositiv e. By the Couran t-Swartz theorem, β ( K w ) = max { z t B ( K w ) z : z ∈ R n , k z k = 1 } = x t A ( K w ) x (9.1) and β ( K u ) = max { z t A ( B u ) z : z ∈ R n , k z k = 1 } ≥ x t B ( K u ) x. (9.2) By 9.10 , x t B ( K w ( a, b, ε ) x = x t B ( K w ) x + 2( x 1 − x 2 )( ε · x S ) = β ( K w ) + 2( x 1 − x 2 )( ε · x S ) . (9.3) By ( h 4), K u = K w ( v 1 , v 2 , ε ) is isomorphic to K w ( v 2 , v 1 , ε 0 ), where ε 0 = r 1 − r 2 + ε . Since b y ( h 3), r 1 ≥ r 2 ≥ ε ≥ 0, w e hav e ε 0 ≥ 0. Therefore w e can assume that x 1 ≥ x 2 . By ( h 3), u and ε are non-negative functions. Since x is p ositiv e, ob viously x | S is also p ositiv e. Therefore ( x 1 − x 2 )( ε · x S ) ≥ 0. Now from (9.1), (9.2), and (9.3) we hav e β ( K u ) ≥ β ( K w ).  67 Notice that if w is a function from E ( K ) to { 0 , 1 } , then K w is an ordinary graph G , an ( x, y , ε )-op eration on K w is a K xy -op eration on G , and ( v 1 , v 2 , ε )-op eration is symmetric on K w if and only if the K xy -op eration on G is symmetric. The ab o v e results on a K xy -op eration for a graph G can b e generalized to a symmetric ( x, y , ε )-op eration for a weigh ted graph K w . Let S p ( K w ) = ( V ( K ) , sup ( w )). It turns out that 9.11 is also true for λ ( K w ) of a bipartite w eigh ted graph K w . Let L ( K w ) = R ( K w ) − A ( K w ) and L + ( K w ) = R ( K w ) + A ( K w ). Accordingly , let L ( λ, K w ) = det ( λI − L ( K w )) and L + ( λ, K w ) = det ( λI − L + ( K w )). Let n b e the num b er of v ertices of K w . Let bip ( K w ) = n + 1 − s , where s − 1 is the degree of the p olynomial L ( λ, K w ) − L + ( λ, K w ), and so L ( λ, K w ) = L + ( λ, K w ) if and only if bip ( K w ) = n + 1. Let odc ( K w ) denote the length of a shortest o dd cycle in S p ( K w ) if any exists, and put odc ( K w ) = n + 1, otherwise, and so K w is bipartite if and only if odc ( K w ) = n + 1. Thus, K w is bipartite if and only if bip ( K w ) = odc ( K w ) = n + 1. Let λ + ( K w ) denote the maximum eigenv alue of L + ( K w ) and, as b efore, λ (( K w ) the maxim um eigen v alue of L ( K w ). It turns out that the follow ing holds. 9.12 [19] L et K w b e a weighte d gr aph. Then bip ( K w ) = odc ( K w ) , and so K w is bip artite if and ony if L ( λ, K w ) = L + ( λ, K w ) . In p articular, if K w is bip artite, then λ (( K w ) = λ + ( K w ) . F rom 9.11 and 9.12 , applied to the non-negativ e matrix L + ( K w ), we hav e: 9.13 [19] L et K u and K w b e weighte d gr aphs. Supp ose that K w is obtaine d fr om K u by a symmetric ( v 1 , v 2 , ε ) -op er ation and that K w is bip artite. Then λ ( K w ) ≥ λ ( K u ) . No w using 9.13 , we can obtain the following generalization of 6.41 for a symmetric K xy -op eration. 9.14 [19] L et G ∈ G and G 0 b e obtaine d fr om G by a symmetric K xy -op er ation. If G 0 is a bip artite gr aph, then G  φ G 0 . The notion of a vertex comparable (or threshold) graph can also b e naturally gen- eralized to w eigh ted graphs and digraphs. Let, as ab o v e, K w b e a weigh ted graph with the v ertex set V , and so w is a function from  V 2  to R . F or x, y ∈ V , x 6 = y , w e write x E y if w ( xv ) ≤ w ( y v ) for every v ∈ V \ { x, y } . W e call K w a vertex c omp ar able ( or thr eshold ) weighte d gr aph if for every t wo distinct v ertices x, y of K w either x E y or y E x . 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