Turan Graphs, Stability Number, and Fibonacci Index

T ur´ an Graphs, Stabilit y Num ber, and Fib onacci Index V ´ eronique Bruy ` ere ∗ Hadrien M ´ elot ∗ , † 22 F ebruary , 2008 Abstract. The Fibo nacci index of a graph is the n um ber of its stable sets. T his para meter is widely studied and has a pplications in chemical gr aph theory . In this pap er , we establish tight upp er b ounds for the Fib onacci index in ter ms of the stability n um ber and the o rder of gener al graphs a nd co nnected gra phs. T ur´ an gr aphs freq ue ntly app ear in ex tremal gra ph theory . W e show that T ur ´ an gr aphs and a c o nnected v ar iant of them are also extr e ma l for these particular problems. Keywor ds: Stable sets; Fib onacci index ; Mer rifield-Simmons index ; T ur´ an g raph; α -critica l gra ph. 1 In tro duction The Fib onacci ind ex F ( G ) of a graph G w as introdu ced in 1982 by Pro din ger and Tic hy [20] as the num b er of stable sets in G . In 1989, Merrifi eld an d S immons [16] int ro duced indep endently this parameter in the chemistry literature 1 . They sho w ed th at there exist correlations b et w een the b oiling p oint and the Fib onacci index of a molecular graph . Since, the Fib onacci index has b een widely studied, esp ecially durin g the last few years. The ma jorit y of these recen t resu lts app eared in c hemical graph theory [12, 1 3, 21, 23– 25] and in extremal graph theory [9, 11, 17– 19]. In this literature, sev eral results are b ounds for F ( G ) among graphs in particular classes. Lo wer and upp er b oun ds ins id e the classes of general graphs, connected graphs, and trees are well k n o w n (see Section 2). Several authors give a characte rization of trees with maxim um Fib onacci index inside the class T ( n, k ) of trees with order n and a fix ed parameter k . F or example, L i et al. [13] determin e su c h trees when k is the diameter; Heub erger an d W agner [9] when k is the maxim um degree; and W ang et al. [25] when k is the num b er of p end ing ve rtices. Un icyclic graphs are also inv estigated in similar w a ys [17, 18, 24]. The Fib onacci index and the s tability num b er of a grap h are b oth related to stable sets. Hence, it is n atur al to use the stabilit y n um b er as a p arameter to d etermin e b ounds for F ( G ). Let G ( n, α ) and C ( n, α ) b e the classes of – resp ectiv ely general and connected – graphs with order n an d stabilit y n um b er α . The lo wer b ound for th e Fib onacci index is kn o w n for graphs in these classes. Indeed, P edersen and V estergaard [18] giv e a simp le pro of to sho w that if G ∈ G ( n, α ) or G ∈ C ( n, α ), then F ( G ) ≥ 2 α + n − α . Equalit y o ccurs if and only if G is a complete split graph (see Section 2). In this article, we determine upp er b ounds for F ( G ) in th e classes G ( n, α ) an d C ( n, α ). In b oth cases, the b ound is tigh t for ev ery p ossible v alue of α and n and the extremal graphs are charac terized. A T ur´ an graph is the u nion of disj oin t balanced cliques. T ur´ an graph s fr equen tly app ear in extremal graph theory . F or example, the well-kno wn Theorem of T ur´ an [22] states that these graphs ha ve minim um size inside G ( n, α ). W e show in S ection 3 that T u r ´ an graph s hav e also maximum Fib onacci ∗ Department of Theoretical Computer Science, Universit ´ e de Mons-Hainaut, Avenue du Champ d e Mars 6, B-7000 Mons, Belgium. † Charg ´ e de R echerc hes F.R .S .-FNRS. Corresp onding aut h or. E-mail: hadrien.m elot@umh.a c.be . 1 The Fib onacci index is called the Fibonacci num b er by Prodinger and Tich y [20]. Merrifield and S immons introduced it as the σ -index [16], also k now n as the Merrifield-S immons index. 