Fibonacci Index and Stability Number of Graphs: a Polyhedral Study

Fib onacci Index and Stabilit y Num b er of Graphs: a P olyhedral Study V ´ eronique Bruy ` ere ∗ Hadrien M ´ elot ∗ , † No vem ber 10, 2018 Abstract. The Fib onacci index of a graph is the num b er of its stable sets. This parameter is widely studied and has applications in c hemical graph theory . In this paper, we establish tight upp er b ounds for the Fib onacci index in terms of the stability num ber and the order of general graphs and connected graphs. T ur´ an graphs frequen tly app ear in extremal graph theory . W e show that T ur´ an graphs and a connected v ariant of them are also extremal for these particular problems. W e also make a p olyhedral study by establishing all the optimal linear inequalities for the stabilit y n umber and the Fibonacci index, inside the classes of general and connected graphs of order n . Keywor ds: Stable se t; Fibonacci index; Merrifield-Simmons index; T ur´ an graph; α -critical graph; GraPHedron. 1 In tro duction The Fib onacci index F ( G ) of a graph G w as in troduced in 1982 b y Pro dinger and Tic h y [21] as the num b er of stable sets in G . In 1989, Merrifield and Simmons [17] in tro duced inde- p enden tly this parameter in the c hemistry literature 1 . They show ed that there exist cor- relations b et ween the boiling point and the Fib onacci index of a molecular graph. Since, the Fib onacci index has b een widely studied, esp ecially during the last few years. The ma jority of these recent results app eared in chemical graph theory [13, 14, 22, 24–26] and in extremal graph theory [10, 12, 18–20]. In this literature, sev eral results are b ounds for F ( G ) among graphs in particular classes. Lo wer and upp er b ounds inside the classes of general graphs, connected graphs, and trees are well kno wn (see Section 2). Sev eral authors giv e a c haracterization of trees with max- im um Fib onacci index inside the class T ( n, k ) of trees with order n and a fixed parameter k . F or example, Li et al. [14] determine such trees when k is the diameter; Heuberger and ∗ Departmen t of Theoretical Computer Science, Univ ersit ´ e de Mons-Hainaut, Av en ue du Champ de Mars 6, B-7000 Mons, Belgium. † Charg ´ e de Rec herc hes F.R.S.-FNRS. Corresponding author. E-mail: hadrien.melot@umh.ac.be . 1 The Fib onacci index is called the Fibonacci num b er by Prodinger and Tic hy [21]. Merrifield and Simmons in tro duced it as the σ -index [17], also kno wn as the Merrifield-Simmons index. 1 W agner [10] when k is the maximum degree; and W ang et al. [26] when k is the n umber of p ending v ertices. Unicyclic graphs are also in vestigated in similar wa ys [18, 19, 25]. The Fib onacci index and the stability n umber of a graph are b oth related to stable sets. Hence, it is natural to use the stabilit y num b er as a parameter to determine bounds 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 and stabilit y n umber α . The lo wer b ound for the Fib onacci index is kno wn for graphs in these classes. Indeed, Pedersen and V estergaard [19] giv e a simple pro of to show 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, w e determine upp er bounds for F ( G ) in the classes G ( n, α ) and C ( n, α ). In b oth cases, the b ound is tigh t for every p ossible v alue of α and n and the extremal graphs are characterized. A T ur´ an graph is the union of disjoin t balanced cliques. T ur´ an graphs frequen tly app ear in extremal graph theory . F or example, the well-kno wn Theorem of T ur´ an [23] states that these graphs ha ve minimum size inside G ( n, α ). W e show in Section 3 that T ur´ an graphs ha ve also maxim um Fib onacci index inside G ( n, α ). Observe that remo ving an edge in a graph strictly increases its Fibonacci index. Indeed, all existing stable sets remain and there is at least one more new stable set: the t w o v ertices inciden t to the deleted edge. Therefore, w e might 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 true (see Sections 2 and 6). The pro of uses structural properties of α -critical graphs. Graphs in C ( n, α ) which maximize F ( G ) are c haracterized in Section 4. W e call them T ur´ an-connected graphs since they are a connected v arian t of T ur´ an graphs. It is interesting to note that these graphs again minimize the size inside C ( n, α ). Hence, our results lead to questions ab out the relations betw een the Fib onacci index, the stabilit y num ber, the size and the order of graphs. These questions are summarized in Section 6. In Section 5, w e further extend our results b y a p olyhedral study of the relations among the stability num b er and the Fib onacci index. Indeed, w e state all the optimal linear inequalities for the stability num b er and the Fib onacci index, inside the classes of general and connected graphs of order n . The ma jor part of the results of this article has b een published in Ref. [4]. 2 Basic prop erties In this section, we supp ose that the reader is familiar with usual 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 prop erties of suc h graphs, used in the next sections. W e end with some basic prop erties of the Fibonacci index of a graph. 