Linear Algebra and its Applications

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1 Linear Algebra and its Applications 41 (2009) Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: wwwelseviercom/locate/laa On graphs whose signless Laplacian index does not exceed 45 Jianfeng Wang a,b,, Qiongxiang Huang a, Francesco Belardo c, Enzo M Li Marzi c a College of Mathematics and System Science, Xinjiang University, Urumqi 80046, PR China b Department of Mathematics, Qinghai Normal University, Xining, Qinghai , PR China c Department of Mathematics, University of Messina, Sant Agata, Messina, Italy A R T I C L E I N F O A B S T R A C T Article history: Received November 2008 Accepted 16 February 2009 Available online 28 March 2009 Submitted by RA Brualdi AMS classification: 05C50 Keywords: Signless Laplacian matrix Q-Spectrum Q-Index Spectral radius Let A G and D G be respectively the adjacency matrix and the degree matrix of a graph G The signless Laplacian matrix of G is defined as Q G = D G + A G The Q-spectrum of G is the set of the eigenvalues together with their multiplicities of Q G The Q-index of G is the maximum eigenvalue of Q G The possibilities for developing a spectral theory of graphs based on the signless Laplacian matrices were discussed by Cvetkovic et al [D Cvetkovic, P Rowlinson, SK Simic, Signless Laplacians of finite graphs, Linear Algebra Appl 42 (2007) ] In the latter paper the authors determine the graphs whose Q-index is in the interval [0, 4] In this paper, we investigate some properties of Q-spectra of graphs, especially for the it points of the Q-index By using these results, we characterize respectively the structures of graphs whose the Q-index lies in the intervals (4, ], (2 + ( 5, ɛ + 2] and (ɛ + 2, 45], where ɛ = 1 1 (54 6 ) + (54 + ) 6 ) Elsevier Inc All rights reserved 1 Introduction All graphs considered here are simple, undirected and finite Let G = G(V(G), E(G)) be a graph with vertex set V(G) ={v 1, v 2,, v n } and edge set E(G) Let matrix A G be the adjacency matrix of G The Supported by the National Science Foundation of China (No ) and INdAM (Italy) Corresponding author addresses: jfwang4@yahoocomcn (J Wang), huangqx@xjueducn (Q Huang), fbelardo@gmailcom (F Belardo), emarzi@dipmatunimeit (EM Li Marzi) /$ - see front matter 2009 Elsevier Inc All rights reserved doi:101016/jlaa

2 J Wang et al / Linear Algebra and its Applications 41 (2009) signless Laplacian matrix of G is defined as Q G = Q(G) = D G + A G, where D G = diag(d 1, d 2,, d n ) with d i = d(v i ) being the degree of vertex v i of G (1 i n) In the sequel we prefer to adopt Q(G) for the signless Laplacian matrix of G Since Q(G) is real, symmetric and positive semidefinite, all of its eigenvalues are non-negative numbers Assume that q 1 (G) q 2 (G) q n (G) 0 are the eigenvalues of Q(G) and call them Q-eigenvalues of G The Q-spectrum of G consists of the Q-eigenvalues together with their multiplicities The maximum Q-eigenvalue q 1 (G) of G is called the Q-index of G and it will be denoted by κ(g) Moreover, since Q(G) is non-negative, then the eigenvector associated with κ(g) can be taken to be non-negative In addition, if G is connected (ie if Q(G) is irreducible), then κ(g) is of multiplicity one and its corresponding eigenvector can be taken to be positive Such an eigenvector is called the Q-Perron eigenvector of G, which is denoted by x = (x 1, x 2,, x n ) T (not necessarily a unit one) Hence, we arrive at κx i = d i x i + x j (i = 1, 2,, n), (1) j i where the summation is over all neighbors j of the vertex i As a matter of fact, (1) isaneigenvalue equation for the ith vertex (associated with the Q-index) We now introduce some notation In what follows, by ϕ(b) = ϕ(b, q) = det(qi n B) (we omit the variable if it is clear from the context) we denote the characteristic polynomial of a real square matrix B with order n, where I n is the identity matrix In particular, if B = Q(G), we also write ϕ(q(g), q) as ϕ(g, q) = ϕ(g) and call it the Q-polynomial of G (note, ϕ(g) is also written as Q G (λ) in [8]) To avoid confusion we adopt the former one Let Q v (G) be the principal matrix of Q(G) formed by deleting the row and column corresponding to vertex v As usual, let P n, C n and K 1,n 1 be respectively the path, the cycle and the star of order n ForagraphG, Δ(G) stands for the largest vertex degree in G As pointed out in the recent survey [8], there are very few research papers concerning the signless Laplacian matrices of graphs Cvetković et al [8,10] discussed possibilities for developing a spectral theory of graphs based on this matrix, and gave some reasons why it is superior to other graph matrices such as the adjacency, Laplacian and Seidel matrix To learn more details we refer the readers to their surveys Characterizing the structures of graphs by the eigenvalues of some graph matrix is an attractive study ( field Some important results have been obtained already For the largest eigenvalue ρ [0, 2], 2, 2 + ] ( 5 and 2 + ] 5, 2 2 of the graph adjacency matrix, the corresponding graphs are respectively determined by Smith [21], Cvetković et al [7], Brouwer and Neumaier [] and Woo and Neumaier [2] For the second largest eigenvalue λ 2 [ ] 0, 1 and third largest eigenvalue λ ) of the graph adjacency matrix, the corresponding graphs are found by Cao and Hong (, [4,5] In this paper, we mainly investigate the properties of the Q-index of graphs In Section 2 we give some useful lemmas and results In Section ( we characterize respectively the structures of graphs whose Q-index lies in the intervals 4, 2 + ] ( 5, 2 + ] 5, ɛ + 2 and (ɛ + 2, 45], where ( ( ɛ = 1 54 ) ( 54 + ) ) In particular we show that almost all [ graphs whose index of the adjacency matrix lies in 2, 2 + ) 5 have their Q-index in [4, ɛ + 2) Finally, in the last section, we determine the it point for the Q-index of some special classes of graphs 2 Some useful lemmas and results A matrix often studied in literature is the matrix L G = D G A G, known as Laplacian matrix of a graph G Many researchers studied the relations between the Laplacian matrix and the signless Laplacian matrix of a graph One of the most important results is the following lemma (see, for example [8]or[17]) Lemma 21 In bipartite graphs, the Q-polynomial is equal to the characteristic polynomial of the Laplacian

