The Signless Laplacian or Adjacency Spectral Radius of Bicyclic

Graphs and Combinatorics DOI 10.1007/s00373-014-1477-2 ORIGINAL PAPER

The Signless Laplacian or Adjacency Spectral Radius of Bicyclic Graphs with Given Number of Cut Edges Zhen-Mu Hong · Yi-Zheng Fan

Received: 2 November 2013 / Revised: 30 July 2014 © Springer Japan 2014

Abstract Let B(n, r ) be the set of all bicyclic graphs with n vertices and r cut edges. In this paper we determine the unique graph with maximal adjacency spectral radius or signless Laplacian spectral radius among all graphs in B(n, r ). Keywords Bicyclic graph · Signless Laplacian · Adjacency matrix · Spectral radius · Cut edge Mathematics Subject Classification

05C50, 15A18

1 Introduction Let G = (V, E) be a simple graph with vertex set V = V (G) = {v1 , v2 , . . . , vn } and edge set E = E(G). The adjacency matrix of G is defined to be a (0, 1)-matrix A(G) = [ai j ]n×n , where ai j = 1 if vi is adjacent to v j , ai j = 0 otherwise. Denote by D(G) = diag(dG (v1 ), dG (v2 ), . . . , dG (vn )), the diagonal matrix of vertex degrees, where dG (vi ) denotes the degree of vi . The matrix Q(G) = D(G)+ A(G) is called the signless Laplacian matrix of G (for example, see [11]), which may be first introduced

Supported by National Natural Science Foundation of China (11071002, 11371028), Program for New Century Excellent Talents in University (NCET-10-0001), Key Project of Chinese Ministry of Education (210091), Specialized Research Fund for the Doctoral Program of Higher Education (20103401110002), Scientific Research Fund for Fostering Distinguished Young Scholars of Anhui University (KJJQ1001). Z.-M. Hong · Y.-Z. Fan (B) School of Mathematical Sciences, Anhui University, Hefei 230601, People’s Republic of China e-mail: [email protected] Z.-M. Hong e-mail: [email protected]

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in the book [8] and is also called an unoriented Laplacian matrix ([18]), or a Q matrix ([11]). Denote by ρ A (G) (respectively, ρ Q (G)) the largest eigenvalue or the spectral radius of A(G) (respectively, Q(G)) and call it the adjacency spectral radius (respectively, signless Laplacian spectral radius) of G. If, in addition, G is connected, then A(G) is irreducible; and by Perron-Frobenius Theorem, ρ A (G) is simple and there exists a unique (up to a multiple) corresponding positive eigenvector, usually called the Perron vector of A(G). Similar results holds for Q(G). There are many results in literatures concerning the spectral radius of the adjacency matrix of a graph, which are involved with the work in two directions: one for the bounds of spectral radius (see e.g. [15,30,40]), and one for the structure of graphs with extreme spectral radius subject to one or more given parameters, such as order and size [3,4,31,32], number of cut vertices [2], number of cut edges [26], number of pendant vertices [38]. One can also refer to [8,9] for basis results on the spectrum of adjacency matrix of a graph. Recently the signless Laplacian spectrum of a graph has attracted the attention of researchers. The papers [11–14] give a survey on this work. The bounds of signless Laplacian spectral radius can be found in [10,20,27,37], and the relations between the spectral radius and graph parameters are discussed in [5,18,19,25,36,39,42]. The least signless Laplacian eigenvalues is also studied; see e.g. [6,17]. Other work can be found in [16,34] for the spectral integrality, [29,35] for the isospectral problem, and [28] for the spectral spread. For more papers, one can refer to [14] and the references therein. A tree of order n contains exactly n − 1 cut edges (that is, each edge is a cut edge). For unicyclic graphs of order n, fixing number of cut edges means fixing girth (i.e. the minimal length of cycles in the graph). But this does not holds for bicyclic graphs. Belardo et al. [1] characterize the maximum adjacency spectral radius among unicyclic graphs subject to fixed girth. Zhai et al. [41] determined the unique graph with the maximal adjacency spectral radius among bicyclic graphs with fixed girth. In this paper, we consider the adjacency spectral radius of bicyclic graphs with given number of cut edges. Using the relation between the signless Laplacian spectrum of a graph and the adjacency spectrum of the subdivision of the graph, we get a corresponding result on the signless Laplacian spectral radius of bicyclic graphs with given number of cut edges.