1 index inside G ( n , α ). Observ e that removing an edge in a graph str ictly increases its Fib onacci ind ex. Indeed, all existing s table sets remain and there is at least one m ore n ew stable set: the t wo v ertices inciden t to the deleted edge. Th erefore, w e migh t hav e the intuition that the upp er b ound for F ( G ) is a simple consequence of the Theorem of T ur ´ an. How ev er, we show that it is not tr u e (see Sections 2 and 5). Th e pr o of uses structural pr op erties of α -critical graphs. Graphs in C ( n, α ) whic h maximize F ( G ) are c haracterized in Section 4. W e call them T u r´ an - connected graph s since they are a connected v ariant of T u r´ an graphs. It is in teresting to note that these graphs again minimize the size inside C ( n, α ). Hence, our r esults lead to questions ab out the relations b et ween the Fib on acci index, the stabilit y n um b er, the size and the order of graphs. These questions are su mmarized in Section 5. 2 Basic p rop erties In this section, we sup p ose that the reader is familiar with u s ual notions of graph theory (w e refer to Berge [1] for more details). First, we fix our terminology and notation. W e then recall the notion of α -critical graphs and giv e p r op erties of such graph s, used in the next sections. W e end w ith s ome basic prop erties of the Fib onacci ind ex of a graph. 2.1 Notations Let G = ( V , E ) b e a simple and undirected graph order n ( G ) = | V | and size m ( G ) = | E | . F or a v ertex v ∈ V ( G ), we denote b y N ( v ) the neigh b orho o d of v ; its closed neighborh o o d is defin ed as N ( v ) = N ( v ) ∪ { v } . The degree of a ve rtex v is denoted by d ( v ) and th e maxim um degree of G by ∆( G ). W e use notation G ≃ H when G and H are isomorph ic graphs. T he complement of G is denoted b y G . The stability numb er α ( G ) of a graph G is the num b er of vertice s of a maximum stable set of G . Clearly , 1 ≤ α ( G ) ≤ n ( G ), and 1 ≤ α ( G ) ≤ n ( G ) − 1 when G is connected. Definition 1. W e denote b y G v the induced su bgraph ob tained by removing a v ertex v from a grap h G . Similarly , the graph G N ( v ) is the ind uced sub graph obtained by remo ving the closed n eigh b orho o d of v . Finally , the graph obtained by r emo vin g an edge e from G is den oted by G e . Classical graphs of order n are used in this article: the complete graph K n , the path P n , the cycle C n , the star S n (comp osed b y one ve rtex adjacen t to n − 1 v ertices of d egree 1) and the complete split graph CS n,α (comp osed of a stable set of α vertice s, a clique of n − α v ertices and eac h v ertex of the stable set is adjacen t to eac h vertex of the clique). The complete sp lit graph CS 7 , 3 is depicted in Figure 1. W e also deeply study the t w o classes of T ur´ an graphs and T ur ´ an-connected graphs. A T ur´ an gr aph T n,α is a graph of order n and a stabilit y n um b er α suc h th at 1 ≤ α ≤ n , that is d efi ned as follo ws. It is the union of α disjoint balanced cliques (that is, suc h th at their ord er s differ from at most one) [22]. These cliques ha v e thus ⌈ n α ⌉ or ⌊ n α ⌋ ve rtices. W e no w defi n e a T ur´ an-c onne cte d g r aph TC n,α with n v ertices and a stabilit y num b er α w here 1 ≤ α ≤ n − 1. It is constructed from the T u r´ an graph T n,α with α − 1 additional edges. Let v b e a vertex of one clique of size ⌈ n α ⌉ , the add itional edges link v and one vertex of eac h remaining cliques. Note that, f or eac h of the t w o classes of graphs defined ab o v e, there is only one graph with giv en v alues of n and α , up to isomorphism . Example 1. Figure 1 sho ws the T ur ´ an graph T 7 , 3 and the T u r´ an-connected graph TC 7 , 3 . When α = 1, w e obs erv e that T n, 1 ≃ TC n, 1 ≃ CS n, 1 ≃ K n . When α = n , we hav e T n,n ≃ CS n,n ≃ K n , and w hen α = n − 1, we hav e TC n,n − 1 ≃ CS n,n − 1 ≃ S n . 