3 164 J Wang et al / Linear Algebra and its Applications 41 (2009) By applying the Perron Frobenius theory of non-negative matrices (see, for example, Section 0 of [6]), the lemma below follows from Theorems 06 and 07 of [6] Lemma 22 Let H be a proper subgraph of a connected graph G Then κ(h) <κ(g) Let G be a graph of order n and size m In the following, let φ(g, λ) (or simply φ(g), if the variable is clear from the context) and ρ(g) denote the characteristic polynomial and the largest eigenvalue of its adjacency matrix (ρ is also known as the A-index), respectively Let L(G) and S(G) be, respectively, its line graph and subdivision graph, where S(G) is obtained from G by inserting a new vertex in each edge It was proved that (see [8]): φ(l(g), λ) = (λ + 2) m n ϕ(g, λ + 2), (2) Next, we study the relation between the Q-eigenvalues of G and those of φ(s(g) (see also [10]) The following lemma from matrix theory can be found in [6, p 62] Lemma 2 If M is a nonsingular square matrix, then M N P Q = M Q PM 1 N For the sake of completeness, we give a proof for the well-known lemma below that originally appeared in [6, p 6] (see also [9]) Lemma 24 Let G be a graph of order n and size m, and S(G) be the subdivision graph of G Then φ(s(g), λ) = λ m n ϕ(g, λ 2 ) () Proof Obviously, S(G) is a bipartite graph( of order ) n + m Without loss of generality, the adjacency 0 R matrix of S(G) can be written as A S(G) = T, where R is the incident matrix of G and R T is the R 0 transpose of R From Lemma 2 and Q(G) = RR T (see [8]), we get φ(s(g), λ) = λi m+n A S(G) = λi m R T R λi = λi m λi n R I m n λ RT = λ m n λ 2 I n RR T =λ m n λ 2 I n Q(G) =λ m n ϕ(g, λ 2 ) This completes the proof The graphs whose Q-index is in [0,4] are characterized by Cvetković et al [8] Lemma 25 Let G be a graph Then the following statements hold: (i) κ(g) = 0 if and only if G has no edges; (ii) 0 <κ(g) <4 if and only if all components of G are paths; (iii) For a connected graph G we have κ(g) = 4 if and only if G is a cycle C n or K 1, Cvetković et al investigated the bounds for Q-index of graphs The lemma below can be found in [8,9] Notice that the second item in the original paper requires that the order of graphs is at least 4 In fact, a straightforward calculation shows that it still holds for n = 2, Lemma 26 Let G be a graph on n vertices with vertex degrees d 1, d 2,, d n : (i) min { d i + d j ij E(G) } κ(g) max { d i + d j ij E(G) } For a connected graph G, then equality holds in either of these inequalities if and only if G is a regular or semi-regular bipartite graph

4 J Wang et al / Linear Algebra and its Applications 41 (2009) (ii) If G is a connected graph of order n 2, then κ(g) (G) + 1 with equality if and only if G is the star K 1,n 1 Hoffman and Smith [12] defined an internal path of a graph G as a walk v 0, v 1,, v k (k 1) where the vertices v 1,, v k are distinct (v 0, v k need not be distinct), d(v 0 )>2, d(v k )>2 and d(v i ) = 2 whenever 0 < i < k, with d(v) as the degree of vertex v in G In accordance with the definition, there are two types of internal paths (see Fig 1) Their well-known result about the internal path for the adjacency spectra (see [12]or[6, p 79]) has been studied A similar result for the Q-spectra has been obtained by several authors independently For instance in [10] this result is proved via the relation of Lemma 24, and in [14] it is shown for the internal path to be a path (ie Type ii) Here we give a direct and complete proof based on the original one given by Hoffmann In what follows we depict the two types of internal paths: Type i v 0 = v k, d(v 0 ) and d(v i ) = 2 for 0 < i < k Type ii v 0 /= v k, d(v 0 ), d(v k ) and d(v i ) = 2 for 0 < i < k In order to prove the following theorem about internal path for Q-spectra, we introduce two useful lemmas, the first of which is taken from matrix theory (see, for example [24]) Lemma 27 Let x be a positive vector and A a real symmetric matrix with α as its largest eigenvalue If Ax >βx, then α>βif Ax <βx, then α<β Lemma 28 Let G be a connected graph and σ an automorphism of G If x is the Q-Perron vector associated to κ(g), then σ(i) = j leads to x i = x j Proof Let P be the permutation matrix corresponding to σ Then κx = Q(G)x = P T Q(G)Px Hence, κ(px) = Q(G)(Px), ie Px is also a Q-Perron vector corresponding to Q(G) Since Px is a positive vector with the same norm as x, then the essential uniqueness of the Q-Perron vector indicates that Px = x Theorem 21 Let uv be an edge of the connected graph G and let G uv be obtained from G by subdividing the edge uv of G: (i) If G = C n, then κ(g uv ) = κ(g) = 4 (ii) If uv is not in an internal path of G /= C n, then κ(g uv )>κ(g) (iii) If uv belongs to an internal path of G, then κ(g uv )<κ(g) Proof Case (i) follows from Lemma 25(iii) Let us consider case (ii) If G uv is obtained from G by subdividing an edge not lying in an internal path, then G is a proper subgraph of G uv and by Lemma 22 we have κ(g uv )>κ(g) This completes the case (ii) Let us consider case (iii) Let Q(G uv ) = Q and Q(G) = Q be the signless Laplacian matrices of G uv and G, respectively, and κ(g uv ) = κ, κ(g) = κletn be the order of G, then n + 1 is the order of G uv Let x = (x 0, x 1,, x n 1 ) T R n be the Q-Perron eigenvector for G and let us consider its components x 1, x 2,, x r related to the vertices v 1, v 2,, v r of the internal path of G For convenience, set v i = i (i = 0, 1,, k) Letω be the vertex of G uv subdividing the edge uv Without loss of generality, let u = t and v = t + 1 By the types of internal path, we distinguish the following two cases: Fig 1 The two types of internal path

