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On the spectral radius of bipartite graphs which are nearly complete
Journal of Inequalities and Applications volume 2013, Article number: 121 (2013)
For with and , let be the bipartite graph with partite sets and such that any two edges and are not adjacent if and only if there exists a positive integer k with such that and . Under these circumstances, Chen et al. (Linear Algebra Appl. 432:606-614, 2010) presented the following conjecture:
Assume that , , , and . Then whether it is true that
In this paper, we prove this conjecture for the range .
Let G be a (simple) graph with the vertex and edge sets given by and , respectively. The adjacency matrix of G on n vertices is an matrix whose entries are given by
Since is symmetric, all the eigenvalues of are real. In fact, the eigenvalues of are called eigenvalues of the graph G. We can list the eigenvalues of the graph G in a non-increasing order as follows:
The largest eigenvalue is often called the spectral radius of G.
Throughout this paper, we will consider only finite, simple, undirected, bipartite graphs. So, let us suppose that is such a bipartite graph, where , are two sets of vertices and E is the set of edges defined as a subset of . As a usual notation, the degrees of vertices and will be denoted by and , respectively. For the integers satisfying and , let us denote the bipartite graph G by with the above partite sets U and V such that and are not adjacent if and only if there exists a with such that and .
In the literature, upper bounds for the spectral radius in terms of various parameters for unweighted and weighted graphs have been widely investigated [1–10]. As a special case, in , Chen et al. studied the spectral radius of bipartite graphs which are close to a complete bipartite graph. For partite sets U and V having , and , in the same reference, the authors also gave an affirmative answer to the conjecture [, Conjecture 1.2] by taking into account of a bipartite graph. Furthermore, refining the same conjecture for the number of edges is at least , there still exists the following conjecture.
Conjecture 1 
For positive integers p, q and k satisfying and , let G be a bipartite graph with partite sets U and V having and , and . Then
2 Main result
The following lemma will be needed for the proof of our main result.
Lemma 1 
Let be the spectral radius of the bipartite graph . Then
We now present an upper bound on the spectral radius of the bipartite graph G.
Theorem 1 For positive integers p, q and k satisfying and , let G be a bipartite graph with partite sets U and V having and , and . If , then
with equality if and only if .
Proof Let be an eigenvector of corresponding to an eigenvalue . For the sets U and V, let and , respectively. Also, let us suppose that is the vertex having minimum degree in V. Then we have
Considering (2), we get
However, from (3) and (4), we clearly obtain
which can be written shortly as
Since is the vertex with the minimum degree in V and the total number of edges in bipartite graph G is , we have
For , from (2) we get
In other words, by (6),
From (5) and (7), we get
Let us consider a function
Thus is a decreasing function on . Since , from (8), we get the required result (1).
Suppose now that equality holds in (1). Then all inequalities in the above argument must become equalities. Thus we have . From the equality in (3), we get
From the equality in (4), we get
From the equality in (7), we get
Hence we conclude that .
Conversely, by Lemma 1, one can easily see that the equality holds in (1) for the graph . □
Remark 1 In Theorem 1, we proved Conjecture 1 for the range . However, this conjecture is still open for the range .
Berman A, Zhang XD: On the spectral radius of graphs with cut vertices. J. Comb. Theory, Ser. B 2001, 83: 233–240. 10.1006/jctb.2001.2052
Brualdi RA, Hoffman AJ:On the spectral radius of a matrix. Linear Algebra Appl. 1985, 65: 133–146.
Chen YF, Fu HL, Kim IJ, Stehr E, Watts B: On the largest eigenvalues of bipartite graphs which are nearly complete. Linear Algebra Appl. 2010, 432: 606–614. 10.1016/j.laa.2009.09.008
Cvetković D, Doob M, Sachs H: Spectra of Graphs. Academic Press, New York; 1980.
Cvetković D, Rowlinson P: The largest eigenvalue of a graph: a survey. Linear Multilinear Algebra 1990, 28: 3–33. 10.1080/03081089008818026
Das KC, Kumar P: Bounds on the greatest eigenvalue of graphs. Indian J. Pure Appl. Math. 2003, 34(6):917–925.
Das KC, Kumar P: Some new bounds on the spectral radius of graphs. Discrete Math. 2004, 281: 149–161. 10.1016/j.disc.2003.08.005
Das KC, Bapat RB: A sharp upper bound on the spectral radius of weighted graphs. Discrete Math. 2008, 308: 3180–3186. 10.1016/j.disc.2007.06.020
Hong Y: Bounds of eigenvalues of graphs. Discrete Math. 1993, 123: 65–74. 10.1016/0012-365X(93)90007-G
Stanley RP: A bound on the spectral radius of graphs with e edges. Linear Algebra Appl. 1987, 67: 267–269.
Bhattacharya A, Friedland S, Peled UN: On the first eigenvalue of bipartite graphs. Electron. J. Comb. 2008., 15: Article ID #R144
Das KC: On conjectures involving second largest signless Laplacian eigenvalue of graphs. Linear Algebra Appl. 2010, 432: 3018–3029. 10.1016/j.laa.2010.01.005
Das KC: Conjectures on index and algebraic connectivity of graphs. Linear Algebra Appl. 2010, 433: 1666–1673. 10.1016/j.laa.2010.06.012
Das KC: Proofs of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs. Linear Algebra Appl. 2011, 435: 2420–2424. 10.1016/j.laa.2010.12.018
Das KC: Proof of conjectures involving the largest and the smallest signless Laplacian eigenvalues of graphs. Discrete Math. 2012, 312: 992–998. 10.1016/j.disc.2011.10.030
Das KC: Proof of conjectures on adjacency eigenvalues of graphs. Discrete Math. 2013, 313(1):19–25. 10.1016/j.disc.2012.09.017
Dedicated to Professor Hari M Srivastava.
The first author is supported by BK21 Math Modeling HRD Div. Sungkyunkwan University, Suwon, Republic of Korea, and the other authors are partially supported by Research Project Offices of Uludag (2012-15 and 2012-19) and Selcuk Universities.
The authors declare that they have no competing interests.
All authors completed the paper together. Moreover, all authors read and approved the final manuscript.
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Das, K.C., Cangul, I.N., Maden, A.D. et al. On the spectral radius of bipartite graphs which are nearly complete. J Inequal Appl 2013, 121 (2013). https://doi.org/10.1186/1029-242X-2013-121
- bipartite graph
- adjacency matrix
- spectral radius