2 Preliminaries Let G be a graph. Denote by (G) and δ(G) the maximum degree and the minimum degree of vertices of G, respectively. A pendant vertex of G is a vertex of degree 1, and a pendant edge of G is one incident to a pendant vertex. A cut edge of a connected graph is one whose deletion yields the resulting graph disconnected. A connected graph with n vertices and n + 1 edges is called bicyclic. Denote by B(n, r ) the set of bicyclic graphs with n vertices and r cut edges. Let uv be an edge of a connected graph G and let G u,v be the graph obtained from G by subdividing the edge uv, or obtained from G − uv by adding a new vertex w together with two edges wu, wv. The subdivision graph S(G) of a graph G is obtained

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from G by subdividing all edges of G. An internal path is a path or a cycle, in which the initial and terminal vertices have degree at least 3 and the internal vertices have degree 2. Denote by K n , Cn and Pn the complete graph, the cycle and the path all on n vertices respectively, and by K m,n for the complete bipartite graph with bipartition of the vertex set consisting of two subsets of size m, n respectively. Let G be a graph on n vertices. A vector X ∈ Rn is called to be defined on G, if there is a 1–1 map ϕ from V (G) to the entries of X ; simply written X (u) = ϕ(u) for each u ∈ V (G). We often say X (u) is a value of the vertex u given by X . If X is an eigenvector of A(G) or Q(G), then it is naturally defined on V (G), i.e. X (u) is the entry of X corresponding to the vertex u. One can find that X T A(G)X = 2

X (u)X (v), X T Q(G)X =

uv∈E(G)

[X (u) + X (v)]2 , (2.1)

uv∈E(G)

and λ is an eigenvalue of A(G) or Q(G) corresponding to an eigenvector X if and only if x = 0 and λX (v) =

X (u),

for each v ∈ V (G),

(2.2)

u∈N G (v)

or [λ − dG (v)]X (v) =

X (u),

for each v ∈ V (G),

(2.3)

u∈N G (v)

where N G (v) denotes the neighborhood of v in G. Denote by PA (G, λ) = det(λI − A(G)) and PQ (G, λ) = det(λI − Q(G)) the characteristic polynomial of A(G) and Q(G) respectively, where I denote the identity matrix of suitable size. As noted in [10], for a graph with n vertices and m edges, the following formula appears implicitly in the literature (see e.g., [8, p.63] or [43]): PA (S(G), λ) = λm−n PQ (G, λ2 ).

(2.4)

Hence, ρ A (S(G)) = ρ Q (G). At the last of this section, we list some results which will be used in Sect. 3 of this paper. Lemma 2.1 [38] Let G be a connected graph containing two vertices u, v, and let X be a Perron vector of A(G). Suppose that v1 , v2 , . . . , vs are vertices in N G (v)\(N G (u) ∪ be the graph obtained from G by deleting the edges vvi and adding the {u}). Let G > ρ A (G). edges uvi for i = 1, 2, . . . , s. If X (u) ≥ X (v), then ρ A (G) Lemma 2.2 [23] Let G be a connected graph containing two vertices u, v, and let X be a Perron vector of Q(G). Suppose that v1 , v2 , . . . , vs are vertices in N G (v)\(N G (u) ∪ be the graph obtained from G by deleting the edges vvi and adding the {u}). Let G > ρ Q (G). edges uvi for i = 1, 2, . . . , s. If X (u) ≥ X (v), then ρ Q (G)

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Denote by Wn (n ≥ 6) the tree on n vertices obtained from a path Pn−4 by attaching two new pendent edges to each of its endvertices. Lemma 2.3 [22] Let G be a connected graph with uv ∈ E(G). If uv belongs to an internal path of G and G = Wn , then ρ A (G u,v ) < ρ A (G). Lemma 2.4 [12] Let G be a connected graph with uv ∈ E(G). If uv belongs to an internal path of G, then ρ Q (G u,v ) < ρ Q (G). Lemma 2.5 [11] Let G be a graph. Then min [dG (u) + dG (v)] ≤ ρ Q (G) ≤ max [dG (u) + dG (v)].

uv∈E(G)

uv∈E(G)

If in addition G is connected, then either equality holds if and only if G is regular or semi-regular bipartite. Lemma 2.6 [10] If G is a connected graph of order n ≥ 4 and with maximum degree (G), then ρ Q (G) ≥ (G) + 1 with equality if and only if G = K 1,n−1 . Lemma 2.7 [8] Let G be a graph containing a vertex u and let C (u) be the set of all cycles of G containing u. Then PA (G, λ) = λPA (G −u, λ)−

PA (G −u −v, λ)−2

v∈N G (u)

PA (G −V (Z ), λ).