2 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 neighborho o d of v ; its closed neighborho o d is defined as N ( v ) = N ( v ) ∪ { v } . The degree of a vertex v is denoted by d ( v ) and the maxim um degree of G b y ∆( G ). W e use notation G ' H when G and H are isomorphic graphs. The complement of G is denoted by G . The stability numb er α ( G ) of a graph G is the num b er of v ertices 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 by G v the induced subgraph obtained b y remo ving a v ertex v from a graph G . Similarly , the graph G N [ v ] is the induced subgraph obtained by removing the closed neigh b orhoo d of v . Finally , the graph obtained b y remo ving an edge e from G is denoted b y 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 by one vertex adjacent to n − 1 vertices of degree 1) and the complete split graph CS n,α (comp osed of a s table set of α vertices, a clique of n − α vertices and each v ertex of the stable set is adjacent to each vertex of the clique). The complete split 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 num b er α such that 1 ≤ α ≤ n , that is defined as follows. It is the union of α disjoint balanced cliques (that is, such that their orders differ from at most one) [23]. These cliques hav e thus d n α e or b n α c vertices. W e no w define a T ur´ an-c onne cte d gr aph TC n,α with n vertices and a stability num ber α where 1 ≤ α ≤ n − 1. It is constructed from the T ur´ an graph T n,α with α − 1 additional edges. Let v b e a v ertex of one clique of size d n α e , the additional edges link v and one vertex of eac h remaining cliques. Note that, for eac h of the t wo 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 shows the T ur´ an graph T 7 , 3 and the T ur´ an-connected graph TC 7 , 3 . When α = 1, we observe 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 when α = n − 1, w e hav e TC n,n − 1 ' CS n,n − 1 ' S n . Figure 1: The graphs CS 7 , 3 , T 7 , 3 and TC 7 , 3 3 2.2 α -critical graphs W e recall the notion of α -critical graphs [7, 11, 15]. An 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. These graphs pla y an imp ortan t role in extremal graph theory [11], and also in our pro ofs. Example 2. Simple examples of α -critical graphs are complete graphs and o dd cycles. T ur´ an graphs are also α -critical. On the contrary , T ur´ an-connected graph are not α - critical, except when α = 1. W e state some in teresting properties of α -critical graphs. Lemma 1. L et G b e an α -critic al gr aph. If G is c onne cte d, then the gr aph G v is c onne cte d for al l vertic es v of G . Pr o of. W e use t wo known results on α -critical graphs (see, e.g., [15, Chapter 12]). If a v ertex v of an α -critical graph has degree 1, then v and its neighbor w form a connected comp onen t of the graph. Every v ertex of degree at least 2 in an α -critical graph is con tained in a cycle. Hence, by the first result, the minimum degree of G equals 2, except if G ' K 2 . Clearly G v is connected b y 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 containing v . Then, there exist in G tw o maxim um stable sets S and S 0 , suc h that S con tains v , but not w , and S 0 con tains w , but not v (see, e.g., [15, Chapter 12]). Thus, α ( G ) = α ( G v ) due to the existence of S 0 . The set S av oids eac h v ertex 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 heck that this stable set is maxim um. 2.3 Fib onacci index Let us no w recall the Fib onacci index of a graph [17, 21]. The Fib onac ci index F ( G ) of a graph G is the n umber of all the stable sets in G , including the empty set. The follo wing lemma ab out F ( G ) is w ell-known (see [9, 14, 21]). It is used intensiv 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 ) . 4 Example 3. W e ha v 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 that the sequence of Fib onacci n umbers f n is f 0 = 0 , f 1 = 1 and f n = f n − 1 + f n − 2 for n > 1). Pro dinger and Tic h y [21] give simple lo wer and upper b ounds for the Fib onacci index. W e recall these b ounds 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 quality if and only if G ' P n (lower b ound) and G ' S n (upp er b ound). W e denote b y G ( n, α ) the class of general graphs with order n and stability num b er α ; and b y C ( n, α ) the class of connected graphs with order n and stabilit y n um b er α . Pedersen and V estergaard [19] characterize graphs with minimum Fib onacci index as indicated in the following theorem. Theorem 5. L et 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 this article is the study of graphs with maximum Fibonacci index inside the t wo classes G ( n, α ) and C ( n, α ). The system GraPHedron [16] allo ws a formal framework to conjecture optimal relations among a set of graph in v arian ts. Thanks to this system, graphs with maxim um Fib onacci index inside each of the t wo previous classes ha ve been computed for small v alues of n [8]. W e observ e that these graphs are isomorphic to T ur´ an graphs for the class G ( n, α ), and to T ur´ 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 ve maxim um Fibonacci index. Recall that the classical Theorem of T ur´ an [23] states that T ur´ an graphs T n,α ha ve minim um size inside G ( n, α ). W e migh t think that T ur´ an graphs hav e maximum Fibonacci index inside G ( n, α ) as a direct corollary of the Theorem of T ur´ an and Lemma 3. This argumen t is not correct since removing an α -critical edge increases the stability num ber. Therefore, Lemma 3 only implies that graphs with maxim um Fibonacci index inside G ( n, α ) are α -critical graphs. In Section 6, we mak e further observ ations on the relations b et ween the size and the Fib onacci index inside the classes G ( n, α ) and C ( n, α ). 