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Linear Algebra and its Applications 474 2015 30 43 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.

Article history: Received 1 May 2014 Accepted 28 January 2015 Available online xxxx Submitted by S.

1 Linear Algebra and its Applications Contents lists available at ScienceDirect Linear Algebra and its Applications Spectral radius of bipartite graphs Chia-an Liu, Chih-wen Weng Department of Applied Mathematics, National Chiao-Tung University, Hsinchu, Taiwan a r t i c l e i n f o a b s t r a c t Article history: Received 1 May 2014 Accepted 28 January 2015 Available online xxxx Submitted by S. Friedland MSC: 05C50 15A18 Keywords: Bipartite graph Adjacency matrix Spectral radius Degree sequence Let k, p, q be positive integers with k <p < q+1. We prove that the maximum spectral radius of a simple bipartite graph obtained from the complete bipartite graph K p,q of bipartition orders p and q by deleting k edges is attained when the deleted edges are all incident on a common vertex which is located in the partite set of order q. Our method is based on new sharp upper bounds on the spectral radius of bipartite graphs in terms of their degree sequences Elsevier Inc. All rights reserved. 1. Introduction Let G be a simple graph of order n. The adjacency matrix A = a ij of G is a 0-1 square matrix of order n with rows and columns indexed by the vertex set VGof G such that for any i, j VG, a ij =1iff i, j are adjacent in G. The spectral radius ρg of G is the largest eigenvalue of the adjacency matrix A of G. Brualdi and Hoffman proposed the problem of finding the maximum spectral radius of a graph with precisely e edges in 1976 [3, p. 438], and ten years later they gave a * Corresponding author. addresses: twister.imm96g@g2.nctu.edu.tw C.-a. Liu, weng@math.nctu.edu.tw C.-w. Weng / 2015 Elsevier Inc. All rights reserved.

2 C.-a. Liu, C.-w. Weng / Linear Algebra and its Applications conjecture in [6] that the maximum spectral radius of a graph with e edges is attained by taking a complete graph and adding a new vertex which is adjacent to a corresponding number of vertices in the complete graph. This conjecture was proved by Peter Rowlinson in [16]. See [18,10] also for the proof of partial cases of this conjecture. The next problem is then to determine graphs with maximum spectral radius in the class of connected graphs with n vertices and e edges. The cases e n + 5 when n is sufficiently large are settled by Brualdi and Solheid [7], and the cases e n = r 2 1by F.K. Bell [1]. The bipartite graphs analogue of the Brualdi Hoffman conjecture was settled by A. Bhattacharya, S. Friedland, and U.N. Peled [4] with the following statement: For a connected bipartite graph G, ρg e with equality iff G is a complete bipartite graph. Moreover, they proposed the problem to determine graphs with maximum spectral radius in the class of bipartite graphs with bipartition orders p and q, and e edges. They then gave Conjecture 1.1 below. From now on the graphs considered are simple bipartite. Let Kp, q, e denote the family of e-edge subgraphs of the complete bipartite graph K p,q with bipartition orders p and q. Conjecture 1.1. Let 1 < e < pqbe integers. An extremal graph that solves max ρg G Kp,q,e is obtained from a complete bipartite graph by adding one vertex and a corresponding number of edges. Moreover, in [4, Theorem 8.1] Conjecture 1.1 was proved in the case that e = st 1 for some positive integers s, t satisfying 2 s p < t q t + t 1 s 1. They also indicated that the only extremal graph is obtained from K s,t by deleting an edge. Conjecture 1.1 does not indicate that the adding vertex goes into which partite set of a complete bipartite graph. For e pq maxp, q resp. e pq minp, q, let e K p,q resp. Kp,q e denote the graph which is obtained from K p,q by deleting pq e edges incident on a common vertex in the partite set of order no larger than resp. no less than that of the other partite set. Then the extremal graph in Conjecture 1.1 is either e K s,t or Ks,t e for some positive integers s p and t q which meet the constraints of the number of edges. Fig. 1 gives two such graphs. In 2010 [8], Yi-Fan Chen, Hung-Lin Fu, In-Jae Kim, Eryn Stehr and Brendon Watts determined ρkp,q e and gave an affirmative answer to Conjecture 1.1 when e = pq 2. Furthermore, they refined Conjecture 1.1 for the case when the number of edges is at least pq minp, q +1to the following conjecture. Conjecture 1.2. Suppose 0 < pq e < minp, q. Then for G Kp, q, e, ρg ρk e p,q.

