On the spectral radius of bipartite graphs which are nearly complete
© Das et al.; licensee Springer 2013
Received: 19 December 2012
Accepted: 24 February 2013
Published: 21 March 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 .
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 
2 Main result
The following lemma will be needed for the proof of our main result.
Lemma 1 
We now present an upper bound on the spectral radius of the bipartite graph G.
with equality if and only if .
Thus is a decreasing function on . Since , from (8), we get the required result (1).
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 .
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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