5 There is another interesting prop ert y of T ur´ an graphs related to stable sets. Bysko v [5] establish that T ur´ an graphs ha v e maximum num b er of maximal stable sets inside G ( n, α ). The Fibonacci index coun ts not only the maximal stable sets but all the stable sets. Hence, the fact that T ur´ an graphs maximize F ( G ) cannot b e simply derived from the result of Bysk ov. 3 General graphs In this section, we study graphs with maxim um Fib onacci index inside the class G ( n, α ). These graphs are said to b e extr emal . F or fixed v alues of n and α , w e show that there is one extremal graph up to isomorphism, the T ur´ an graph T n,α (see Theorem 8). Before establishing this result, w e need some auxiliary results. W e denote b y f T ( n, α ) the Fib onacci index of the T ur´ 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 following inductive formula. 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 vertex of T n,α with maximum degree. Thus v is in a  n α  -clique. As α < n , the v ertex v is not isolated. Therefore T v n,α ' T n − 1 ,α . As α ≥ 2, the graph T N [ v ] n,α has at least one vertex, and T N [ v ] n,α ' T n − d n α e ,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 ). Indeed, the cases α = 1 and α = n are trivial, and the term f T ( n −  n α  , α − 1) is alwa ys strictly p ositiv e when 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 no w state the upp er b ound on the Fib onacci index of graphs in the class G ( n, α ). Theorem 8. L et 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 . 6 Pr o of. The cases α = 1 and α = n are straigh tforward. Indeed 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 thus n ≥ 3. W e now prov e b y induction on n that if G is extremal, then it is isomorphic to T n,α . The graph G is α -critical. Otherwise, there exists an edge e ∈ E ( G ) suc h that α ( G ) = α ( G e ), and b y Lemma 3, F ( G ) < F ( G e ). This is a contradiction with G b eing extremal. Let us compute F ( G ) thanks to Lemma 3. Let v ∈ V ( G ) of maxim um degree ∆. The v ertex v is not isolated since α < n . Thus by Lemma 2, α ( G v ) = α and α ( G N [ v ] ) = α − 1. On the other hand, If χ is the chromatic n umber of G , it is well-kno wn that n ≤ χ . α (see, e.g., Berge [1]), and that χ ≤ ∆ + 1 (see Brooks [3]). It follo ws that n ( G N [ v ] ) = n − ∆ − 1 ≤ n − l n α m . (1) Note that n ( G N [ v ] ) ≥ 1 since α ≥ 2. W e can apply the induction hypothesis 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 induction, = 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 equalit y holds everywhere. 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 − d n α e ,α − 1 . Coming bac k to G from G v and G N [ v ] and recalling that v has maxim um degree, it follo ws that G ' T n,α . Corollary 7 states that f T ( n, α ) is increasing in n . It w as an easy consequence of Lemma 6. The function f T ( n, α ) is also increasing in α . Theorem 8 can b e used to pro ve this fact easily as shown no w. 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. Moreov er, 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). 7 4 Connected graphs W e now consider graphs with maximum Fib onacci index inside the class C ( n, α ). Such graphs are called extr emal . If G is connected, the b ound of Theorem 8 is clearly not tigh t, except when α = 1, that is, when G is a complete graph. W e are going to prov 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, we need preliminary results and definitions to pro v e this theorem. W e denote b y f TC ( n, α ) the Fibonacci index of the T ur´ an-connected graph TC n,α . An inductiv e form ula for its v alue is given 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 0 , α 0 ) if 2 ≤ α ≤ n − 2 , wher e n 0 = n −  n α  − α + 1 and α 0 = min( n 0 , α − 1) . Pr o of. The cases α = 1 and α = n − 1 are trivial by Lemma 4. Supp ose now that 2 ≤ α ≤ n − 2. Let v b e a v ertex of maxim um degree in TC n,α . W e apply Lemma 3 to compute F ( TC n,α ). Observe that the graphs TC v n,α and TC N [ v ] n,α are both T ur´ an graphs when 2 ≤ α ≤ n − 2. The graph TC v n,α is isomorphic to T n − 1 ,α . Let us sho w that TC N [ v ] n,α is isomorphic to T n 0 ,α 0 . By definition of a T ur´ an-connected graph, d ( v ) is equal to  n α  + α − 2. Th us n ( TC N [ v ] n,α ) = n − d ( v ) − 1 = n 0 . 