3 32 C.-a. Liu, C.-w. Weng / Linear Algebra and its Applications Fig. 1. The graphs K 5 2,3, 5 K 2,3 and 5 K 2,4. The paper is organized as follows. Preliminary contents are in Section 2. Theorem 3.3 in Section 3 presents a series of sharp upper bounds of ρg in terms of the degree sequence of G. Some special cases of Theorem 3.3 are further investigated in Section 4 in which Corollary 4.2 is the most useful in this paper. We prove Conjecture 1.2 as an application of Corollary 4.2 in Section 5. Finally we propose another conjecture which is a general refinement of Conjecture 1.1 in Section Preliminary Basic results are provided in this section for later use. Lemma 2.1. See [4, Proposition 2.1]. Let G be a simple bipartite graph with e edges. Then ρg e with equality iff G is a disjoint union of a complete bipartite graph and isolated vertices. Let G be a simple bipartite graph with bipartition orders p and q, and degree sequences d 1 d 2 d p and d 1 d 2 d q respectively. We say that G is biregular if d 1 = d p and d 1 = d q. Lemma 2.2. See [2, Lemma 2.1]. Let G be a simple connected bipartite graph. Then with equality iff G is biregular. ρg d 1 d 1 Let M be a real matrix of the following block form M 1,1 M 1,m M =.., M m,1 M m,m

4 C.-a. Liu, C.-w. Weng / Linear Algebra and its Applications where the diagonal blocks M i,i are square. Let b ij denote the average row-sums of M i,j, i.e. b ij is the sum of entries in M i,j divided by the number of rows. Then B =b ij is called a quotient matrix of M. If in addition for each pair i, j, M i,j has constant row-sum, then B is called an equitable quotient matrix of M. The following lemma is direct from the definition of matrix multiplication [5, Chapter 2]. Lemma 2.3. Let B be an equitable quotient matrix of M with an eigenvalue θ. Then M also has the eigenvalue θ. The following lemma is a part of the Perron Frobenius Theorem [15, Chapter 2]. Lemma 2.4. If M is a nonnegative n n matrix with largest eigenvalue ρm and rowsums r 1, r 2,..., r n, then ρm max 1 i n r i. Moreover, if M is irreducible then the above equality holds if and only if the row-sums of M are all equal. 3. A series of sharp upper bounds of ρg We give a series of sharp upper bounds of ρg in terms of the degree sequence of a bipartite graph G in this section. The following set-up is for the description of extremal graphs of our upper bounds. Definition 3.1. Let H, H be two bipartite graphs with given ordered bipartitions VH = X Y and VH = X Y, where VH VH = φ. The bipartite sum H + H of H and H with respect to the given ordered bipartitions is the graph obtained from H and H by adding an edge between x and y for each pair x, y X Y X Y. Example 3.2. Let N s,t denote the bipartite graph with bipartition orders s, t and without any edges. Then for p q and e meeting the desired constraints, e K p,q = K p 1,q pq+e + N 1,pq e and K e p,q = K p pq+e,q 1 + N pq e,1. Theorem 3.3. Let G be a simple bipartite graph with bipartition orders p and q, and corresponding degree sequences d 1 d 2 d p and d 1 d 2 d q. For 1 s p and 1 t q, let X s,t = d s d t + s 1 i=1 d i d s + t 1 j=1 d j d t and Y s,t = s 1 i=1 d i d s j=1 d j d t. Then the spectral radius X s,t + Xs,t 2 4Y s,t ρg φ s,t :=. 2

5 34 C.-a. Liu, C.-w. Weng / Linear Algebra and its Applications Furthermore, if G is connected then the above equality holds if and only if there exist nonnegative integers s <sand t <t, and a biregular graph H of bipartition orders p s and q t respectively such that G = K s,t + H. Before proving Theorem 3.3, we mention some simple properties of φ s,t. Lemma 3.4. i φ 1,1 = d 1 d 1. ii If d s = d s then φ s,t = φ s,t. If d t = d t then φ s,t = φ s,t. iii with equality iff φ 2 s,t = e. iv φ 4 s,t X s,t φ 2 s,t + Y s,t =0. φ 2 s,t max d i d s, d j d t i=1 Proof. i, ii, iv are immediate from the definition of φ s,t. Clearly d s d t =0if and only if j=1 max d i d s, d j d t = e. i=1 j=1 Hence iii follows by using X s,t d s d t =0to simplify φ s,t. s 1 i=1 d i d s + t 1 j=1 d j d twith equality iff We set up notations for the use in the proof of Theorem 3.3. For 1 k s 1, let d 1+ td k d s x k = φ 2 s,t s 1 i=1 d i d s, if φ2 s,t > s 1 i=1 d i d s ; 1, if φ 2 s,t = s 1 i=1 d i d s, 3.1 and for 1 l t 1let d s d l 1+ d t x l = φ 2 s,t t 1 j=1 d j d t, if φ2 s,t > t 1 j=1 d j d t; 1, if φ 2 s,t = t 1 j=1 d j d t. 3.2 Note that x k, x l 1 because of Lemma 3.4iii. The relations between the above parameters are given in the following.