Z ∈C (u)

At the end of this section, we compare the adjacency or signless Laplacian spectral radius of two bicyclic graphs of order n with n − 5 cut edges. Denote by G(3, 3; n − 5) the graph obtained from two triangles sharing a common vertex by attaching n − 5 n−5 the graph obtained from K 2,3 by pendant edges to the common vertex, and K 2,3 attaching n − 5 pendant edges to a vertex of degree 3, where n ≥ 5; see Fig. 1. Proposition 2.8 The adjacency or signless Laplacian spectral radius of G(3, 3; n−5) n−5 is larger than that of K 2,3 , i.e., n−5 n−5 ), ρ Q (G(3, 3; n − 5)) > ρ Q (K 2,3 ). ρ A (G(3, 3; n − 5)) > ρ A (K 2,3

Proof By Lemma 2.7, ρ A (G(3, 3; n − 5)) is the maximum root of ϕ1 (λ) = λ6 − (n + 1)λ4 − 4λ3 + (2n − 5)λ2 + 4λ − (n − 5), n−5 and ρ A (K 2,3 ) is the maximum root of

ϕ2 (λ) = λ4 − (n + 1)λ2 + 3(n − 5).

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Fig. 1 Two bicyclic graphs with n − 5 cut edges n−5 In fact, ρ A (K 2,3 ) is also the maximum root of λ2 ϕ2 (λ). Observe that

ϕ1 (λ) − λ2 ϕ2 (λ) = −4λ(λ2 − 1) − (n − 10)λ2 − (n − 5). √ n−5 So, when λ ≥ ρ A (K 2,3 ) ≥ ρ A (K 2,3 ) = 6, ϕ1 (λ) − λ2 ϕ2 (λ) < 0, which implies n−5 that ρ A (G(3, 3; n − 5)) > ρ A (K 2,3 ). n−5 By Lemma 2.6, ρ Q (G(3, 3; n − 5)) > n. By Lemma 2.5, ρ Q (K 2,3 ) ≤ n. So n−5 ρ Q (G(3, 3; n − 5)) > ρ Q (K 2,3 ). 3 Characterization of the Extremal Graph Denote by G(n 1 , n 2 , . . . , n k ; r ) the graph obtained from k cycles Cn 1 , Cn 2 , . . . , Cn k sharing a common vertex by attaching r pendant edges to the common vertex, and by H (n 1 , n 2 , . . . , n k ; r ) the graph obtained from G(n 1 , n 2 , . . . , n k ; r ) by subdividing each of its pendant edges, where n 1 ≥ n 2 ≥ · · · ≥ n k ≥ 3; see Fig. 2. The technique used in the proof of Theorem 3.1 is adopted from [33] with a little modification. , n 2 , . . . , n k ) be the adjacency spectral radius of the Theorem 3.1 Let ρ = ρ A (n 1 k n i is fixed and all n ν s are fixed except, say n i and graph G(n 1 , n 2 , . . . , n k ; r ). If i=1 n j , then ρ is strictly increasing in |n i − n j |. Proof For convenience, let G := G(n 1 , n 2 , . . . , n k ; r ). Let u be the common vertex of the cycles in G and v1 , v2 , . . . , vr be the pendant vertices of G. Let X = (X (v1 ), . . . , X (vr ), X (u), X 11 , . . . , X n11 −1 , . . . , X 1s , . . . , X ns s −1 , . . . , X 1k , . . . , X nk k −1 ) be the Perron vector of A(G), where X ij corresponds to the jth vertex on the cycle Cn i (counting along the cycle clockwise from u but except u) for i = 1, 2, . . . , k, j = 1, 2, . . . , n i − 1.

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Fig. 2 The graphs G(n 1 , n 2 , . . . , n k ; r ) and H (n 1 , n 2 , . . . , n k ; r )

Noting that X > 0 as X is the Perron vector of A(G). So by (2.2) on pendant vertices vi ’s, we get ρ X (vi ) = X (u), i = 1, 2, . . . , r.

(3.1)

By symmetry (or see [21, Lemma 3.3, p.166]) we have X is = X ns s −i , i = 1, . . . , n s − 1; s = 1, . . . , k.