5 166 J Wang et al / Linear Algebra and its Applications 41 (2009) Case 1 G contains the internal path of Type i By Lemma 28 we have x i = x k i+1, where i = 0, 1,, k 1 By the parity of k, we distinguish two subcases If k = 2t, then x t = x t+1 Note that the cycle C k is a proper subgraph of G, and so κ>κ(c k ) = 4 Take the positive vector y = (x 0,, x t, x ω, x t+1,, x n 1 ) R n+1, where x ω = x t+1 Then we have that (Q y) r = (κy) r for r V(G uv )\{ω} Since (Q y) ω = x t + 2x ω + x t+1 = 4x ω <κx ω = (κy) ω, then Q y <κy By Lemma 27(ii) we get κ <κ If k = 2t + 1, then x t = x t+2 Similarly, take y = (x 0,, x t, x ω, x t+1,, x n 1 ) R n+1, where x ω = x t+1 we also have (Q y) r = (κy) r for r V(G uv )\{ω} Fromκx t+1 = x t + 2x t+1 + x t+2 = 2x t + 2x t+1, it follows (κ 2)x t+1 = 2x t, and thus x t+1 < x t (since κ>4) Hence, (Q y) ω = x t + 2x ω + x t+1 < x t+2 + 2x ω + x t = x t+2 + 2x t+1 + x t = κx t+1 = κx ω which leads to Q y <κy and so κ <κbylemma 27(ii) Case 2 G contains the internal path of Type ii It has been shown by Lemma 27 in [14] This completes the proof Finally, we introduce some formulas to calculate the Q-polynomials of some graphs based on the following formulas (also known as Schwenk formulas, see [19]) Lemma 29 (Schwenk formulas) Let G be a (simple) graph Denote by C(v)(C(e)) the set of all cycles in G containing a vertex v (resp an edge e = uv) Then we have: (i) φ(g, λ) = λφ(g v, λ) w v φ(g v w, λ) 2 C C(v) φ(g V(C), λ), (ii) φ(g, λ) = φ(g e, λ) φ(g v u, λ) 2 C C(e) φ(g V(C), λ) We assume that φ(g, λ) = 1 if G is the empty graph (ie with no vertices) Lemma 210 Let G be a graph and Q v (G) the principal submatrix of Q(G) obtained by deleting the row and column related to v and ϕ(q v (G), λ) its characteristic polynomial Then the following formula holds: φ(s(g) v, λ) = λ m n+1 ϕ(q v (G), λ 2 ) Proof Let R be the vertex edge incidence matrix of G and let R v be the submatrix of R obtained by deleting the row related to vertex v Then the adjacency matrix of S(G) v is ( ) 0 R T v R v 0 So φ(s(g) v, λ) = λi m R v Rv T λi n 1 Since R v R T v = Q v(g) then we get the assertion = λ m n+1 λ 2 In 1 R v R T v Theorem 22 Let G 1 and G 2 be two vertex-disjoint graphs, and G the graph obtained from G 1 and G 2 by joining a vertex u of G 1 toavertexvofg 2 by an edge Then ϕ(g) = ϕ(g 1 )ϕ(g 2 ) ϕ(g 1 )ϕ(q v (G 2 )) ϕ(g 2 )ϕ(q u (G 1 )) (4) Proof Consider S(G) and let w be the vertex inserted in the edge uv of G in S(G) Since uv is a bridge in G then there are not cycles through it in S(G) Then by Schwenk formula (i) applied at vertex w,we have: φ(s(g), λ) = λφ(s(g 1 ), λ)φ(s(g 2 ), λ) φ(s(g 1 ), λ)φ(s(g 2 ) v, λ) φ(s(g 2 ), λ)φ(s(g 1 ) u, λ) Since φ(s(g), λ) = λ m n ϕ(g, λ 2 ) and by Lemma 210 we get the assertion

6 J Wang et al / Linear Algebra and its Applications 41 (2009) Remark 21 Formula (4) is equivalent to Formula (6) in [10] By using Formula (4) of Theorem 22 it is possible to obtain the results on it points in a form close to the adjacency variant (see, for example, Section 4) Clearly by using Formula (6) in [10], we can obtain the same results Corollary 21 Let G, G 1, G 2,, G k be k + 1 disjoint connected graphs and v V(G), v i V(G i )(i = 1, 2,, k) Let H k be a graph obtained from G 1, G 2,, G k and G by adding k new edges to join vertex v to v 1, v 2,, v k, respectively Then k k k ϕ(h k ) = ϕ(g) (ϕ(g i ) ϕ(q vi (G i ))) ϕ(q v (G)) ϕ(g i ) (ϕ(g j ) ϕ(q vj (G j ))) i=1 i=1 Proof By induction on k we prove the corollary If k = 1, from Theorem 22 we obtain that the result holds Assume next that k 2 Let G i (v) be the graph obtained from G i by attaching a new pendant edge v i v at v i (i = 1, 2,, k) Then ϕ(q v (G i (v))) = ϕ(g i ) ϕ(q vi (G i )) (i = 1, 2,, k) Setting u = v k and v = v to H k in Theorem 22, by the induction hypothesis, we arrive at ϕ(h k, q) = ϕ(h k, q) = det(qi Q(H k )) k 1 = ϕ(h k 1 )(ϕ(g k ) ϕ(q vk (G k ))) ϕ(g k )ϕ(q v (G)) ϕ(q v (G i (v))) k 1 = ϕ(g) (ϕ(g i ) ϕ(q vi (G i ))) ϕ(q v (G)) i=1 i=1 k 1 j=1 j /=i i=1 ϕ(g i ) i=1 j /=i [ ϕ(g k ) ϕ(q vk (G k )) ] k 1 ϕ(g k )ϕ(q v (G)) (ϕ(g i ) ϕ(q vi (G i ))) i=1 i=1 (ϕ(g j ) ϕ(q vj (G j ))) i=1 k k k = ϕ(g) (ϕ(g i ) ϕ(q vi (G i ))) ϕ(q v (G)) ϕ(g i ) (ϕ(g j ) ϕ(q vj (G j ))) j=1 j /=i This completes the proof Graphs whose Q-index does not exceed 45 In [2] Woo and Neumaier determined the graphs whose largest eigenvalue (of the adjacent matrix) ρ(g) is bounded by 2 2 They defined the following graphs: an open quipu is a tree of maximum valency such that all vertices of degree lie on a path; a closed quipu is a connected graph of maximum valency such that all vertices of degree lie on a circuit, and no other circuit exists; a dagger is a path with a -claw attached to an end vertex By S 1, S 2 and S we denote the sets of open quipus, closed quipus and daggers, respectively Then they proved the following results: Theorem 1 A connected graph G whose largest eigenvalue ρ(g) satisfies 2 <ρ(g) 2 2 belongs to the set S 1 S 2 S Note that the subdivision operation on a graph does not change the maximum degree Since the maximum degree of a dagger is 4, the Q-index of a dagger is at least 5 (see Lemma 26(ii)) Clearly, the subdivision of an open quipu or a closed quipu is still an open quipu or a closed quipu respectively