6 C.-a. Liu, C.-w. Weng / Linear Algebra and its Applications Lemma 3.5. i Suppose φ 2 s,t > s 1 a=1 d a d s. Then 1 x i for 1 i s 1, and d i d t + d h d t+ x k 1d i = φ 2 s,t d s d t + d h d t+ x k 1d s = φ 2 s,t. ii Suppose φ 2 s,t > t 1 b=1 d b d t. Then 1 d s d j + d h d s + x l 1d j = φ 2 s,t x j for 1 j t 1, and l=1 d s d t + d h d s + x l 1d t = φ 2 s,t. l=1 Proof. Referring to 3.1 and Lemma 3.4iv, 1 d i d t + d h d t+ x k 1d i x i 1 = φ 2 s,t s 1 d k d s +d td i d s d i d t + d h d t d h d t d k d s = φ 2 s,t φ 2 s,t for 1 i s 1, and d s d t + d h d t+ x k 1d s = 1 φ 2 s,t s 1 d k d s = φ 2 s,t. φ 2 s,t d s d t + d h d t d h d t d k d s Hence i follows. Similarly, referring to 3.2 and Lemma 3.4iv we have ii.

7 36 C.-a. Liu, C.-w. Weng / Linear Algebra and its Applications Let U = {u i 1 i p} and V = {v j 1 j q} be the bipartition of G such that the degree sequences d 1 d 2 d p and d 1 d 2 d q, respectively are according to the list. For 1 i, j p, let n ij denote the numbers of common neighbors of u i and u j, i.e., n ij = Gu i Gu j where Gu is the set of neighbors of the vertex u in G. Similarly, for 1 i, j q let n ij = Gv i Gv j. Since G is bipartite, the adjacency matrix A and its square A 2 look like the following in block form: 0 B A = B T, A 2 = 0 BB T 0 n ij 1 i,j p 0 0 B T = B 0 n ij i,j q We have the following properties of n ij and n ij. Lemma 3.6. i For 1 i p and 1 j q, n ii = d i and n jj = d j. ii For 1 i, j p, n ij d i with equality if and only if Gu j Gu i. iii For 1 i, j q, n ij d i with equality if and only if Gv j Gv i. iv For 1 i p, v For 1 j q, p n ik = q n jk = j: u i v j EG i: u i v j EG d j d i t +1d t + d h. d i d j s +1d s + d h. Proof. i iii are immediate from the definition of n ij. Counting the pairs u k, v j such that v j Gu i Gu k in two orders j, k and k, j, we have the first equality in iv. The second inequality of iv is clear since d j is non-increasing. v is similar to iv. Proof of Theorem 3.3. It is well known that ρa 2 = ρa 2. In the following we will show that ρa 2 φ 2 s,t. Let U = diagx 1,x 2,,x s 1, 1,, 1,x }{{} 1,x 2,,x t 1, 1,, 1 }{{} p q be a diagonal matrix of order p + q. Let C = U 1 A 2 U. Then A 2 and C are similar and with the same spectrum. Let r 1,, r p, r 1,, r q be the row-sums of C. Referring to 3.3, we have