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 0 . Otherwise, TC N [ v ] n,α ' K n 0 and α ( TC N [ v ] n,α ) = n 0 ≤ α − 1. Therefore α ( TC N [ v ] n,α ) = min( n 0 , 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 0 , α 0 ) . Definition 2. A bridge in a connected graph G is an edge e ∈ E ( G ) such that the graph G e is no more connected. T o a bridge e = v 1 v 2 of G which 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 tw o connected comp onen ts 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 gr aph. If G is extr emal, then either G is α -critic al or G has an α -critic al de c omp osition. 8 Pr o of. W e supp ose that G is not α -critical and we sho w that it must contain an α -critical decomp osition. Let e b e an α -safe edge of G . Then e must b e a bridge. Otherwise, the graph G e is connected, has the same order and stability num b er as G and satisfies F ( G e ) > F ( G ) by Lemma 3. This is a con tradiction with G b eing extremal. Therefore G contains at least one α -safe bridge defining a decomposition of G . Let us choose a decomposition D ( G 1 , v 1 , G 2 , v 2 ) such that G 1 is of minimum order. Then, G 1 is α -critical. Otherwise, G 1 con tains an α -safe bridge e 0 = w 1 w 2 , since the edges of G are α -critical or α -safe bridges b y the first part 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 0 , such that v 1 ∈ V ( H 2 ). Then n ( H 1 ) < n ( G 1 ), which is a con tradiction. Hence the decomposition 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 ' TC 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 ve by induction 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 induction (in a wa y to av oid the extremal graph C 5 ), we consider all connected graphs with up to 6 v ertices as the basis of the induction. F or these basic cases, w e refer to the rep ort of an exhaustive automated v erification [8]. W e th us supp ose that n ≥ 7. W e know b y Lemma 11 that either G has an α -critical decomp osition or G is α -critical. W e consider now these t wo situations. 1) G has an α -critical decomp osition. W e prov e in three steps that G ' TC n,α : ( i ) W e establish that for every decomp osition D ( G 1 , v 1 , G 2 , v 2 ), the graph G i is extremal and is isomorphic to a T ur´ an-connected graph such that d ( v i ) = ∆( G i ), for i = 1 , 2. ( ii ) W e sho w 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 stability num ber. W e pro ve that G 1 ' TC n 1 ,α 1 suc h that d ( v 1 ) = ∆( G 1 ). The argument is identical for G 2 . By Lemma 3, we hav e F ( G ) = F ( G 1 ) F ( G v 2 2 ) + F ( G v 1 1 ) F ( G N [ v 2 ] 2 ) . By the induction hypothesis, 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 b y Theorem 8 and Corollary 9, F ( G v 1 1 ) ≤ f T ( n 1 − 1 , α 1 ). It follows 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) 9 As G is supp osed to b e extremal, equality 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). How ev er, F ( G v 1 1 ) = F ( P 4 ) < f T (4 , 2). By (2), this leads to a con tradiction with G b eing extremal. Th us, G 1 m ust b e isomorphic to TC n 1 ,α 1 . Moreov er, v 1 is a vertex of maxim um degree of G 1 . Otherwise, G v 1 1 cannot b e isomorphic to the graph T n 1 − 1 ,α 1 . ( ii ) The 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 no w supp ose that G has an α -critical decomp 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 bridge, it is 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 v ertex of maximum 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 w e ha v e, 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 function is maximized when n 1 = n 2 . That is, when G 1 and G 2 are balanced cliques. This appears if and only if G ' TC n, 2 . Th us w e supp ose that α ≥ 3. In other w ords, G con tains at least three cliques: the clique G 1 of order n 1 ; the clique H con taining v 2 and a clique H 0 in G 2 link ed to H by an α -safe bridge v 2 v 3 . Let k = n − n 1 α − 1 , then the order of H is d k e and the order of H 0 is d k e or b k c (recall that G 2 ' TC n − n 1 ,α − 1 ). These cliques are represented in Figure 2. v 1 G 1 v 2 H v 3 H ′ Figure 2: Cliques in the graph G T o prov e that G is isomorphic to a T ur´ an-connected graph, it remains to show that the clique G 1 is balanced with the cliques H and H 0 . W e consider the decomp osition defined b y the α -safe bridge v 2 v 3 . By ( i ), G 1 and H are cliques of a T ur´ an-connected graph, and H is a clique with maximum order in this graph (recall that v 2 is a vertex of maximum degree in G 2 ). Therefore d k e − 1 ≤ n 1 ≤ d k e , showing that G 1 is balanced with H and H 0 . 2) G is α -critical. Under this h yp othesis, w e pro ve that G is a complete graph, and th us is isomorphic to a T ur´ an-connected graph. 