(3.2)

In addition the vertex u can be considered as an initial or terminal vertex of each cycle. So for convenience we denote X (u) as X 01 = X 02 = · · · = X 0k = X (u) = X n11 = X n22 = · · · = X nk k .

(3.3)

Thus, applying (2.2) to the vertices of each cycle, we have s s X i+2 − ρ X i+1 + X is = 0,

i = 0, . . . , n s − 2;

(3.4)

and applying (2.2) to the vertex u we get X (u) ρ X (u) = 2 X 11 + X 12 + · · · + X 1k + r . ρ

(3.5)

The last condition (3.5) may be regarded as boundary conditions for (3.4). Solving (3.4) for a fixed s, we get X is = ps β i + qs β −i , where β = (ρ +

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ρ 2 − 4)/2 > 1.

(3.6)


From (3.2) it follows that qs = ps β n s , and therefore we get X is = ps β i + β n s −i .

(3.7)

From (3.3) X (u) = p1 1 + β n 1 = p2 1 + β n 2 = · · · = pk 1 + β n k so we have ps = c

k

1 + βn j .

(3.8)

j=1 j=s

Thus, if c from (3.8) is normalized to 1, X is =

k

β i + β n s −i 1 + βn j , i = 0, 1, . . . , n s . 1 + β ns

(3.9)

j=1

It is easy to see that (3.8) is valid even if s = k and n k = 1. Putting ρ = 2 cosh(2t) (t > 0) or equivalently β = e2t in (3.6), after some usual transformations on (3.5) we get k r cosh(n s − 2)t − = 0. cosh(2t) − cosh(n s t) 4 cosh(2t)

(3.10)

s=1

Now, we shall establish the behavior of ρ = ρ A (n 1 , n 2 , . . . , n k ) under the assumptions of the theorem. Since ρ = 2 cosh(2t) (t > 0), we should pay attention to t. Consequently, we will examine t = t (n 1 , . . . , n i , . . . , n j , . . . , n k ) allowing only n i and n j to change, while keeping their sum fixed (n i + n j = d, for convenience); if so, t depends only on n i and d − n i , or, in other words, it is a function of |n i − n j |. Let k r cosh(n s − 2)t − . F(n 1 , n 2 , . . . , n k , t) := cosh(2t) − cosh(n s t) 4 cosh(2t)

(3.11)

s=1

Deriving F with respect to t and n i we get n s sinh(2t)+2 cosh(n s t) sinh(n s −2)t ∂F r sinh(2t) + > 0, = 2 sinh(2t) + ∂t cosh2 (n s t) 2 cosh2 (2t) k

s=1

(3.12)

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and 1 ∂F 1 = t sinh(2t) − . ∂n i cosh2 (n i t) cosh2 (d − n i )t Since

(3.13)

∂F ∂F ∂t , we conclude =− / ∂n i ∂n i ∂t ∂t ∂t > 0 for n i > n j , < 0 for n i < n j . ∂n i ∂n i

(3.14)

Since ρ = 2 cosh(2t), from (3.14) we conclude that ρ is an increasing function in |n i − n j |, completing the proof of theorem. Corollary 3.2 Let ρ be the adjacency spectral radius of G(k, l; r ). If k ≥ l ≥ 4, then ρ A (G(k, l; r )) < ρ A (G(k + 1, l − 1; r )). Theorem 3.3 Let ρ = ρ A (n 1 , n 2 , . . . , n k ) be the adjacency spectral radius of the k n i is fixed and all n ν s are fixed except, say n i graph H (n 1 , n 2 , . . . , n k ; r ). If i=1 and n j , then ρ is an increasing function in |n i − n j |. Proof By a similar processing as in the proof of Theorem 3.1, we can complete the proof of theorem, where the corresponding parts of (3.1), (3.5), (3.9), (3.10) and (3.11) are listed in the following: (ρ 2 − 1)X (vi ) = X (u),

i = 1, 2, . . . , r,

ρ X (u) = 2(X 11 + · · · + X 1k ) + r

cosh(2t) −

ρ X (u) , ρ2 − 1

k r cosh(2t) cosh(n s − 2)t − = 0, cosh(n s t) 4 cosh2 (2t) − 1 s=1

F(n 1 , n 1 , . . . , n k , t) = cosh(2t) −

k cosh(n s −2)t s=1

cosh(n s t)