8 C.-a. Liu, C.-w. Weng / Linear Algebra and its Applications r i = x k x i n ik + p k=s 1 x i n ik = 1 x i p n ik + 1 x k 1n ik x i for 1 i s 1; 3.4 p p r i = x k n ik + n ik = n ik + x k 1n ik for s i p; 3.5 r j = x l x l=1 j n jl + k=s q l=t 1 x j n jl = 1 x j q n jl + 1 x x l 1n jl l=1 j l=1 for 1 j t 1; 3.6 q q r j = x ln jl + n jl = n jl + x l 1n jl for t j q. 3.7 l=1 l=t l=1 If φ 2 s,t = s 1 a=1 d a d s then x k =1for 1 k s 1by 3.1 and φ 2 s,t = e by Lemma 3.4iii. Hence 3.4 and 3.5 become r i = p n ik = l=1 j: u i v j EG d j e = φ 2 s,t 3.8 for 1 i p. Suppose φ 2 s,t > s 1 a=1 d a d s. Referring to 3.4 and 3.5, for 1 i s 1 and for s i p r i 1 d i t +1d t + x i d h + 1 x k 1d i = φ 2 s,t 3.9 x i r i d i t +1d t + d h + x k 1d i 3.10 d s t +1d t + d h + x k 1d s = φ 2 s,t, 3.11 where the inequalities are from Lemma 3.6ii iv and the non-increasing of degree sequence, and the equalities are from Lemma 3.5i. Thus, r i φ 2 s,t for 1 i p. Similarly, referring to 3.6, 3.7, Lemma 3.6iii v, the non-increasing of degree sequence, and Lemma 3.5ii we have r j φ2 s,t for 1 j q. Hence ρa 2 = ρc φ 2 s,t by Lemma 2.4. To verify the second part of Theorem 3.3, assume that G is connected. We prove the sufficient conditions of ρg = φ s,t. If s =0or t =0then G is biregular. By Lemmas 2.2 and 3.4i ii, ρg = d 1 d 1 = φ s,t. Suppose s =0and t 1. Then d 1 = d p and

9 38 C.-a. Liu, C.-w. Weng / Linear Algebra and its Applications p = d 1 = d t d t +1 = d q. We take the equatable quotient matrix E of A with respect to the partition {{1,..., p}, {p +1,..., p + t }, {p + t +1,..., p + q}}. Hence 0 t d s t E = p 0 0. d t 0 0 The eigenvalues of E are 0and ± d s d t + t p d t=±φ s,t. By Lemma 2.3, φ s,t is also an eigenvalue of A. Since ρg φ s,t has been shown in the first part, we have ρg = φ s,t. Similarly for the case s 1and t =0. Suppose s 1and t 1. Then q = d 1 = d s d s +1 = d p and p = d 1 = d t d t +1 = d q. We take the equatable quotient matrix F of A with respect to the partition {{1,..., s }, {s +1,..., p}, {p + 1,..., p + t }, {p + t +1,..., p + q}}. Hence Then the eigenvalues of F are 0 0 t q t 0 0 t d s t F = s p s 0 0. s d t s 0 0 X s,t ± Xs,t 2 4Y s,t ±. 2 We see φ s,t is an eigenvalue of F, and by Lemma 2.3 φ s,t is also an eigenvalue of A. Hence ρg = φ s,t. Here we complete the proof of the sufficient conditions of φ s,t = ρg. To prove the necessary conditions of ρg = φ s,t, suppose ρg = φ s,t. Then by Lemma 2.4 r i = r j = φ2 s,t for 1 i p and 1 j q. Let s <sand t <tbe the smallest nonnegative integers such that d s +1 = d s and d t +1 = d t, respectively. We prove either d 1 = d p or q = d 1 = d s >d s +1 = d p in the following. The connectedness of G implies d s d t > 0so that φ 2 s,t > max d i d s, d j d t i=1 by Lemma 3.4iii. Hence the equalities in 3.9 to 3.11 all hold. The choose of s and the equalities in 3.11 imply that d s +1 = d s = d p. If s =0then d 1 = d p. Suppose s 1. For 1 i s, since d i >d s and φ 2 s,t > s 1 a=1 d a d s, we have x i > 1by 3.1. The equalities in 3.9 imply n ik = d i and then Gu k Gu i by Lemma 3.6ii for 1 k s and 1 i s 1. Similarly the equalities in 3.10 imply Gu k Gu i for 1 k s and s i p by Lemma 3.6ii. That is, j=1