10 Supp ose that G is not complete. Let v b e a vertex of G with a maximum degree d ( v ) = ∆. As G is connected and α -critical, the graph G v is connected b y Lemma 1. By Lemma 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 the induction h yp othesis and Theorem 8, w e 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. How ever, 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 b y the graph G v ' TC n − 1 ,α and an additional v ertex v connected to TC n − 1 ,α b y ∆ edges. There must b e an edge b etw een v and a vertex v 0 of maximum degree in G v , otherwise G N [ v ] is not isomorphic to a T ur´ an graph. The vertex v 0 is adjacent to  n − 1 α  + α − 2 v ertices in G v and it is adjacen t to v , that is, d ( v 0 ) =  n − 1 α  + α − 1 . It follows that ∆ ≥ d ( v 0 ) >  n − 1 α  (3) as G is not a complete graph. On the other hand, v is adjacent to each vertex of some clique H of G v since G N [ v ] has a stabilit y num b er α − 1. As this clique has order at most  n − 1 α  , v must b e adjacen t to a v ertex w / ∈ H by (3). W e observ e that the edge v w is α -safe. This is impossible as G is α -critical. It follows that G is a complete graph and the proof is completed. The study of the maxim um Fib onacci index inside the class T ( n, α ) of trees with order n and stability num b er α is strongly related to the study done in this section for the c lass C ( n, α ). Indeed, due to the fact that trees are bipartite, a tree in T ( n, α ) has alwa ys a stabilit y n umber α ≥ n 2 . Moreov er, the T ur´ an-connected graph TC n,α is a tree when α ≥ n 2 . Therefore, the upp er bound on the Fib onacci index for connected graphs is also v alid for trees. W e th us get the next corollary with in addition the 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 remains to compute the exact v alue of f TC ( n, α ). When α ≥ n 2 , the graph TC n,α is comp osed by one cen tral v ertex v of degree α and α p ending paths of length 1 or 2 attached 11 to v . An extremity of a pending path of length 2 is a v ertex w such that w / ∈ N ( v ). Th us there are x = n − α − 1 p ending paths of length 2 since N ( v ) has size α + 1, and there are y = α − x = 2 α − n + 1 p ending paths of length 1. W e apply 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 . W e conclude this section by sho wing that the function f TC ( n, α ) is strictly increasing in n and α , as already stated for the function f T ( n, α ) (see Corollaries 7 and 9). Prop osition 14. The function f TC ( n, α ) is strictly incr e asing in n and α . Pr o of. W e first pro ve that f TC ( n, α ) is strictly increasing in n when α is fixed. The cases α = 1 and α = n − 1 are obvious by Lemma 10 and we supp ose that 2 ≤ α ≤ n − 2. Let n 0 = n −  n α  − α + 1 and α 0 = min( n 0 , α − 1). Also, w e note n 00 = n + 1 −  n +1 α  − α + 1 and α 00 = min( n 00 , α − 1). Observe that n 0 ≤ n 00 and α 0 ≤ α 00 . W e ha ve f TC ( n, α ) = f T ( n − 1 , α ) + f T ( n 0 , α 0 ) b y Lemma 10, < f T ( n, α ) + f T ( n 00 , α 00 ) b y Corollaries 7 and 9, = f TC ( n + 1 , α ) b y Lemma 10. Therefore, f TC ( n, α ) < f TC ( n + 1 , α ). W e no w prov e that f TC ( n, α ) is strictly increasing in α when n is fixed. Let 2 ≤ α ≤ n − 2. Ob viously , f TC ( n, 1) < f TC ( n, α ) < f TC ( n, n − 1) by Lemma 4. W e consider tw o cases. a) If α < n 2 , then TC n,α con tains at least one clique H of size at least 3 and the remaining cliques are of size at least 2. Supp ose that G is the graph obtained from TC n,α b y remo ving an edge inside H . Then, G is connected and α ( G ) = α + 1. Moreo v er, Lemma 3 and Theorem 12 ensure that f TC ( n, α ) < F ( G ) < f TC ( n, α + 1) and the result follows. b) Supp ose no w that α ≥ n 2 . In this case, TC n,α and TC n,α +1 are trees. Let x = n − α − 1, x 0 = n − α − 2, y = 2 α − n + 1, and y 0 = 2 α − n + 3. Then, f TC ( n, α + 1) − f TC ( n, α ) = 3 x 0 2 y 0 + 2 x 0 − 3 x 2 y − 2 x b y Corollary 13, = 3 x − 1 2 y − 2 x − 1 . As α ≤ n − 2, w e ha ve that x − 1 ≥ 0. Thus, 2 x − 1 ≤ 3 x − 1 . Morev ov er, as α ≥ n 2 , w e ha ve that y ≥ 0 and thus 2 y ≥ 1. It follows that 3 x − 1 2 y − 2 x − 1 ≥ 0. The case of equalit y with 0 happ ens when b oth x − 1 = 0 and y = 0, that is, when α = 1. This nev er holds since α ≥ 2. Therefore f TC ( n, α ) is strictly increasing in α . 12 5 P olyhedral study In the previous sections, we ha ve stated that the graphs with maximum Fib onacci index inside the classes G ( n, α ) and C ( n, α ) are isomorphic to T ur´ an graphs and T ur´ an-connected graphs resp ectiv ely (see Theorems 8 and 12). These results ha ve b een suggested thanks to the system GraPHedron [8]. In this section, we further push the use of the system GraPHedron as outlined in [16]. Indeed, this framework allo ws to suggest the set of all optimal linear inequalities among the stabilit y num b er and the Fib onacci index for graphs inside the class G ( n ) of general graphs of order n and the class C ( n ) of connected graphs of order n . That is, it allows to determine for small v alues of n the complete description of the p olytop es P G ( n ) = con v { ( x, y ) | ∃ G ∈ G ( n ) , α ( G ) = x, F ( G ) = y } , (4) P C ( n ) = con v { ( x, y ) | ∃ G ∈ C ( n ) , α ( G ) = x, F ( G ) = y } , (5) where c onv denotes the con v ex h ull. K 10 S 10 K 10 Stability Number Fibonacci index 0 1 2 3 4 5 6 7 8 9 10 0 103 206 309 412 515 618 721 824 927 K 10 S 10 Stability Number Fibonacci index 0 1 2 3 4 5 6 7 8 9 0 52 104 156 208 260 312 364 416 468 Figure 3: The polytop es P G (10) (left) and P C (10) (righ t) F or example, Figure 3 sho ws the p olytopes P G ( n ) and P C ( n ) when n = 10, as given in the rep orts created by GraPHedron [8]. In these represen tations, w e associate to a p oint ( x, y ) the set of all graphs with a stability num ber x and a Fib onacci index y , and we sa y that the p oin t ( x, y ) c orr esp onds to these graphs. F or instance, in Figure 3, the point (1 , 11) corresp onds to the graph K 10 , whereas the p oin t (9 , 2 9 + 1) corresponds to the graph S 10 . 