−

r cosh(2t) , 4 cosh2 (2t)−1

n s sinh(2t) + 2 cosh(n s t) sinh(n s − 2)t ∂F = 2 sinh(2t) + ∂t cosh2 (n s t)

(3.1 )

(3.5 )

(3.10 )

(3.11 )

k

(3.12 )

s=1

+

2r sinh(2t)(4 cosh2 (2t) + 1) > 0. (4 cosh2 (2t) − 1)2

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Corollary 3.4 Let ρ be the adjacency spectral radius of H (k, l; r ). If k ≥ l ≥ 4, then ρ A (H (k, l; r )) < ρ A (H (k + 1, l − 1; r )). Corollary 3.5 Let μ be the signless Laplacian spectral radius of G(k, l; r ). If k ≥ l ≥ 4, then ρ Q (G(k, l; r )) < ρ Q (G(k + 1, l − 1; r )). Proof Let S1 = S(G(k, l; r )) and S2 = S(G(k + 1, l − 1; r )). It is easy to know that S1 ∼ = H (2k, 2l; r ) and S2 ∼ = H (2k + 2, 2l − 2; r ). By Corollary 3.4, we have ρ A (S1 ) < ρ A (S2 ) and consequently by (2.4) ρ Q (G(k, l; r )) < ρ Q (G(k + 1, l − 1; r )) as required. Next, we shall prove that G(n − r − 2, 3; r ) is the unique graph with maximal adjacency spectral radius or signless Laplacian spectral radius among all graphs in B(n, r ) for each r (0 ≤ r ≤ n − 5), respectively. Theorem 3.6 Let G ∈ B(n, r ), where 0 ≤ r ≤ n − 5. Then ρ A (G) ≤ ρ A (G(n − r − 2, 3; r )), with equality if and only if G = G(n − r − 2, 3; r ). Proof Suppose that G has maximal adjacency spectral radius among all graphs in B(n, r ). Let X be a unit Perron vector of A(G). We assert that any two cycles of G share at least one common vertex. Otherwise, G is obtained from two unicyclic graphs, say G 1 and G 2 , connected by a path, say P with end-vertices u, w, where u, w belong to the cycles of G 1 , G 2 respectively. We may assume that X (u) ≥ X (w). Now attaching the graph G 2 to the vertex u, we get a graph G ∗ ∈ B(n, r ). However by Lemma 2.1, ρ A (G ∗ ) > ρ A (G), a contradiction. Now, we distinguish the following two cases: Case 1. Two cycles of G share exactly one common vertex, say u. In this case G contains G ∞ as an induced subgraph; see Fig. 3. Thus G can be considered as two unicyclic subgraph, say G 1 and G 2 , sharing with the vertex u. We will prove that