2 Figure 1: T he graph s CS 7 , 3 , T 7 , 3 and TC 7 , 3 2.2 α -critical graphs W e recall the notion of α -critical graph s [6, 10, 14]. A n edge e of a graph G is α -critic al if α ( G e ) > α ( G ), otherwise it is called α -safe . A graph is said to b e α -critic al if all its edges are α -critical . By con ven tion, a graph with no edge is also α -critical. Th ese graphs play an imp ortant role in extremal graph theory [10], and also in ou r p ro ofs. Example 2. Simp le examples of α -critical graph s are complete graphs and o dd cycles. T ur´ an graph s are also α -critical. On the con trary , T u r´ an-connected graph are not α -critical, except when α = 1. W e s tate some interesting prop erties of α -critical graph s. Lemma 1. L et G b e an α -critic al gr aph. If G is c onne cte d, then the g r aph G v is c onne cte d for al l vertic es v of G . Pr o of. W e u se tw o kno wn results on α -critical graphs (see, e.g., [14, Chapter 12]). If a ve rtex v of an α -critical graph has d egree 1, then v and its neigh b or w form a connected comp onen t of the graph . Ev ery ve rtex of degree at least 2 in an α -critical graph is con tained in a cycle. Hence, by the first result, the min im u m degree of G equals 2, except if G ≃ K 2 . Clearly G v is connected by the second result or when G ≃ K 2 . Lemma 2. L et G b e an α -critic al gr aph. L et v b e any vertex of G which is not isolate d. Then, α ( G ) = α ( G v ) = α ( G N ( v ) ) + 1 . Pr o of. Let e = v w b e an edge of G con taining v . Then , there exist in G t w o maxim um stable sets S and S ′ , such that S cont ains v , but not w , and S ′ con tains w , bu t not v (see, e.g., [14, Chapter 12]). Th us, α ( G ) = α ( G v ) d u e to th e existence of S ′ . The set S av oids eac h ve rtex of N ( v ). Hence, S \ { v } is a stable set of the graph G N ( v ) of size α ( G ) − 1. It is easy to c hec k that this stable set is maximum. 2.3 Fib onacci index Let us now recall the Fib onacci ind ex of a graph [16, 20]. Th e Fib onac ci index F ( G ) of a graph G is the num b er of all the stable sets in G , including the empt y set. The follo wing lemma ab out F ( G ) is w ell-kno wn (see [8, 13, 20]). It is used in tensiv ely through the article. Lemma 3. L et G b e a gr aph. • L et e b e an e dge of G , then F ( G ) < F ( G e ) . • L et v b e a vertex of G , then F ( G ) = F ( G v ) + F ( G N ( v ) ) . • If G is the union of k disjoint gr aphs G i , 1 ≤ i ≤ k , then F ( G ) = Q k i =1 F ( G i ) . Example 3. W e hav e F ( K n ) = n + 1, F ( K n ) = 2 n , F ( S n ) = 2 n − 1 + 1 and F ( P n ) = f n +2 (recall th at the sequence of Fib onacci num b ers f n is f 0 = 0 , f 1 = 1 and f n = f n − 1 + f n − 2 for n > 1). 3 Pro dinger and Tic h y [20] giv e simple lo w er and up p er b oun ds for the Fib onacci index. W e recall these b ound s in the next lemma. Lemma 4. L et G b e a gr aph of or der n . • Then n + 1 ≤ F ( G ) ≤ 2 n with e quality if and only if G ≃ K n (lower b ound) and G ≃ K n (upp er b ound). • If G is c onne cte d, then n + 1 ≤ F ( G ) ≤ 2 n − 1 + 1 with e quality if and only if G ≃ K n (lower b ound) and G ≃ S n (upp er b ound). • If G is a tr e e, then f n +2 ≤ F ( G ) ≤ 2 n − 1 + 1 with e qu ality if and only if G ≃ P n (lower b ound) and G ≃ S n (upp er b ound). W e d enote by G ( n, α ) the class of general graph s with order n and stabilit y num b er α ; and b y C ( n, α ) the class of connected graphs with order n and s tabilit y num b er α . P ed er s en and V estergaard [18] c h aracterize graphs with minimum Fib on acci index as indicated in the follo wing theorem. Theorem 5. L e t G b e a gr aph inside G ( n, α ) or C ( n, α ) , then F ( G ) ≥ 2 α + n − α, with e quality if and only if G ≃ CS n,α . The aim of th is article is the stu