13 In this section, w e mak e a polyhedral study in a w ay to giv e a complete description of the p olytop es P G ( n ) and P C ( n ) for all (sufficiently large) v alues of n . More precisely , we are going to describ e the fac et defining ine qualities of b oth p olytop es P G ( n ) and P C ( n ) , that is, their minimal system of linear inequalities. Let us fix some notation: L n ( x ) = 2 n − n − 1 n − 1 ( x − 1) + n + 1 , L 0 n ( x ) = 2 n − 1 − n n − 2 ( x − 1) + n + 1 . The following Theorems 15 and 16 give the complete description of P G ( n ) and P C ( n ) . These theorems will be pro v ed at the end of this section, after some preliminary results. Theorem 15. L et n ≥ 5 . Then the p olytop e P G ( n ) has n fac ets define d by the ine qualities y ≥  2 k − 1  x + 2 k (1 − k ) + n, for k = 1 , 2 , . . . , n − 1 , (6) y ≤ L n ( x ) . (7) Theorem 16. L et n ≥ 8 . Then the p olytop e P C ( n ) has n − 1 fac ets define d by the ine qualities y ≥  2 k − 1  x + 2 k (1 − k ) + n, for k = 1 , 2 , . . . , n − 2 , (8) y ≤ L 0 n ( x ) . (9) W e first make some comments. In Figure 4, the t wo p olytopes P G ( n ) and P C ( n ) are dra wn together. This gives a graphical summary of the main res ults stated in Theorems 5, 8, 12, 15 and 16: • blac k p oints corresp ond to T ur´ an graphs and ha v e maxim um y -v alue among general graphs by Theorem 8; • grey p oin ts corresp ond to T ur´ an-connected graphs and ha v e maximum y -v alue among connected graphs b y Theorem 12; • white p oin ts correspond to complete split graphs and hav e minimum y -v alue among general and connected graphs by Theorem 5; • the n facets of P G ( n ) are the n − 1 lines joining t wo consecutive p oints corresp onding to complete split graphs, and the line y = L n ( x ) joining the tw o p oin ts corresp onding to K n and K n (see Theorem 15); • the n − 1 facets of P C ( n ) are the n − 2 lines joining t wo consecutiv e p oin ts corresp onding to (connected) complete split graphs, and the line y = L 0 n ( x ) joining the t wo points corresp onding to K n and S n (see Theorem 16). 14 x y Stability Number Fibonacci index y = L (x) n y = L (x) ’ n 1 n/2 n-1 n n+1 2 n 2 n-1 +1 K n S n K n Figure 4: Representation of P G ( n ) and P C ( n ) together In the next lemma, we establish the inequalities (7) and (9). Lemma 17. The ine quality y ≥  2 k − 1  x + 2 k (1 − k ) + n, (10) defines a fac et of P G ( n ) for k = 1 , 2 , . . . , n − 1 , and a fac et of P C ( n ) for k = 1 , 2 , . . . , n − 2 . Pr o of. W e know b y Theorem 5 that the p oin ts which ha ve minimum y -v alues are those corresp onding to complete split graphs. These p oints are ( x, 2 x + n − x ) , whic h are conv exly independent as the function 2 x + n − x is strictly con vex in x . Therefore these p oints are vertices of P G ( n ) and P C ( n ) , for x = 1 , 2 , . . . , n − 1, and x = 1 , 2 , . . . , n − 2, 15 resp ectiv ely . Moreo v er, there can b e no other p olytope vertices b et w een tw o consecutive p oin ts b ecause x is increasing b y step of 1, and there exists a complete split graph for eac h p ossible v alue of x . Ineq. (10) can then be derived by computing the equation of the line passing by tw o con- secutiv e points  k , 2 k + n − k  and  k + 1 , 2 k +1 + n − k − 1  . It is obvious that Ineq. (10) is facet defining since these points are tw o indep enden t polytop e vertices. W e no w consider the class G ( n ) and study in more details ho w p oints ( x, y ) corre- sp onding to graphs G with α ( G ) = x and F ( G ) = y are situated with resp ect to the line y = L n ( x ). Lemma 18. L et n and α b e inte gers such that n ≥ 7 and 2 ≤ α ≤ n , then f T ( n, α ) α − 1 ≤ 2 n n − 1 · (11) Pr o of. W e consider three cases α = n , α = 2 and 3 ≤ α ≤ n − 1. Let q = n −  n α  b e the order of the graph obtained b y removing a clique of maximal size in T n,α . ( i ) Supp ose that α = n . In this case, both sides of Ineq. (11) are equal and the result trivially holds. ( ii ) Supp ose that α = 2. If n is ev en, f T ( n, 2) = n 2 4 + n + 1 and if n is o dd, f T ( n, 2) = n 2 4 + n + 3 4 . Hence f T ( n, 2)( n − 1) ≤  n 2 4 + n + 1  ( n − 1) . The latter function is cubic, and thus strictly less than 2 n when n ≥ 7. The result holds in case α = 2. ( iii ) Supp ose no w that 3 ≤ α ≤ n − 1. The proof will use an induction on n . If 7 ≤ n ≤ 10, Ineq. (11) can b e chec k ed b y easy computation and we assume that n ≥ 11. By Lemma 6, w e hav e f T ( n, α ) α − 1 = f T ( n − 1 , α ) α − 1 + f T ( q , α − 1) α − 1 ≤ f T ( n − 1 , α ) α − 1 + f T ( q , α − 1) α − 2 · W e can use induction for f T ( n − 1 , α ) / ( α − 1) b ecause either