Fig. 3 The graphs G ∞ and G θ

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X (u) > X (v) for any other vertex v of G. Assume, on the contrary, there exists a vertex v (v = u) such that X (v) ≥ X (u). We may assume that v belongs to G 1 , and be the graph obtained from G 1 by attaching G 2 at the vertex v. By Lemma 2.1, let G > ρ A (G), a contradiction. ρ A (G) Next we show that each pendant edge of G is incident to u. Otherwise, attaching this pendant edge to u, we obtain a new graph in B(n, r ) but with larger spectral radius, a contradiction. Hence G = G(k, l; r ) for some k, l. By Corollary 3.2, we have G = G(n − r − 2, 3; r ). Case 2. Any two cycles of G share more than one vertices. In this case there exists an induced subgraph G θ of G which is obtained from three vertex disjoint paths P1 , P2 and P3 (at most one of them is of length 1) by identifying their initial vertices and terminal vertices into u and w respectively; see Fig. 3. We assert that each of the three paths is of length < 3. Otherwise, we may assume that P1 has length ≥ 3. Let u 1 , w1 be the internal vertices of P1 adjacent to u, w respectively. Without loss of generality, assume that X (u) ≥ X (w). Now deleting the in B(n, r ) which holds that edge ww1 and adding a new edge uw1 , we get a graph G > ρ A (G) by Lemma 2.1, a contradiction. Therefore G θ is one of the following ρ A (G) graphs: one being two triangles sharing with one common edge, the another being the graph K 2,3 . But for the former case, the graph G has n − 4 cut edges, contradictory to the assumption. So G θ = K 2,3 . Let u, w be two vertices of degree 3 and v1 , v2 , v3 be three vertices of degree 2 in G θ . We first assert that X (vi ) < X (u) where 1 ≤ i ≤ 3. Otherwise, we may assume that X (v1 ) ≥ X (u). Deleting the edge uv2 and adding a new edge v1 v2 , we get a new graph in B(n, r ) but with larger spectral radius by Lemma 2.1, a contradiction. Similarly, X (vi ) < X (w) where 1 ≤ i ≤ 3. Without loss of generality, assume that X (u) ≥ X (w). Therefore G is obtained from G θ by attaching a tree T at u. Otherwise, removing each possible tree at v1 , v2 , v3 or w and attaching them to u, a new graph in B(n, r ) with larger spectral radius is obtained, a contradiction. Next, we prove that X (u) > X (v) for any other vertex v of T . Otherwise, assume be the graph obtained from T by attaching G θ at v does not hold the inequality, let G > ρ A (G), a contradiction. Again by Lemma 2.1, the vertex v. By Lemma 2.1, ρ A (G) n−5 . each pendant vertex of G is adjacent to u. Hence, G = K 2,3 n−5 But by Proposition 2.8, ρ A (K 2,3 ) < ρ A (G(3, 3; n − 5)), which implies case 2 can not happen. By above discussions, the proof is completed. By Lemma 2.2, Proposition 2.8 and Corollary 3.5, we get a similar result on the signless Laplacian spectral radius of a graph by a proof similar to the one used in Theorem 3.6. Theorem 3.7 Let G be a connected bicyclic graph with n vertices and r cut edges (0 ≤ r ≤ n − 5). Then ρ Q (G) ≤ ρ Q (G(n − r − 2, 3; r )) and the equality holds if and only if G = G(n − r − 2, 3; r ). Remark In case 2 of the proof of Theorem 3.6, we first assert G θ = K 2,3 which n−5 implies G is a bicyclic graph with girth 4, then we say G = K 2,3 . In fact, in [41,

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Graphs and Combinatorics n−5 Theorem t1.1], the authors have given the result that K 2,3 is the unique graph with maximal spectral radius among all bicyclic graphs on n vertices with girth 4. In order to keep the consistency of the proof in Theorem 3.6 and that in Theorem 3.7, therefore we provide the proof here. ∗∗ For every integer n ≥ 4, let K 1,n−1 be the graph obtained from the cycle C4 by first adding a chord and then attaching n − 4 pendant edges to one end of the chord. Brualdi ∗∗ is the unique graph among all bicyclic graphs of order and Solheid [4] proved K 1,n−1 ∗∗ is exactly the unique n whose adjacency spectral radius is maximal. Hence K 1,n−1 graph with maximal adjacency spectral radius among all graphs in B(n, n − 4). In ∗∗ is the unique graph with maximal signless Laplacian [18], Fan et.al. proved that K 1,n−1 spectral radius among all bicyclic graphs of order n, which is also the unique graph with maximal signless Laplacian spectral radius among all graphs in B(n, n − 4).

Theorem 3.8 Let G be a graph with maximal adjacency spectral radius (signless Laplacian spectral radius, respectively) among all graphs in B(n, r ), then ρ A (G) (ρ Q (G), respectively) is strictly increasing in r . ∗∗ Proof By the discussions prior to this theorem, we have ρ A (K 1,n−1 ) > ρ A (G(3, 3; n− ∗∗ 5)) and ρ Q (K 1,n−1 ) > ρ Q (G(3, 3; n − 5)). It is sufficient to consider 0 ≤ r ≤ n − 6. By Theorem 3.6, we have G = G(n − r − 2, 3; r ). Contracting the cycle Cn−r −2 in G(n − r − 2, 3; r ) to Cn−r −3 , we obtain a new graph G(n − r − 3, 3; r ). By Lemma 2.3, ρ A (G(n − r − 2, 3; r )) < ρ A (G(n − r − 3, 3; r )). Adding a pendant edge to the vertex u of G(n − r − 3, 3; r ), we get a graph which is isomorphism to G(n−r −2, 3; r ), where u is as labelled in Fig. 2. Since G(n−r −3, 3; r ) is a subgraph of G(n − r − 3, 3; r + 1), ρ A (G(n − r − 3, 3; r )) < ρ A (G(n − r − 3, 3; r + 1)). Again by Theorem 3.6, the first result follows. Similarly, by Lemma 2.4 and Theorem 3.7, the corresponding result on the signless Laplacian spectral radius can be obtained.

Acknowledgments We greatly thank the referees for careful reading and helpful suggestions that led to many improvements.

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