d y of graphs with maxim um Fib onacci index inside the t w o classes G ( n, α ) and C ( n, α ). The s ystem GraPHedron [15] allo ws a formal framewo rk to conjecture optimal relations among a set of grap h inv ariants. Thanks to this system, graphs with maxim u m Fib onacci index inside eac h of the tw o previous classes hav e b een computed for small v alues of n [7]. W e ob s erv e that these graphs are isomorphic to T ur´ an graph s for the class G ( n, α ), and to T u r´ an -connected graphs for the class C ( n, α ). F or the class C ( n, α ), there is one exception when n = 5 and α = 2: b oth the cycle C 5 and the graph TC 5 , 2 ha v e maxim u m Fib onacci ind ex. Recall that the classical Theorem of T u r´ an [22] states that T u r ´ an graph s T n,α ha v e min im um size inside G ( n, α ). W e migh t think that T ur´ an graph s ha v e maxim u m Fib onacci ind ex inside G ( n, α ) as a direct corollary of the Theorem of T ur´ an and Lemma 3. Th is argument is n ot correct since remo vin g an α -critical edge in creases the stability num b er. Therefore, Lemma 3 only implies that graphs with maximum Fib onacci index inside G ( n, α ) are α -critical graphs. In Section 5, w e make further observ ations on the relations b et w een the size and the Fib onacci index in side the classes G ( n, α ) and C ( n, α ). There is another interesting pr op ert y of T ur´ an graphs related to stable sets. Bysko v [4] establish that T u r´ an graphs ha ve maximum num b er of maximal stable sets insid e G ( n, α ). The Fib onacci index coun ts not only the maximal stable sets but all the stable sets. Hence, the f act that T ur´ an graphs maximize F ( G ) cannot b e simply derived fr om the result of Bysko v. 3 General graphs In this section, we stud y graphs with maxim um Fib onacci index inside the class G ( n, α ). These graphs are said to b e extr emal . F or fi xed v alues of n and α , w e sho w th at there is one extremal graph u p to isomorphism, the T ur´ an graph T n,α (see Theorem 8). Before establishing this result, we need some auxiliary r esults. W e denote by f T ( n, α ) the Fib onacci index of the T u r´ an graph T n,α . By Lemma 3, its v alue is equal to f T ( n, α ) = l n α m + 1  p j n α k + 1  α − p , where p = ( n mo d α ). W e ha v e also the follo win g indu ctive formula. 4 Lemma 6. L et n and α b e inte gers such that 1 ≤ α ≤ n . Then f T ( n, α ) =    n + 1 if α = 1 , 2 n if α = n, f T ( n − 1 , α ) + f T ( n −  n α  , α − 1) if 2 ≤ α ≤ n − 1 . Pr o of. The cases α = 1 and α = n are trivial (see Example 3). Supp ose 2 ≤ α ≤ n − 1. Let v b e a v ertex of T n,α with maximum degree. Thus v is in a  n α  -clique. As α < n , the v ertex v is n ot isolated. Therefore T v n,α ≃ T n − 1 ,α . As α ≥ 2, the graph T N ( v ) n,α has at least one v ertex, and T N ( v ) n,α ≃ T n − ⌈ n α ⌉ ,α − 1 . By Lemma 3, we obtain f T ( n, α ) = f T ( n − 1 , α ) + f T ( n − l n α m , α − 1) . A consequence of Lemma 6 is that f T ( n − 1 , α ) < f T ( n, α ). In deed, the cases α = 1 and α = n are trivial, and the term f T ( n −  n α  , α − 1) is alw ays strictly p ositiv e wh en 2 ≤ α ≤ n − 1. Corollary 7. The function f T ( n, α ) is strictly incr e asing in n when α is fixe d. W e n o w state the up p er b ound on the Fib on acci index of grap h s in the class G ( n, α ). Theorem 8. L e t G b e a gr aph of or der n with a stability numb er α , then F ( G ) ≤ f T ( n, α ) , with e quality if and only if G ≃ T n,α . Pr o of. The cases α = 1 and α = n are straigh tforw ard . In deed G ≃ T n, 1 when α = 1, and G ≃ T n,n when α = n . W e can assume that 2 ≤ α ≤ n − 1, and th us n ≥ 3. W e n o w p ro v e by induction on n that if G is extremal, then it is isomorph ic to T n,α . The graph G is α -critical. Otherwise, there exists an ed ge e ∈ E ( G ) suc h that α ( G ) = α ( G e ), and b y Lemma 3, F ( G ) < F ( G e ). This is a con tr ad iction with G b eing extremal. Let u s compute F ( G ) thanks to Lemm a 3. Let v ∈ V ( G ) of m axim u m degree ∆. The vertex v is not isolated since α < n . Thus b y Lemma 2, α ( G v ) = α and α ( G N ( v ) ) = α − 1. On the other h and, If χ is the c hromatic n um b er of G , it is well-kno wn that n ≤ χ . α (see, e.g., Berge [1]), and that χ ≤ ∆ + 1 (see Bro oks [3]). It follo ws that n ( G N ( v ) ) = n − ∆ − 1 ≤ n − l n α m . (1) Note that n ( G N ( v ) ) ≥ 1 sin ce α ≥ 2. W e can app ly the in duction hyp othesis on the graphs G v and G N ( v ) . W e obtain f T ( n, α ) ≤ F ( G ) as G is extremal, = F ( G v ) + F ( G N ( v ) ) , b y Lemma 3, ≤ f T ( n ( G v ) , α ( G v )) + f T ( n ( G N ( v ) ) , α ( G N ( v ) )) , b y indu ction, = f T ( n − 1 , α ) + f T ( n − ∆ − 1 , α − 1) , ≤ f T ( n − 1 , α ) + f T ( n −  n α  , α − 1) , b y Eq. (1) and Corollary 7, = f T ( n, α ) b y Lemma 6. Hence equality h olds eve rywhere. In particular, by induction, the graphs G v , G N ( v ) are extremal, and G v ≃ T n − 1 ,α , G N ( v ) ≃ T n − ⌈ n α ⌉ ,α − 1 . Coming bac k to G fr om G v and G N ( v ) and recalling that v has maxim u m d egree, it follo ws that G ≃ T n,α . 5 Corollary 7 states that f T ( n, α ) is in cr easing in n . It was an easy consequence of Lemma 6. T h e function f T ( n, α ) is also increasing in α . Theorem 8 can b e used to pr o ve this fact easily as shown now. Corollary 9. The function f T ( n, α ) is strictly incr e asing in α when n is fixe d. Pr o of. Supp ose 2 ≤ α ≤ n − 1. By Lemma 4 it is clear that f T ( n, 1) < f T ( n, α ) < f T ( n, n ). No w, let e b e an edge of T n,α . Clearly α ( T e n,α ) = α + 1. Moreo ver, by Lemma 3 and Theorem 8, F ( T n,α ) < F ( T e n,α ) < F ( T n,α +1 ) . Therefore, f T ( n, α ) < f T ( n, α + 1). 4 Connected graphs W e no w consider graphs with maxim u m Fib onacci index insid e the class C ( n, α ). Su c h graphs are called extr emal . I f G is connected, the b ound of Th eorem 8 is clearly not tigh t, except when α = 1, that is, when G is a complete graph. W e are going to pro v e that there is one extremal graph up to isomorphism, the T ur´ an-connected graph TC n,α , with the exception of the cycle C 5 (see Theorem 12). First, w e need pr eliminary results and d efinitions to p ro ve this theorem. W e denote b y f TC ( n, α ) the Fib onacci index of th e T u r´ an -connected graph TC n,α . An in ductiv e form ula for its v alue is give n in the next lemma. Lemma 10. L et n and α b e inte gers such that 1 ≤ α ≤ n − 1 . Then f TC ( n, α ) =    n + 1 if α = 1 , 2 n − 1 + 1 if α = n − 1 , f T ( n − 1 , α ) + f T ( n ′ , α ′ ) if 2 ≤ α ≤ n − 2 , wher e n ′ = n −  n α  − α + 1 and α ′ = min( n ′ , α − 1) . Pr o of. The cases α = 1 and α = n − 1 are trivial by Lemma 4. Supp ose no w that 2 ≤ α ≤ n − 2. Let v b e a vertex of maximum degree in TC n,α . W e app ly Lemma 3 to compute F ( TC n,α ). Ob serv e that the graphs TC v n,α and TC N ( v ) n,α are b oth T ur´ an graph s w hen 2 ≤ α ≤ n − 2. The grap h TC v n,α is isomorphic to T n − 1 ,α . Let us sh o w that TC N ( v ) n,α is isomorphic to T n ′ ,α ′ . By definition of a T u r ´ an-connected graph, d ( v ) is equal to  n α  + α − 2. Thus n ( TC N ( v ) n,α ) = n − d ( v ) − 1 = n ′ . If α < n 2 , then TC n,α has a clique of order at least 3 and α ( TC N ( v ) n,α ) = α − 1 ≤ n ′ . Otherwise, TC N ( v ) n,α ≃ K n ′ and α ( TC N ( v ) n,α ) = n ′ ≤ α − 1. Therefore α ( TC N ( v ) n,α ) = m in ( n ′ , α − 1) in b oth cases. By Lemma 3, these observ ations leads to f TC ( n, α ) = f T ( n − 1 , α ) + f T ( n ′ , α ′ ) . Definition 2. A bridge in a connected graph G is an edge e ∈ E ( G ) suc h that the graph G e is n o more connected. T o a br idge e = v 1 v 2 of G whic h is α -safe, we asso ciate a de c omp osition D ( G 1 , v 1 , G 2 , v 2 ) such that v 1 ∈ V ( G 1 ), v 2 ∈ V ( G 2 ), and G 1 , G 2 are the t w o connected comp onents of G e . A decomp osition is said to b e α - critic al if G 1 is α -critical. Lemma 11. L et G b e a c onne cte d g r aph. If G is extr emal, then e ither G is α -critic al or G has an α -critic al de c omp osition. 