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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)
Abstract
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 .
MSC:05C05, 05C50.
1 Introduction
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 [3], 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 [[11], 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 [3]
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
We note that similar conjectures in this topic have been resolved by the first author in the papers [12–16]. In here, as the main goal, we present the proof of Conjecture 1 for the range .
2 Main result
The following lemma will be needed for the proof of our main result.
Lemma 1 [3]
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
Now,
Considering (2), we get
and
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),
that is,
From (5) and (7), we get
that is,
Let us consider a function
Then
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 .
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Acknowledgements
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.
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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
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DOI: https://doi.org/10.1186/1029-242X-2013-121