7 168 J Wang et al / Linear Algebra and its Applications 41 (2009) Fig 2 Graphs T a,b,c, Q a,b,c, and H a,b By using Lemma 24 we immediately obtain the following theorem, which describes the structures of graphs whose Q-index does not exceed 45: Theorem 2 Let G be a connected graph whose largest eigenvalue κ(g) satisfies 4 <κ(g) <45 Then G is an open or a closed quipu In this section, we will consider a subdivision of the interval (4,45] and we describe the structure of graphs whose Q-index belongs to a subinterval of (4,45] More precisely, ( let G 1, G 2 and G be respectively the sets of connected graphs of order n whose Q-index lies in 4, 2 + ] ( 5, 2 + ] 5, ɛ + 2 and ) (ɛ + 2, 45], where ɛ = 1 ( ( 54 6 depicted in Fig 2 ) 1 + ( ) 1 Some graphs used in this section are Remark 1 In the above figure and in the rest of the paper as well, by the symmetry of the above graphs and without loss of generality, we set a b c in T a,b,c, a c in Q a,b,c and a b in H a,b Our main result is the following: Theorem (Main result) Let the set G i (i = 1, 2, ) be as defined above Then (i) G 1 ={T 1,1,n n 5}, G 1 S 1 ; (ii) G 2 S 1 consists of the following graphs: (a) T 1,b,c for c b 2; (b) Q a,b,c for b f (a, c), where f (a, c) is an appropriate integral function (iii) G S 1 \(G 1 G 2 ) S 2 In order to prove the above theorem, it is enough to show Theorems 5, 6 and 8 However the set G 2 and G are not completely described so we pose the following open problem: Problem 1 Determine an accurate condition for G S 1 S 2 such that G G 2 or G G Now we make progress towards the main result step by step Note that some proofs depends on it points for the Q-index which are calculated in Section 4 Theorem 4 None of graphs can attain κ(g) = 2 + 5, or κ(g) = ɛ + 2, or κ(g) = 45 Proof Recall that any graph polynomial is monic and with integer coefficients For contradiction we assume that there exists a graph G satisfying κ(g) = 2 + 5orκ(G) = ɛ + 2orκ(G) = 45 If

8 J Wang et al / Linear Algebra and its Applications 41 (2009) κ(g) = 2 + 5, consider then S(G)By()wehaveρ(S(G)) = and its minimal polynomial (in Z[x]) contains complex roots, which is absurd If κ(g) = ɛ + 2orκ(G) = 45 the minimal polynomial in (Z[x]) is not monic, that is again absurd Then 2 + 5, ɛ + 2 and 45 cannot be the Q-index of any graph Lemma 1 If G G 1 G 2, then Δ(G) = Furthermore, there are at most two vertices of degree in G Proof If Δ(G) = 1, then G = P 2 IfΔ(G) = 2, then G is a path or a cycle From Lemma 25 we have κ(g) 4 If Δ(G) 4, then K 1,4 is subgraph of G From Lemma 22 we get κ(g) κ(k 1,4 ) = 5 Hence, Δ(G) = This completes the first part Now we show the second part If G has at least three vertices of degree, then it must have H a,b as subgraph for some a, b 1 (note that G cannot have any cycle as subgraph, see the next lemma), and we find from Lemma 22 and Theorem 21(ii) that κ(g) κ(h a,b )>κ(h n,n )>κ(t 1,n,n ) for all n > b By Corollary 4 we get κ(g) n κ(t 1,n,n ) = ɛ + 2 This completes the proof Lemma 2 If G G 1 G 2, then G is a tree Proof If G is not a tree, then G must contain a cycle C r as a proper subgraph Since G is a connected graph with Δ(G) = (see Lemma 1), then K 1 (C r ) (see Theorem 41 for its definition) is a subgraph of G From Lemma 22 and Corollary 41 it follows that κ(g) κ(k 1 (C r )) > ɛ + 2 By Theorem 22 we get ϕ(t 2,2,2 ) = q(q 2 6q 7)(q 2 q + 1) 2 which yields κ(t 2,2,2 ) = + 2 >ɛ+ 2, and thus the following lemma holds Lemma Under the conditions in above lemma, then T 2,2,2 is not a subgraph of G Theorem 5 G 1 ={T 1,1,n n 5} Proof Let G G 1 From Lemmas 1 and 2, it follows that G is a tree having at most two vertices of degree Δ(G) = If G has one vertex of degree Δ(G) =, then G is a tree of type T a,b,c Since T 2,2,2 is not a subgraph of G (see Lemma ), then a must be equal to 1 If b = 1, then G = T 1,1,n Since κ(t 1,1,1 ) = 4, then by Lemma 22 and Corollary 42 we conclude that κ(t 1,1,1 )<κ(t 1,1,n )<2 + 5 if and only if n 5 Let b 2 By calculation we get ϕ(t 1,2,2 ) = q(q 2)(q 2 5q + )(q 2 q + 1) and so κ(t 1,2,2 ) = 1 2 (5 + 1) >2 + 5, which implies for c b 2 that κ(t 1,b,c ) κ(t 1,2,2 )>2 + 5, since T 1,2,2 is subgraph of T 1,b,c If G has two vertices of degree, then by Lemma we conclude that G must be a tree of type Q a,b,c From Theorem 21(iii) and Lemma 22 we get for any r > b that κ(q a,b,c )>κ(q a,r,c )>κ(t 1,1,r ), which implies by Corollary 42 that κ(q a,b,c ) r κ(t 1,1,r) = This ends the proof