6 Pr o of. W e supp ose th at G is not α -critical and we sh ow that it must con tain an α -critical d ecomp osition. Let e b e an α -safe edge of G . Then e m ust b e a bridge. Otherwise, th e graph G e is connected, has the same order and stabilit y num b er as G and satisfies F ( G e ) > F ( G ) b y Lemma 3. This is a cont radiction w ith G b eing extremal. Therefore G con tains at least one α -safe br idge defi ning a decomp osition of G . Let us c ho ose a decomp osition D ( G 1 , v 1 , G 2 , v 2 ) suc h that G 1 is of minim um order. T hen, G 1 is α -critical. Otherwise, G 1 con tains an α -safe bridge e ′ = w 1 w 2 , since the edges of G are α -critical or α -safe bridges by the fi r st p art of the pro of. Let D ( H 1 , w 1 , H 2 , w 2 ) b e the decomp osition of G defined b y e ′ , such th at v 1 ∈ V ( H 2 ). Then n ( H 1 ) < n ( G 1 ), w hic h is a contradictio n. Hence the d ecomp osition D ( G 1 , v 1 , G 2 , v 2 ) is α -critical. Theorem 12. L et G b e a c onne cte d gr aph of or der n with a stability numb er α , then F ( G ) ≤ f TC ( n, α ) , with e quality if and only if G ≃ T C n,α when ( n, α ) 6 = (5 , 2) , and G ≃ TC 5 , 2 or G ≃ C 5 when ( n, α ) = (5 , 2) . Pr o of. W e pro v e by in duction on n that if G is extremal, then it is isomorphic to TC n,α or C 5 . T o handle more easily the general case of the in duction (in a w a y to a v oid the extremal graph C 5 ), we consider all connected graphs with u p to 6 v ertices as the basis of the indu ction. F or these b asic cases, w e r efer to the r ep ort of an exhaus tiv e automated verificatio n [7]. W e thus sup p ose that n ≥ 7. W e know by Lemma 11 that either G has an α -critical decomp osition or G is α -critical. W e consider no w these t w o situations. 1) G has an α -critical decomp osition. W e p ro v e in three steps that G ≃ TC n,α : ( i ) W e establish that for ev ery decomp osition D ( G 1 , v 1 , G 2 , v 2 ), the graph G i is extremal and is isomorp hic to a T u r´ an - connected graph su c h that d ( v i ) = ∆( G i ), for i = 1 , 2. ( ii ) W e show that if such a decomp osition is α -critical, then G 1 is a clique. ( iii ) W e pro v e that G is itself isomorphic to a T ur´ an-connected graph. ( i ) F or the first step, let D ( G 1 , v 1 , G 2 , v 2 ) b e a decomp osition of G , n 1 b e the order of G 1 , and α 1 its stabilit y num b er. W e p ro ve that G 1 ≃ TC n 1 ,α 1 suc h that d ( v 1 ) = ∆ ( G 1 ). The argumen t is iden tical for G 2 . By Lemma 3, we h a ve F ( G ) = F ( G 1 ) F ( G v 2 2 ) + F ( G v 1 1 ) F ( G N ( v 2 ) 2 ) . By the ind uction h yp othesis, F ( G 1 ) ≤ f TC ( n 1 , α 1 ). The graph G v 1 1 has an order n 1 − 1 and a stabilit y n um b er ≤ α 1 . Hence by Theorem 8 and Corollary 9, F ( G v 1 1 ) ≤ f T ( n 1 − 1 , α 1 ). It follo ws that F ( G ) ≤ f TC ( n 1 , α 1 ) F ( G v 2 2 ) + f T ( n 1 − 1 , α 1 ) F ( G N ( v 2 ) 2 ) . (2) As G is su p p osed to b e extremal, equalit y o ccurs. It means that G v 1 1 ≃ T n 1 − 1 ,α 1 and G 1 is extremal. If G 1 is isomorphic to C 5 , then n 1 = 5, α 1 = 2 and F ( G 1 ) = f TC (5 , 2). Ho we v er , F ( G v 1 1 ) = F ( P 4 ) < f T (4 , 2). By (2), this leads to a contradictio n with G b eing extremal. Thus, G 1 m ust b e isomorphic to TC n 1 ,α 1 . Moreo v er, v 1 is a vertex of maximum degree of G 1 . Oth erwise, G v 1 1 cannot b e isomorphic to the graph T n 1 − 1 ,α 