10 C.-a. Liu, C.-w. Weng / Linear Algebra and its Applications Gu 1 =Gu 2 = = Gu s Gu i for s +1 i p. Due to the connectedness of G, d 1 = d s = q. The result follows. Similarly, either d 1 = d q or p = d 1 = d t >d t +1 = d q. Clearly that the graphs with those degree sequences are K s,t + H for some biregular graph H of bipartition orders p s and q t respectively. Here we complete the proof for the necessary conditions of φ s,t = ρg, and also for Theorem 3.3. Remark 3.7. Other previous results shown by the style of the above proof can be found in [17,14,9,13]. Similar earlier results are referred to [6,7,18,11,12]. 4. A few special cases of Theorem 3.3 In this section we study some special cases of φ s,t in Theorem 3.3. We follow the notations in Theorem 3.3. As φ 1,1 = d 1 d 1 in Lemma 3.4i, Theorem 3.3 provides another proof of ρg d 1 d 1 in Lemma 2.2. Applying Theorem 3.3 and simplifying the formula φ s,t in cases s, t = 1, q and s, t = p, 1, we have the following corollary. Corollary 4.1. i ρg φ 1,q = e q d 1 d q. ii ρg φ p,1 = e p d 1 d p. We can quickly observe that and X p,q = d p d q +e pd p +e qd q=2e pd p + qd q d p d q 4.1 Hence we have the following corollary. Corollary 4.2. Y p,q =e pd p e qd q e pd p + qd q d p d q+ pd p + qd q d p d q 2 4d p d qpq e ρg 2. By adding an isolated vertex if necessary, we might assume d p =0and find φ p,q = e from Corollary 4.2. Hence Theorem 3.3 provides another proof of ρg e in Lemma 2.1.

11 40 C.-a. Liu, C.-w. Weng / Linear Algebra and its Applications Proof of Conjecture 1.2 When e, p, q are fixed, the formula 2e pd p + qd φ p,q d p,d q d p d q+ q= pd p + qd q d p d q 2 4d p d qpq e obtained in Corollary 4.2 is a 2-variable function. The following lemma will provide shape of the function φ p,q d p, d q. Lemma 5.1. If 1 d q p 1 and qd q e then φ p,q d p,d q d p < 0. Proof. Referring to 5.1, it suffices to show that 2e pd p + qd q d p d d q+ pd p + qd q d p d q 2 4d p d qpq e p = p + d q + pd p + qd q d p d qp d q 2d qpq e pd p + qd q d p d q 2 4d p d qpq e 5.2 is negative. If qd q = e then 5.2 has negative value 2d q p. Indeed if the numerator of the fraction in 5.2 is not positive then 5.2 has negative value. Thus assume that it is positive and qd q <e. From simple computation to have the fact that pd p + qd q d p d q 2d q pq e 2 p d pd p + qd q d p d q 2 4d p d qpq e q = 4d 2 q pq e p d q 2 qd q e < 0, we find that the fraction in 5.2 is strictly less than p d q, so the value in 5.2 is negative. Remark 5.2. From Example 3.2, if p q then the graphs e K p,q = K p 1,q pq+e + N 1,pq e and K e p,q = K p pq+e,q 1 +N pq e,1 satisfy the equalities in Theorem 3.3. Hence ρ e K p,q = φ p,q q pq + e, p 1 and ρk e p,q = φ p,q q 1, p pq + e; the latter is expanded as

12 C.-a. Liu, C.-w. Weng / Linear Algebra and its Applications ρk e p,q = e + e 2 4q 1p pq + epq e by 5.1. Lemma 5.3. Suppose 0 < pq e < minp, q, 1 d p q 1, 1 d q p 1 and d p + d q = e p 1q Then φ p,q d p,d q ρk e p,q. Proof. From symmetry, we can assume p q. Referring to 5.1 and 5.3, we only need to show that e pd p + qd q d p d q+ pd p + qd q d p d q 2 4d p d qpq e 5.5 e 2 4q 1p pq + epq e. 5.6 From 5.4, we have e pd p + qd q d p d q=p d q 1q d p and d p d q = d p + d q 2 [2d p d p + d q] 2 4 e p 1q 12 [2q 1 e p 1q 1] 2 4 =q 1p pq + e. 5.8 Hence Eq. 5.5 is at most e pd p + qd q d p d q+ pd p + qd q d p d q 2 4q 1p pq + epq e. 5.9 Set a = e pd p + qd q d p d qand b = 4q 1p pq + epq e. Note that a 0by 5.7 and b 0by the relations between p, q, e. Using the fact that e2 b e a 2 b e 2 e a 2 = a 5.10 from the concave property of the function y = x, we find the value in 5.9 is at most that in 5.6 and the result follows.