we fall in case ( i ) when α = n − 1, or we sta y in cas e ( iii ). W e can also use induction for f T ( q , α − 1) / ( α − 2). Indeed, if α − 1 = 2 or α − 1 = q , we fall in cases ( ii ) and ( i ), resp ectiv ely . Otherwise, notice that q = n − l n α m > n − l n 3 m ≥ n − n + 2 3 · (12) 16 Hence, q ≥ 7 when n ≥ 11, and w e fall in case ( iii ). It follo ws that f T ( n, α ) α − 1 ≤ 2 n − 1 n − 2 + 2 q q − 1 · As 2 q / ( q − 1) is an increasing function, it is maximum when q = n − 2. This leads to f T ( n, α ) α − 1 ≤ 2 n − 1 n − 2 + 2 n − 2 n − 3 ≤ 2 n − 1 n − 3 + 2 n − 2 n − 3 = 3 · 2 n 4( n − 3) · T o finish the pro of, one has to c hec k if 3 4( n − 3) ≤ 1 n − 1 · This is the case when n ≥ 9. Lemma 19. L et G b e a gr aph of or der n ≥ 5 with a stability numb er α and a Fib onac ci index F , then F ≤ 2 n − n − 1 n − 1 ( α − 1) + n + 1 , with e quality if and only if G ' K n or G ' K n . Pr o of. Notice that the righ t hand side of the inequality in this lemma is equal to L n ( α ) (see Figures 3 and 4). The cases α = 1 and α = n are trivial and corresp ond to b oth cases of equalit y with G ' K n and G ' K n , resp ectiv ely . W e now assume that 2 ≤ α ≤ n − 1 and w e prov e the strict inequality F < L n ( α ). By Theorem 8, it suffices to show that f T ( n, α ) < L n ( α ). The cases n = 5 and n = 6 can b e easily c hec ked b y computation and we supp ose that n ≥ 7. T o achiev e this aim, w e use the following geometrical argumen t. F or a fixed v alue of n , we consider tw o lines. The first one is y = L n ( x ) and the second one is the line passing b y the p oin ts (1 , n + 1), ( α , f T ( n, α )) corresp onding to K n and T n,α , resp ectiv ely . The first line has slope 2 n − n − 1 n − 1 , and the second line has slope f T ( n, α ) − ( n + 1) α − 1 · W e no w pro ve that the slop e of the second line is strictly less than the slop e if the first line. As α < n and b y Lemma 18, f T ( n, α ) − ( n + 1) α − 1 < f T ( n, α ) α − 1 − n + 1 n − 1 ≤ 2 n − 1 n − 1 − n + 1 n − 1 , and the result holds. 17 W e now consider the class C ( n ), and we mak e the same kind of computations of done in the t w o previous lemmas, but with resp ect to the line y = L 0 n ( x ). Lemma 20. L et n and α b e inte gers such that n ≥ 11 and 2 ≤ α ≤ n − 4 , then f T ( n, α ) α − 1 ≤ 2 n − 1 n − 2 · (13) Pr o of. The pro of is similar to the pro of of Lemma 18. W e consider three cases α = 2, α = n − 4 and 3 ≤ α ≤ n − 5. Let q = n −  n α  . ( i ) Supp ose that α = 2. Similarly to case ( ii ) in the pro of of Lemma 18, w e hav e that f T ( n, 2) ≤ n 2 4 + n + 1. Hence f T ( n, 2)( n − 2) ≤  n 2 4 + n + 1  ( n − 2) . The latter function is cubic, and th us strictly less than 2 n − 1 when n ≥ 9. ( ii ) Supp ose that α = n − 4. In this case, and as n ≥ 11, the T ur´ an graph T n,n − 4 is isomorphic to the disjoin t union of four graphs K 2 and n − 8 graphs K 1 . Hence, 2 n − 1 n − 2 − f T ( n, n − 4) n − 5 = 2 7 · 2 n − 8 n − 2 − 3 4 · 2 n − 8 n − 5 , = 2 n − 8  (2 7 − 3 4 ) n − (5 · 2 7 − 2 · 3 4 )  ( n − 2)( n − 5) , = 2 n − 8 [47 n − 478] ( n − 2)( n − 5) , whic h is p ositiv e when n ≥ 11. Ineq. (13) holds in this case. ( iii ) Supp ose now that 3 ≤ α ≤ n − 5. W e use an induction on n . If 11 ≤ n ≤ 16, Ineq. (13) can b e chec ked b y computation and we assume that n ≥ 17. Similarly to case ( iii ) in the pro of of Lemma 18, we hav e f T ( n, α ) α − 1 ≤ f T ( n − 1 , α ) α − 1 + f T ( q , α − 1) α − 2 , and w e can use induction for b oth terms. Indeed for f T ( n − 1 , α ) / ( α − 1) w e fall in case ( ii ) when α = n − 5, or w e sta y in case ( iii ). F or f T ( q , α − 1) / ( α − 2), since 3 ≤ α ≤ n − 5, we can chec k that either α − 1 = 2 or α − 1 = q − 4 tw o cases already treated in ( i ) and ( ii ), or 3 ≤ α − 1 ≤ q − 5. In the latter case, we hav e q ≥ 11 when n ≥ 17 b y Ineq. (12). It follows that f T ( n, α ) α − 1 ≤ 2 n − 2 n − 3 + 2 q − 1 q − 2 . 18 As 2 q / ( q − 2) is increasing, it is maximum when q = n − 2. This leads to f T ( n, α ) α − 1 ≤ 2 n − 2 n − 3 + 2 n − 3 n − 4 ≤ 3 · 2 n − 1 4( n − 4) · The pro of is completed b ecause 3 4( n − 4) ≤ 1 n − 2 when n ≥ 10. Lemma 21. L et G b e a c onne cte d gr aph of or der n ≥ 8 with a stability numb er α and a Fib onac ci index F , then F ≤ 2 n − 1 − n n − 2 ( α − 1) + n + 1 , with e quality if and only if G ' K n or G ' S n . Pr o of. Observe that the right hand side of the inequality stated in the lemma is equal to L 0 n ( α ) (see Figure 4). The cases α = 1 and α = n − 1 are trivial and corresp ond to the t w o cases of equalit y . When 2 ≤ α ≤ n − 2, we pro ve the strict inequality F < L 0 n ( α ). The cases n = 8, n = 9 and n = 10 can b e chec k ed b y computation and we therefore supp ose that n ≥ 11. W e consider separately the t wo cases 2 ≤ α ≤ n − 4 and n − 3 ≤ α ≤ n − 2. ( i ) Let 2 ≤ α ≤ n − 4. By Theorem 12, it is enough to show that f TC ( n, α ) < L 0 n ( α ). W e prov e a stronger result, that is, f T ( n, α ) < L 0 n ( α ). The result follows since f TC ( n, α ) ≤ f T ( n, α ). This situation is w ell illustrated in Figure 4 whic h also indicates that the case n − 3 ≤ α ≤ n − 2 has to b e