1 . ( ii ) Th e second step is easy . Let D ( G 1 , v 1 , G 2 , v 2 ) b e an α -critical decomp osition of G , that is, G 1 is α -critical. By ( i ), G 1 is isomorphic to a T ur ´ an-connected graph . The complete graph is the only T ur´ an-connected graph which is α -critical. Therefore, G 1 is a clique. ( iii ) W e n o w su pp ose that G has an α -critical d ecomp osition D ( G 1 , v 1 , G 2 , v 2 ) and w e sho w that G ≃ TC n,α . Let n 1 b e the order of G 1 and α 1 its stabilit y num b er. As v 1 v 2 is an α -safe b ridge, it is 7 clear that n ( G 2 ) = n − n 1 and α ( G 2 ) = α − α 1 . By ( i ) and ( ii ), G 1 is a clique (and th us α 1 = 1), G 2 ≃ TC n − n 1 ,α − 1 , and v 2 is a verte x of maxim um degree in G 2 . If α = 2, then G 2 is also a clique in G . By Lemma 3 and the fact that F ( K n ) = n + 1 we ha ve, F ( G ) = F ( G v 1 ) + F ( G N ( v 1 ) ) , = n 1 ( n − n 1 + 1) + ( n − n 1 ) = n + n n 1 − n 2 1 . When n is fixed, this fun ction is maximized when n 1 = n 2 . That is, wh en G 1 and G 2 are balanced cliques. This app ears if and only if G ≃ TC n, 2 . Th us we supp ose that α ≥ 3. In other words, G con tains at least three cliques: th e clique G 1 of order n 1 ; th e clique H con taining v 2 and a clique H ′ in G 2 link ed to H by an α -safe b ridge v 2 v 3 . Let k = n − n 1 α − 1 , then the order of H is ⌈ k ⌉ and the order of H ′ is ⌈ k ⌉ or ⌊ k ⌋ (recall that G 2 ≃ TC n − n 1 ,α − 1 ). These cliques are rep resen ted in Figure 2. v 1 G 1 v 2 H v 3 H ′ Figure 2: C liques in the graph G T o prov e th at G is isomorphic to a T ur´ an-connected graph, it remains to sho w that the clique G 1 is balanced with the cliques H and H ′ . W e consider the decomp osition defined by the α -safe bridge v 2 v 3 . By ( i ), G 1 and H are cliques of a T u r´ an-connected graph, and H is a clique with maximum order in this graph (recall that v 2 is a v ertex of maxim um d egree in G 2 ). Th erefore ⌈ k ⌉ − 1 ≤ n 1 ≤ ⌈ k ⌉ , sho wing that G 1 is balanced with H and H ′ . 2) G is α -critical. Under this hyp othesis, w e pr o ve th at G is a complete graph , and th u s is isomorphic to a T ur´ an-connected graph . Supp ose that G is not complete. Let v b e a v ertex of G w ith a maximum degree d ( v ) = ∆. As G is connected and α -critical, the graph G v is connected by Lemma 1. By L emma 2, α ( G v ) = α and α ( G N ( v ) ) = α − 1. Moreo ver, n ( G v ) = n − 1 and n ( G N ( v ) ) = n − ∆ − 1. By th e induction hyp othesis and Theorem 8, we get F ( G ) = F ( G v ) + F ( G N ( v ) ) ≤ f TC ( n − 1 , α ) + f T ( n − ∆ − 1 , α − 1) . Therefore, G is extremal if and only if G N ( v ) ≃ T n − ∆ − 1 ,α − 1 and G v is extremal. Ho we v er, G v is not isomorphic to C 5 as n ≥ 7. Thus G v ≃ TC n − 1 ,α . So, the graph G is comp osed by the graph G v ≃ TC n − 1 ,α and an additional v ertex v connected to TC n − 1 ,α b y ∆ edges. There m ust b e an edge b et w een v and a vertex v ′ of maxim um degree in G v , otherwise G N ( v ) is not isomorp hic to a T ur´ an graph. Th e vertex v ′ is adjacen t to  n − 1 α  + α − 2 ve rtices in G v and it is adjacen t to v , that is, d ( v ′ ) =  n − 1 α  + α − 1 . 8 It follo ws that ∆ ≥ d ( v ′ ) >  n − 1 α  (3) as G is not a complete graph. On the other hand, v is adjacen t to eac h v ertex of some clique H of G v since G N ( v ) has a stabilit y n um b er α − 1. As this clique h as order at most  n − 1 α  , v m ust b e adjacent to a vertex w / ∈ H by (3). W e observe that the edge v w is α -safe. This is imp ossible as G is α -critical. It follo ws that G is a complete graph and the pro of is completed. The study of the maxim u m Fib onacci index inside the class T ( n, α ) of trees with order n and stabilit y num b er α is strongly related to the s tudy done in this section for the class C ( n, α ). Indeed, due to the fact th at trees are bipartite, a tree in T ( n, α ) has alwa ys a stabilit y num b er α ≥ n 2 . Moreo v er, the T u r´ an -connected graph TC n,α is a tree wh en α ≥ n 2 . Th erefore, the upp er b ound on the Fib onacci index for connected graphs is also v alid for trees. W e thus get th e next corollary with in addition th e exact v alue of f TC ( n, α ). Corollary 13. L et G b e a tr e e of or der n with a stability numb er α , then F ( G ) ≤ 3 n − α − 1 2 2 α − n +1 + 2 n − α − 1 , with e quality if and only if G ≃ TC n,α . Pr o of. It r emains to compute the exact v alue of f TC ( n, α ). When α ≥ n 2 , the graph TC n,α is comp osed b y one cent ral vertex v of d egree α and α p end ing paths of length 1 or 2 attac hed to v . An extremit y of a p end ing path of length 2 is a v ertex w such that w / ∈ N ( v ). T h us there are x = n − α − 1 p end in g paths of length 2 since N ( v ) has size α + 1, and there are y = α − x = 2 α − n + 1 p endin g p aths of length 1. W e app ly Lemma 3 on v to get f TC ( n, α ) = F ( K 2 ) x F ( K 1 ) y + F ( K 1 ) x = 3 x 2 y + 2 x . 5 Observ ations T ur´ an graphs T n,α ha v e minimum size inside G ( n, α ) by the T heorem of T ur´ an [22]. Ch r istophe et al. [5] giv e a tigh t lo wer b oun d for the connected case of this theorem, and Bougard and J oret [2] c h aracterized th e extremal graph s, wh ic h happ en to con tain the TC n,α graphs as a sub class. By these r esults and Theorems 8 and 12, we can observ e the follo wing relations b etw een graphs with minim um size and maximum Fib onacci index. Th e graphs inside G ( n, α ) m inimizing m ( G ) are exactly those wh ic h maximize F ( G ). This is also true f or the graph s inside C ( n , α ), except that there exist other graphs with minimum size than the T ur´ an-connected graphs. Ho wev er, these observ ations are not a trivial consequence of the fact that F ( G ) < F ( G e ) w here e is any edge of a graph G . As indicated in our pr o ofs, the latter prop erty only implies th at a graph maximizing F ( G ) con tains only α -critical edges (and α -safe bridges for th e connected case). Our pro ofs use a d eep study of the str u cture of th e extremal graphs to obtain T heorems 8 and 12. W e no w giv e additional examples sho wing that the intuitio n that more edges imply fewer stable sets is w rong. Pedersen and V estergaard [18] giv e the follo wing example. Let r b e an integ er su ch th at r ≥ 3, G 1 b e the T u r´ an graph T 2 r,r and G 2 b e the star S 2 r . The graphs G 1 and G 2 ha v e the same order but G 1 has less edges ( r ) than G 2 (2 r − 1). Neverthele ss, obs erv e that F ( G 1 ) = 3 r < F ( G 2 ) = 2 2 r − 1 + 1. This example do es not tak e in to account the stabilit y num b er since α ( G 1 ) = r and α ( G 2 ) = 2 r − 1. W e prop ose a similar example of p airs of graph s with the same order and the same stabilit y num b er (see the graph s G 3 and G 4 on Figure 3). These tw o graphs are inside the class G (6 , 4), how ev er 9 G 3 : G 4 : Figure 3: Gr aphs with s ame order and stabilit y num b er m ( G 3 ) < m ( G 4 ) and F ( G 3 ) < F ( G 4 ). Noti ce that we can get such examples inside G ( n, α ) with n arbitrarily large, by considering the union of s ev eral disjoin t copies of G 3 and G 4 . These remarks and our results suggest some qu estions ab out the relations b et ween th e size, the stabilit y num b er and the Fib onacci index of graph s . What are the lo wer and u pp er b ounds for the Fib onacci index inside the class G ( n, m ) of graphs order n and size m ; or inside the class G ( n, m, α ) of graph s order n , s ize m and stabilit y num b er α ? Are there classes of graph s for whic h more edges alw ays im p ly fewer stable sets? 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