9 170 J Wang et al / Linear Algebra and its Applications 41 (2009) Theorem 6 Let G G 2 Then G is precisely one of the following graphs: (i) T 1,b,c for c b 2; (ii) There exists an integral function f (a, c) such that G = Q a,b,c for any b f (a, c) Proof By Lemmas 1 and 2, G is a tree having at most two vertices of degree Δ(G) = If G has one vertex of degree Δ(G) =, then G is a tree of type T a,b,c Lemma and Theorem 5 show that a = 1 and c b 2 As proved in Theorem 5, it is clear that <κ(t 1,2,2 ) κ(t 1,b,c ) <κ(t 1,r,r ) for arbitrary r > c Hence, by Corollary 4 we have <κ(t 1,b,c ) r κ(t 1,r,r) = ɛ + 2 If G has two vertices of degree, by Lemma we conclude that G must be a tree of type Q a,b,c From Theorem 21(iii) and Lemma 22 we get for any r > b that κ(q a,b,c )>κ(q a,r,c )>κ(t 1,1,r ), which implies by Corollary 42 that κ(q a,b,c ) r κ(t 1,1,r) = For t > max{a, b 1, c}, we get, from Lemma 22, Theorem 44 and Corollary 4, that n κ(q a,b,c) = max κ(p a 2 (P a, P b 1 )), κ(p 2 (P b 1, P c )) b t κ(p 2(P t, P t )) = ɛ + 2 b c Thus, for any given a and c, there exists a constant f (a, c) such that κ(q a,b,c )<ɛ+ 2 if and only if b f (a, c) This completes the proof In general, the values of f (a, c) are difficult to obtain Nevertheless, we try to provide much more information for it (see Theorem 7) Now we introduce the following lemma, whose proof can be found for (i) in [2] and for (ii) in [22] Recall that ρ(g) is the largest eigenvalue of the adjacency matrix of a graph G Lemma 4 Let the graphs N d i and P m,m,d i,j be defined in Fig Then (i) d ρ(n d d ) = α, where α = ɛ + 2; 2 (ii) Set 0 i < j d k 1 2 Then (a) if d = 2k, then ρ ( ) ( P m,m,d i,i+k+1 = ρ P m,m,d (b) if d > 2k, then ρ ( ) ( P m,m,d i,i+k+1 <ρ P m,m,d (c) if d < 2k, then ρ ( ) ( P m,m,d i,i+k+1 >ρ P m,m,d j,j+k+1) ; j,j+k+1) ; j,j+k+1) Fig The graphs N d i and P m,m,d i,j

10 J Wang et al / Linear Algebra and its Applications 41 (2009) Theorem 7 Let Q a,b,c be the graph defined in Fig 2 and r = a + b + c Then (i) if b a + c + 1, then κ(q a,b,c ) ɛ + 2; (ii) if r is sufficiently large, then κ(q a,b,c )<ɛ+ 2 implies b a + c + 1 Proof From Lemma 24 we have that κ(q a,b,c ) ɛ + 2 that is equivalent to ρ(s(q a,b,c )) ɛ + 2 = α Note, S(Q a,b,c ) P 2,2,2(a+b+c) 2a,2(a+b) ρ ( N 2(a+b+c+1) a+b+c 1 By Lemma 4(ii-a) we get that ) ( = ρ P 2,2,2(a+b+c) ) ( 1,2b+1 = =ρ P 2,2,2(a+b+c) 0,2b P 2,2,2(a+b+c) 2a,2(a+b) if and only if b = a + c + 1 Furthermore if s = a + c, by Lemma 4(i) we get s ρ ( N 2(2s+2) 2s N 2(2s+2) ) 2s+2 = α In addition, by Lemma 4(ii-c) we finally get that ρ ( P 2,2,2(a+b+c) 2a,2(a+b) ) α if b a + c + 1 which means r κ(q a,a+c+1,c ) α 2 = ɛ + 2ifb a + c + 1 This ends the proof of (i) Now we show the second one By contradiction we assume b a + c It is just equivalent to the condition of Lemma 4(ii-b) that d > 2k, where the corresponding d = 2(a + b + c) and k = 2b 1 in P 2,2,2(a+b+c) 2a,2(a+b) Then by lemma 4(ii-b) we have for r = a + b + c that ρ ( P 2,2,2(a+b+c) ) ( 2a,2(a+b) >ρ P 2,2,2(a+b+c) 0,2b N 2(a+b+c+1) ) ( a+b+c 1 = ρ N 2(r+1) ) r 1 (5) Note that ρ ( N 2(r 1) ) ( r 1 <ρ N 2(r+1) ) ( r 1 <ρ N 2(r+) ) r+ (since Lemma 22 holds also for ρ(g)) Then α = r ρ ( N 2(r 1) ) r 1 r ρ ( N 2(r+1) ) r 1 r ρ ( N 2(r+) ) r+ = α implying r ρ ( N 2(r+1) ) ( r 1 = α, and and thus ρ P 2,2,2(a+b+c) ) 2a,2(a+b) α meaning κ(q a,b,c ) α 2 = ɛ + 2 Hence, it contradicts the condition of (ii) This finishes the proof Remark 2 The above theorem nearly gives a necessary and sufficient condition for a, b and c such that κ(q a,b,c )<ɛ+ 2, however it can be that for some particular values of a, b and c with b < a + c + 1we have that κ(q a,b,c )<ɛ+ 2 Computer calculations confirm that if b a + c then κ(q a,b,c )>ɛ+ 2 We will then pose the following conjecture Conjecture 1 κ(q a,b,c )<ɛ+ 2 if and only if b a + c + 1 In order to prove the above conjecture it can be useful to rely on subdivision graphs By Theorem 21, we have that κ(q 1,k 1,k 1 )>κ(q 1,k,k 2 ) (6) for any k N Then if one proves that {κ(q 1,k 1,k 1 )} k N is a monotone decreasing sequence then κ(q 1,k 1,k 1 )>ɛ+ 2 for any k and Conjecture 1 holds by (), Lemma 4(ii) and Theorem 21 Clearly (6) is equivalent to ρ ( P 2,2,2(2k 1) 2,2k S(Q 1,k 1,k 1 ) ) >ρ ( P 2,2,2(2k 1) ) ( 1,2k 1 >ρ P 2,2,2(2k 1) 2,2(k+1) S(Q 1,k,k 2 ) ) Then Conjecture 1 can be expressed by the following: Conjecture 2 Let G k = P 2,2,2(2k 1) 1,2k 1 {ρ(g k )} k N is a monotonic decreasing sequence, with k ρ(g k ) = α Furthermore ρ(g k )>αfor any k N Finally, the theorem below follows from Lemma 24 and Theorems 2, 5 and 6: )