13 42 C.-a. Liu, C.-w. Weng / Linear Algebra and its Applications Proof of Conjecture 1.2. By Theorem 3.3, ρg φ p,q d p, d q. Note that the assumption 0 < pq e < minp, q implies 1 d p q 1 and 1 d q p 1. Let e p = e p 1q 1 d q. Clearly that 1 e p d p and qd q e. By Lemma 5.1, φ p,q d p, d q φ p,q e p, d q. With e p in the role of d p in Lemma 5.3, we have φ p,q e p, d q ρkp,q. e This completes the proof. 6. Concluding remark We give a series of sharp upper bounds for the spectral radius of bipartite graphs in Theorem 3.3. One of these upper bounds can be presented only by five variables: the number e of edges, bipartition orders p and q, and the minimal degrees d p and d q in the corresponding partite sets as shown in Corollary 4.2. We apply this bound when three variables e, p, q are fixed to prove Conjecture 1.2, a refinement of Conjecture 1.1 in the assumption that 0 < pq e < minp, q. To conclude this paper we propose the following general refinement of Conjecture 1.1. Conjecture 6.1. Let G Kp, q, e. Then ρg ρk e s,t for some positive integers s p and t q such that 0 st e mins, t. We believe that the function φ p,q d p, d qin 5.1 will still play an important role in solving Conjecture 6.1. Two of the key points might be to investigate the shape of the 4-variable function φ p,q d p, d qwith variables p, q, d p, d q, and to check that for which sequence s, t, d s, d t such that s p and t q and 0 st e mins, t, there exists a bipartite graph H with e edges whose spectral radius satisfying ρh = φ s,t d s, d t, where s, t are the bipartition orders of H and d s and d t are corresponding minimum degrees. Acknowledgements This research is supported by the National Science Council of Taiwan R.O.C. under the project NSC M MY3. References [1] F.K. Bell, On the maximal index of connected graphs, Linear Algebra Appl [2] Abraham Berman, Xiao-Dong Zhang, On the spectral radius of graphs with cut vertices, J. Combin. Theory Ser. B [3] J.-C.Bermond, J.-C.Fournier, M.Las Vergnas, D.Sotteau Eds., Problèmes Combinatoires et Theorie des Graphes, Orsay, 1976, Coll. Int. C.N.R.S., vol. 260, C.N.R.S. Publ., [4] A. Bhattacharya, S. Friedland, U.N. Peled, On the first eigenvalue of bipartite graphs, Electron. J. Combin R144. [5] Andries E. Brouwer, Willem H. Haemers, Spectra of Graphs Monograph, Springer, 2011.

14 C.-a. Liu, C.-w. Weng / Linear Algebra and its Applications [6] R.A. Brualdi, A.J. Hoffman, On the spectral radius of 0, 1-matrices, Linear Algebra Appl [7] R.A. Brualdi, Ernie S. Solheid, On the spectral radius of connected graphs, Publ. Inst. Math., N.S [8] Yi-Fan Chen, Hung-Lin Fu, In-Jae Kim, Eryn Stehr, Brendon Watts, On the largest eigenvalues of bipartite graphs which are nearly complete, Linear Algebra Appl [9] Yingying Chen, Huiqiu Liu, Jinlong Shu, Sharp upper bounds on the distance spectral radius of a graph, Linear Algebra Appl [10] Shmuel Friedland, Bounds on the spectral radius of graphs with e edges, Linear Algebra Appl [11] Yuan Hong, Upper bounds of the spectral radius of graphs in terms of genus, J. Combin. Theory Ser. B [12] Yuan Hong, Jin-Long Shu, Kunfu Fang, A sharp upper bound of the spectral radius of graphs, J. Combin. Theory Ser. B [13] Yu-pei Huang, Chih-wen Weng, Spectral radius and average 2-degree sequence of a graph, Discrete Math. Algorithms Appl , [14] Chia-an Liu, Chih-wen Weng, Spectral radius and degree sequence of a graph, Linear Algebra Appl [15] Henryk Minc, Nonnegative Matrices, John Wiley and Sons Inc., New York, [16] Peter Rowlinson, On the maximal index of graphs with a prescribed number of edges, Linear Algebra Appl [17] Jinlong Shu, Yarong Wu, Sharp upper bounds on the spectral radius of graphs, Linear Algebra Appl [18] Richard P. Stanley, A bound on the spectral radius of graphs with e edges, Linear Algebra Appl

arxiv: v1 [math.co] 23 Feb 2014

arxiv: v1 [math.co] 23 Feb 2014 Spectral Radius of Bipartite Graphs Chia-an Liu a,, Chih-wen Weng b a Department of Applied Mathematics, National Chiao-Tung University, Hsinchu, Taiwan b Department of Applied Mathematics, National Chiao-Tung