treated separately . The argumen t to prov e that f T ( n, α ) < L 0 n ( α ) is the same as in the pro of of Lemma 19. W e sho w that the slop e of the line y = L 0 n ( x ) is strictly greater than the slop e of the line passing b y the t w o points corresp onding to K n and T n,α . As α ≤ n − 4, we ha ve n + 1 α − 1 ≥ n + 1 n − 5 > n n − 2 . This observ ation and Lemma 20 lead to f T ( n, α ) α − 1 − n + 1 α − 1 < f T ( n, α ) α − 1 − n n − 2 ≤ 2 n − 1 n − 2 − n n − 2 , and the announced prop ert y on the slop es is prov ed. 19 ( ii ) Let n − 3 ≤ α ≤ n − 2. By Theorem 12, one has to sho w that f TC ( n, n − 2) < L 0 n ( n − 2) and f TC ( n, n − 3) < L 0 n ( n − 3). It suffices to pro v e that f TC ( n, n − 2) < L 0 n ( n − 3) . Indeed, f TC ( n, n − 3) < f TC ( n, n − 2) b y Corollary 14 and L 0 n ( n − 3) < L 0 n ( n − 2) b ecause the slop e of y = L 0 n ( x ) is strictly p ositiv e. As n ≥ 11, w e ha ve α ≥ n 2 and we use Corollary 13 to compute f TC ( n, n − 2). This leads to L 0 n ( n − 3) − f TC ( n, n − 2) = 2 n − 1 − n n − 2 ( n − 4) + n + 1 − 3 · 2 n − 3 − 2 , = ( n − 10) · 2 n − 3 + n + 2 n − 2 , whic h is strictly p ositiv e when n ≥ 10. W e can now give the pro of of Theorems 15 and 16. Pr o of of The or ems 15 and 16. W e b egin with the pro of for the p olytope P G ( n ) . Looking at Lemma 17, it remains to pro v e that ( i ) Ineq. (7) is facet defining; ( ii ) there are exactly n facet defining inequalities of P G ( n ) . The pro of ( i ) is straightforw ard. Indeed, Lemma 19 ensures that Ineq. (7) is v alid. Moreo ver, the p oin ts (1 , n + 1) and ( n, 2 n ) corresp ond to the graphs K n and K n , respectively . These p oin ts are affinely independent and satisfy Ineq. (7) with equality . Therefore Ineq. (7) is facet defining. F or ( ii ), it suffices to observ e that for any v alue of x = 1 , 2 , . . . , n , there is exactly one v ertex in the p olytop e: the p oint which correspond to the complete split graph CS n,x . It follo ws that P G ( n ) has exactly n v ertices and n facets. The pro of is similar for the p olytope P C ( n ) except that x < n . Indeed, Ineq. (9) is v alid b y Lemma 21, and the p oints satisfying Ineq. (9) with equality correspond to the graphs K n and S n . 6 Observ ations T ur´ an graphs T n,α ha ve minimum size inside G ( n, α ) by the Theorem of T ur´ an [23]. Christophe et al. [6] giv e a tigh t low er b ound for the connected case of this theorem, and Bougard and Joret [2] characterized the extremal graphs, which happ en to con tain the TC n,α graphs as a sub class. By these results and Theorems 8 and 12, we can observ e the following relations betw een graphs with minim um size and maximum Fib onacci index. The graphs inside G ( n, α ) minimizing m ( G ) are exactly those which maximize F ( G ). This is also true for the graphs 20 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 ) where e is an y edge of a graph G . As indicated in our pro ofs, the latter prop ert y only implies that a graph maximizing F ( G ) contains only α -critical edges (and α -safe bridges for the connected case). Our pro ofs use a deep study of the structure of the extremal graphs to obtain Theorems 8 and 12. W e now give additional examples showing that the intuition that more edges imply few er stable sets is wrong. P edersen and V estergaard [19] giv e the following example. Let r b e an in teger such that r ≥ 3, G 1 b e the T ur´ an graph T 2 r,r and G 2 b e the star S 2 r . The graphs G 1 and G 2 ha ve the same order but G 1 has less edges ( r ) than G 2 (2 r − 1). Nev ertheless, observ e that F ( G 1 ) = 3 r < F ( G 2 ) = 2 2 r − 1 + 1. This example do es not take in to accoun t 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 pairs of graphs with the same order and the same stabilit y num b er (see the graphs G 3 and G 4 on Figure 5). These t wo graphs are inside the class G (6 , 4), how ev er m ( G 3 ) < m ( G 4 ) and F ( G 3 ) < F ( G 4 ). Notice that w e can get suc h examples inside G ( n, α ) with n arbitrarily large, b y considering the union of sev eral disjoin t copies of G 3 and G 4 . G 3 : G 4 : Figure 5: Graphs with same order and stability num b er These remarks and our results suggest some questions ab out the relations b et w een the size, the stability num b er and the Fib onacci index of graphs. What are the low er and upp er bounds 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 graphs order n , size m and stability n umber α ? Are there classes of graphs for which more edges alwa ys imply fewer stable sets? W e think that these questions deserv e to be studied. Ac kno wledgmen ts The authors thank Gw ena¨ el Joret for helpful suggestions. References [1] Ber ge, C. The The ory of Gr aphs . Do v er Publications, New Y ork, 2001. 21 [2] Bougard, N., and Joret, G. T ur´ an Theorem and k -connected graphs. J. Gr aph The ory 58 (2008), 1 – 13. [3] Br ooks, R.-L. On colouring the no des of a net work. Pr o c. Cambridge Philos. So c. 37 (1941), 194 – 197. [4] Br uy ` ere, V., and M ´ elot, H. T ur` an Graphs, Stability Number, and Fibonacci In- dex. 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