11 172 J Wang et al / Linear Algebra and its Applications 41 (2009) Theorem 8 A graph belongs to G if and only if it has the same structures as those of graphs in S 1 \(G 1 G 2 ) S 2 4 The it point of the Q-index of graphs The study of it points of eigenvalues of adjacency matrices of graphs was initiated by Hoffman [1] See [15,16,20,11] for more details about the it points In this section, we investigate the it points of the Q-index of some classes of graphs From Lemma 21 we shall see that Theorems 1 in [16] are respectively the corollary of Theorems in this paper, since his theorems are only proved for the bipartite graphs We would mention, however, that we adopt Guo s and Hoffman s ideas to prove these three theorems Recall, for a given graph G, qi n Q(G) =ϕ(q(g), q) = ϕ(q(g)) = ϕ(g) will denote the signless Laplacian characteristic polynomial of G With ϕ(m, q) = ϕ(m), where M is a square matrix, we will denote the characteristic polynomial of M, when matrix M does not represent a graph The lemma below was first proved for the characteristic polynomial of the Laplacian matrix of a path [16] Since a path is bipartite, by Lemma 21 we get the following lemma: Lemma 41 Let a = q 2+ q 2 4q 2 and b = q 2 q 2 4q Then ϕ(p 2 n ) = q q 2 4q (an b n ) Let M n and N n be the matrices of order n obtained from Q(P n+1 ) and Q(P n+2 ) by deleting the row and column corresponding to some end vertex of the path P n+1 and by deleting the rows and columns corresponding to the two end vertices of P n+2, respectively Lemma 42 Let ϕ(p 0 ) = 0, ϕ(m 0 ) = 1 and ϕ(n 0 ) = 1 Then (i) ϕ(m n ) = 1 q ϕ(p n+1) + 1 q ϕ(p n) = 1 q 2 4q (an+1 + a n b n+1 b n ); (ii) ϕ(p n+1 ) = (q 2) ϕ(p n 1 ), where n 1; (iii) ϕ(n n ) = 1 q ϕ(p n+1) = 1 q 2 4q (an+1 b n+1 ); (iv) ϕ(c n ) = 1 q ϕ(p n+1) 1 q ϕ(p 2 = 1 q 2 4q (an+1 b n+1 a n 1 + b n 1 2 q 2 4q) Proof The second equality in each item follows from the first equality and Lemma 41 Now we prove the first equalities From the property of determinant it follows that q q ϕ(m n ) = 0 1 q 2 0 = ϕ(m n 1 ) (7) q 1 From Theorem 22 we get ϕ(p n+1 ) = (q 1) qϕ(m n 1 ), (8) which, together with (7), yields that (i) holds Clearly, (ii) follows from (i) and (8) By induction on n we show (iii) The result is obvious for n = 1, 2 Assume that n By expanding determinant ϕ(n n ) by the first row we get ϕ(n n ) = (q 2)ϕ(N n 1 ) ϕ(n n 2 ) From (ii) and the inductive hypothesis we arrive at ϕ(n n ) = (q 2)ϕ(N n 1 ) ϕ(n n 2 ) = 1 q [(q 2)ϕ(P n) ϕ(p n 1 )] = 1 q ϕ(p n), which ends the proof of (iii) Now we prove the last one Expanding the determinant ϕ(c n ) by the first row, then expanding the two determinants obtained by the first column we obtain that

12 J Wang et al / Linear Algebra and its Applications 41 (2009) q q ϕ(c n ) = 0 1 q q q = (q 2)ϕ(N n 1 ) q q 2 1 q q = (q 2)ϕ(N n 1 ) 2ϕ(N n 2 ) 2 (substitute (iii) into this equality) = q 2 2 q q ϕ(p 2 = 1 q [(q 2)ϕ(P n) ϕ(p n 1 ) ϕ(p n 1 )] 1 = 1 q ϕ(p n+1) 1 q ϕ(p 2 This completes the proof of the lemma Remark 41 It is a remarkable fact that the above equalities can be obtained in an undirect way through (2), () and Lemma 29 Theorem 41 Let G u (C n )(or simply G(C n )) be the graph obtained from two vertex-disjoint graphs G and C n by adding a new edge joining a vertex u of G with a vertex v of C n Then (i) n κ(g u (C n )) = τ u (G) exists and τ u (G) >4 Indeed, τ u (G) is the it point of the Q-index of graph sequence {G u (C n )} (ii) τ u (G) is the largest root of the equation 1 b b2 ϕ(g) ϕ(q 1 b 2 u (G)) = 0, where b = q 2 q 2 4q 2 Proof From Theorem 21(iii) and Lemma 26(ii) we respectively get κ(g u (C n )) > κ(g u (C n+1 )) and κ(g u (C n )) Δ(G u (C n )) + 1, which implies that n κ(g u (C n )) = τ u (G) exists We obtain from Lemma 26(ii) that τ u (G) >4 This completes the proof of (i) Next we show (ii) From Theorem 22 and Q v (C n ) = N n 1 we get ϕ(g u (C n )) = ϕ(g)ϕ(c n ) ϕ(g)ϕ(q v (C n ) ϕ(c n )ϕ(q u (G)) ( = ϕ(c n ) ϕ(g) ϕ(q u (G)) ϕ(g) ϕ(n ) ϕ(c n ) From Lemma 42(ii) and (iv), we obtain for q > 4 (which implies that a > 1 and b = 1 < 1) a ϕ(n n 1 ) a n b n 1 a = = = b n ϕ(c n ) n a n+1 b n+1 a n 1 + b n 1 2 q 2 4q b 2 a 2

13 174 J Wang et al / Linear Algebra and its Applications 41 (2009) Hence, we arrive at ( n ϕ(g) ϕ(q u (G)) ϕ(g) ϕ(n ϕ(c n ) ) b = ϕ(g) ϕ(q u (G)) ϕ(g) 1 b 2 = 1 b b2 ϕ(g) 1 b 2 ϕ(q u (G)) Since τ u (G) >4, we know that τ u (G) is the largest positive root of the following equation: ( ϕ(g) ϕ(q n u (G)) ϕ(g) ϕ(n ) = 0 ϕ(c n ) This completes the proof ( ( Recall that ɛ = 1 54 ) ( ) 1 ) Corollary 41 Let G = K 1, an isolated vertex u, in Theorem 41 Then n κ(k 1(C n )) = ɛ + 2 and κ(k 1 (C n )) > ɛ + 2 Proof By Theorem 41(i) we get that n κ(k 1 (C n )) exists By Lemma 26(ii) κ(k 1 (C n )) > 4 implying n κ(k 1 (C n )) > 4 Since ϕ(k 1 ) = q and ϕ(q u (K 1 )) = ϕ(m 0 ) = 1 (see the condition of Lemma 42), by Theorem 41(ii), we get n κ(k 1 (C n )) is the largest root of the following equation: 1 b b 2 1 b 2 ϕ(k 1 ) ϕ(q u (K 1 )) = 0 that is 1 b b 1 b 2 q 1 = 0, where b = q 2 q 2 4q By computation we get that the largest root of the above equation is ɛ + 2 So 2 we have n κ(k 1 (C n )) = ɛ + 2 By Theorem 21(iii), we have that κ(k 1 (C n )) is strictly decreasing when n increases, which indicates that κ(k 1 (C n )) > ɛ + 2 It has been proved that the Laplacian eigenvalues of the path P n are (see [1]) which, together with Lemma 21, yields the following lemma Lemma 4 Let P n be a path of order n Then n κ(p n ) = 4 2 { 4 sin 2 r 2n π,0 r n 1 } Theorem 42 Let G u (P n )(or simply G(P n )) be the graph obtained from two vertex-disjoint graphs G and P n by adding a new edge joining a vertex u of G with an end vertex of P n Then (i) n κ(g u (P n )) = τ u (G) exists and τ u (G) 4 Indeed, τ u (G) is the it point of the Q-index of graph sequence {G u (P n )} (ii) If τ u (G) >4, then τ u (G) is the largest root of the equation q+ q 2 4q 2q ϕ(g) ϕ(q u (G)) = 0 Proof By Lemmas 22 and 26(i) we respectively get that κ(g u (P n )) < κ(g u (P n+1 )) and κ(g u (P n )) 2Δ(G u (P n )) Thus these two inequalities implies that n κ(g u (P n )) = τ u (G) exists If G u (P n ) is a path, then by Lemma 4 we get τ u (G) = 4 Otherwise, we obtain from Lemma 26(ii) that τ u (G) >4 This completes the proof of (i) Next we show (ii) From Theorem 22 we get ϕ(g u (P n )) = ϕ(g) ϕ(q u (G)) ϕ(g)ϕ(m n 1 ) ( = ϕ(g) ϕ(q u (G)) ϕ(g) ϕ(m )

14 J Wang et al / Linear Algebra and its Applications 41 (2009) From Lemmas 41 and 42(i) we obtain for q > 4 (which implies that a > 1 and b = 1 < 1) that a ϕ(m n 1 ) a n + a n 1 b n b n 1 = = a = q q 2 4q n n q(a n b n (9) ) q 2q Hence, we arrive at ( n ϕ(g) ϕ(q u (G)) ϕ(g) ϕ(m ) = ϕ(g) ϕ(q u (G)) ϕ(g) q q 2 4q 2q = q + q 2 4q ϕ(g) ϕ(q u (G)) 2q Since τ u (G)( >4, we know that τ u (G) is the largest positive root of the following equation: ϕ(g) ϕ(q n u (G)) ϕ(g) ϕ(m ) = 0 This ends the proof Corollary 42 Let G = K 1,2 and u the center of K 1,2 in Theorem 42 Then n κ(k 1,2(P n )) = and κ(k 1,2 (P n )) < Proof From Theorem 42(i) we know that n κ(k 1,2 (P n )) exists A direct calculation shows that κ(k 1,2 (P 2 )) > 4 and so n κ(k 1,2 (P n )) > 4 From Theorem 42(ii) it follows that n κ(k 1,2 (P n )) is the largest root of the following equation: q + q 2 4q ϕ(k 1,2 ) ϕ(q u (K 1,2 )) = 0 2q By substituting ϕ(k 1,2 ) = q(q 1)(q ) and ϕ(q u (K 1,2 )) = (q 1) 2 into the above equation, we obtain by a straightforward calculation that the largest root of the above equation is Theorem 21(ii) shows thatκ(k 1,2 (P n )) is strictly increasing when n increases, which indicates thatκ(k 1,2 (P n )) < Theorem 4 Let G u (P n, P n )(or simply G(P n, P n )) be the graph obtained from a connected non-trivial graph G and two paths P n by adding two new edges joining vertex u of G to vertices u 1 and u 2, respectively, where u 1 is an end vertex of a P n and v 2 is an end vertex of the other P n Then (i) n κ(g u (P n, P n )) = τ u (G) exists and τ u (G) >4 Indeed, τ u (G) is the it point of the Q- index of graph sequence {G u (P n, P n )} (ii) τ u (G) is the largest positive root of the equation q+ q 2 4q 4q ϕ(g) ϕ(q u (G)) = 0 Proof By a similar method as that of Theorem 42, (i) holds Now let us consider (ii) From Corollary 21, weget ϕ(g u (P n, P n )) = ϕ(g)( ϕ(m n 1 )) 2 2ϕ(Q u (G))( ϕ(m n 1 )) ( = 2 ϕ(g) 1 ϕ(m ) 2 ( 2ϕ(Q u (G)) 1 ϕ(m ) Since τ u (G) >4, then τ u (G) is the largest positive root of the following equation: ( ϕ(g) 1 ϕ(m ) 2 ( 2ϕ(Q n u (G)) 1 ϕ(m ) = 0

15 176 J Wang et al / Linear Algebra and its Applications 41 (2009) From (9), for q > 4wearriveat ( ϕ(g) 1 ϕ(m ) 2 ( 2ϕ(Q n u (G)) 1 ϕ(m ) ( ) 2 q + q 2 4q = ϕ(g) 4q 2 ϕ(q u (G) q + q 2 4q q = q + q 2 4q q + q 2 4q ϕ(g) ϕ(q u (G), q 4q which implies that (ii) holds Corollary 4 Let G = P 2 and u be a vertex of P 2 in Theorem 4 Then n κ(p 2(P n, P n )) = ɛ + 2 and κ(p 2 (P n, P n )) < ɛ + 2 Proof From Theorem 4(i) we know that n κ(p 2 (P n, P n )) exists A direct calculation shows that κ(p 2 (P 2, P 2 )) > 4 and so n κ(p 2 (P n, P n )) > 4 From Theorem 4(ii) it follows that n κ(p 2 (P n, P n )) is the largest root of the following equation: q + q 2 4q ϕ(p 2 ) ϕ(q u (P 2 )) = 0 4q By substituting ϕ(p 2 ) = q(q 2) and ϕ(q u (P 2 )) = q 1 into the above equation, we obtain by a straightforward calculation that the largest root of the above equation is ɛ + 2 By Theorem 21(ii) we have that κ(p 2 (P n, P n )) is strictly increasing when n increases, which indicates that κ(p 2 (P n, P n )) < ɛ + 2 Theorem 44 Let X and Y be two vertex-disjoint connected graphs, x a vertex of X and y a vertex of Y Let G = G xy (X, Y; P n ) be the graph obtained from X, Y and a new path P n, v 1 v 2 v n, of length n 1 by adding edges xv 1 and yv n Then κ(g) = max n { n κ(x x(p n )), n κ(y y(p n )) Proof The existence of n κ(x x (P n )) and n κ(y y (P n )) follows from Theorem 42(i) If v 1 v 2 v n is an internal path of G xy (X, Y; P n ), then from Lemma 26(ii) and Theorem 21(iii) we know that n κ(g) exists Otherwise, we get that X x (P n ) or Y y (P n ) is a path So by Theorem 42(i) again, n κ(g) also exists If n κ(g) = 4, then G is a path for any n, and so X x (P n ) and Y y (P n ) are paths Then n κ(g) = n κ(x x (P n )) = n κ(y y (P n )) = 4, and the theorem clearly holds If n κ(g) >4, then applying Theorem 22 twice we obtain ϕ(g) = ϕ(x x (P n ))ϕ(y) ϕ(x x (P n ))ϕ(q y (Y)) ϕ(y)ϕ(q vn (X x (P n ))) = [(ϕ(x) ϕ(q x (X))) ϕ(x)ϕ(m n 1 )] [ ϕ(y) ϕ(q y (Y)) ] ϕ(g 2 ) [ϕ(m n 1 )(ϕ(x) ϕ(q x (X))) ϕ(x)ϕ(n n 2 )] Substituting Lemma 42(iii) into the above equality, we get [( ϕ(g) = ϕ(x) ϕ(q x (X)) ϕ(x) ϕ(m ) (ϕ(y) ϕ(q y (Y))) ϕ(y) ϕ(m ( ϕ(x) ϕ(q x (X)) ϕ(x) ϕ(p )] qϕ(m n 1 ) g n (q) }

16 J Wang et al / Linear Algebra and its Applications 41 (2009) Since n κ(g) >4, then n κ(g) is the largest positive root of the equation n g n(q) = 0 From Lemma 41 and Lemma 42(i), we have for q > 4 (implying a > 1 and b < 1) that ϕ(p n 1 ) n xϕ(m n 1 ) = n a n 1 b n 1 a n + a n 1 b n b n 1 = = q q 2 4q h(q) (10) a 2q By (9) and (10), we get for q > 4 that n g n(q) = (ϕ(x) ϕ(q x (X)) h(q)ϕ(x)) ( ϕ(y) ϕ(q y (Y)) ) h(q)ϕ(y) (ϕ(x) ϕ(q x (X)) h(q)ϕ(x)) = (ϕ(x) ϕ(q x (X)) h(q)ϕ(x)) ( ϕ(y) ϕ(q y (Y)) h(q)ϕ(y) ) = q + q 2 4q ϕ(x) ϕ(q x (X)) q + q 2 4q ϕ(y) ϕ(q y (Y)), 2q 2q which indicates from Theorem 42(ii) that the theorem holds Acknowledgement The authors are grateful to the referee for his many valuable comments and suggestions which led to a significant improvement of the paper References [1] WN Anderson, TD Morley, Eigenvalues of the Laplacian of a graph, Linear and Multilinear Algebra, 18 (1985) [2] F Belardo, EM Li Marzi, SK Simić, Path-like graphs ordered by the index, Int J Algebra 1 (2007) [] AE Brouwer, A Neumaier, The graphs with spectral radius between 2 and 2 + 5, Linear Algebra Appl (1989) [4] DS Cao, Y Hong, Graphs characterized by the second largest eigenvalue, J Graph Theory 17 () (199) 25 1 [5] DS Cao, Y Hong, The distribution of eigenvalues of graphs, Linear Algebra Appl 216 (1995) [6] D Cvetković, M Doob, H Sachs, Spectra of Graphs - Theory and Applications, III revised and enlarged edition, Johan Ambrosius Bart Velarg, Heidelberg - Leipzig, 1995 [7] DM Cvetković, M Doob, I Gutman, On graphs whose spectral radius does not exceed (2 + 5) 1/2, Ars Combin 14 (1982) [8] D Cvetković, P Rowlinson, SK Simić, Signless Laplacians of finite graphs, Linear Algebra Appl 42 (2007) [9] D Cvetković, P Rowlinson, SK Simić, Eigenvalue bounds for the signless Laplacians, Publ Inst Math (Beograd) 81 (95) (2007) [10] D Cvetković, SK Simić, Towards a spectral theory of graphs based on signless Laplacian: I, Publications de L Institut Mathematique Nouvelle série tome, vol 85(99), 2009, pp 1 15 [11] M Doob, The it points of eigenvalues of graphs, Linear Algebra Appl (1989) [12] AJ Hoffman, JH Smith, On the spectral radii of topological equivalent graphs, in: M Fiedker (Ed), Recent Advances in Graph Theory, Academia Praha, 1975, pp [1] AJ Hoffman, On it points on spectral radii of non-negative symmetric integral matrices, in: Y Alavi (Ed), et al, Lecture Notes Math, vol 0, Springer-Verlag, Berlin, 1972, pp [14] LH Feng, Q Li, XD Zhang, Minimizing the Laplacian spectral radius of trees with given matching number,linear Multilinear Algebra 55 (2007) [15] JM Guo, The it points of Laplacian spectra of graphs, Linear Algebra Appl 62 (200) [16] JM Guo, On it points of Laplacian spectral radii of graphs, Linear Algebra Appl 429 (2008) [17] R Grone, R Meris, VS Sunder, The Laplacian spectrum of a graph, SIAM J Matrix Anal Appl 11 (1990) [18] A Neumaier, The second largest eigenvalue of a tree, Linear Algebra Appl 46 (1982) 9 25 [19] AJ Schwenk, Computing the characteristic polynomial of a graph, Graphs Combin, Notes Math 406 (1974)

17 178 J Wang et al / Linear Algebra and its Applications 41 (2009) [20] J Shearer, On the distribution of the maximum eigenvalue of graphs, Linear Algebra Appl (1989) [21] JH Smith, Some properties of the spectrum of a graph, Combinatorial Structures and their Applications, Gordon and Breach, New York, 1970, pp [22] JF Wang, QX Huang, X An, F Belardo, Some notes on graphs whose spectral radius is close to 2 2, Linear Algebra Appl 429 (2008) [2] R Woo, A Neumaier, On graphs whose spectral radius is bounded by 2 2, Graphs Combin 2 (2007) [24] RA Horn, CR Johnson, Matrix Analysis, Cambridge University Press, England, 1986