Open Access

On the spectral radius of bipartite graphs which are nearly complete

  • Kinkar Chandra Das1,
  • Ismail Naci Cangul2Email author,
  • Ayse Dilek Maden3 and
  • Ahmet Sinan Cevik3
Journal of Inequalities and Applications20132013:121

https://doi.org/10.1186/1029-242X-2013-121

Received: 19 December 2012

Accepted: 24 February 2013

Published: 21 March 2013

Abstract

For p , q , r , s , t Z + with r t p and s t q , let G = G ( p , q ; r , s ; t ) be the bipartite graph with partite sets U = { u 1 , , u p } and V = { v 1 , , v q } such that any two edges u i and v j are not adjacent if and only if there exists a positive integer k with 1 k t such that ( k 1 ) r + 1 i k r and ( k 1 ) s + 1 j k s . Under these circumstances, Chen et al. (Linear Algebra Appl. 432:606-614, 2010) presented the following conjecture:

Assume that p q , k < p , | U | = p , | V | = q and | E ( G ) | = p q k . Then whether it is true that

λ 1 ( G ) λ 1 ( G ( p , q ; k , 1 ; 1 ) ) = p q k + p 2 q 2 6 p q k + 4 p k + 4 q k 2 3 k 2 2 .

In this paper, we prove this conjecture for the range min v h V { deg v h } p 1 2 .

MSC:05C05, 05C50.

Keywords

bipartite graphadjacency matrixspectral radius

1 Introduction

Let G be a (simple) graph with the vertex and edge sets given by V ( G ) = { v 1 , v 2 , , v n } and E ( G ) = { v i v j v i  and  v j  are adjacent } , respectively. The adjacency matrix of G on n vertices is an n × n matrix A ( G ) whose entries a i j are given by
a i j = { 1 ; if  v i v j E ( G ) , 0 ; otherwise .
Since A ( G ) is symmetric, all the eigenvalues of A ( G ) are real. In fact, the eigenvalues of A ( G ) are called eigenvalues of the graph G. We can list the eigenvalues of the graph G in a non-increasing order as follows:
λ 1 ( G ) λ 2 ( G ) λ n 1 ( G ) λ n ( G ) .

The largest eigenvalue λ 1 ( G ) 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 G = ( U V , E ) is such a bipartite graph, where U = { u 1 , u 2 , , u p } , V = { v 1 , v 2 , , v q } are two sets of vertices and E is the set of edges defined as a subset of U × V . As a usual notation, the degrees of vertices u i U and v j V will be denoted by deg u i and deg v j , respectively. For the integers p , q , r , s , t Z + satisfying r t p and s t q , let us denote the bipartite graph G by G ( p , q ; r , s ; t ) with the above partite sets U and V such that u i U and v j V are not adjacent if and only if there exists a k Z + with 1 k t such that ( k 1 ) r + 1 i k r and ( k 1 ) s + 1 j k s .

In the literature, upper bounds for the spectral radius in terms of various parameters for unweighted and weighted graphs have been widely investigated [110]. 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 | U | = p , | V | = q and p q , in the same reference, the authors also gave an affirmative answer to the conjecture [[11], Conjecture 1.2] by taking | E ( G ) | = p q 2 into account of a bipartite graph. Furthermore, refining the same conjecture for the number of edges is at least p q p + 1 , there still exists the following conjecture.

Conjecture 1 [3]

For positive integers p, q and k satisfying p q and k < p , let G be a bipartite graph with partite sets U and V having | U | = p and | V | = q , and | E ( G ) | = p q k . Then
λ ( G ) λ ( G ( p , q ; k , 1 ; 1 ) ) = p q k + p 2 q 2 6 p q k + 4 p k + 4 q k 2 3 k 2 2 .

We note that similar conjectures in this topic have been resolved by the first author in the papers [1216]. In here, as the main goal, we present the proof of Conjecture 1 for the range min v h V { deg v h } p 1 2 .

2 Main result

The following lemma will be needed for the proof of our main result.

Lemma 1 [3]

Let λ 1 be the spectral radius of the bipartite graph G ( p , q ; k , 1 ; 1 ) . Then
λ 1 = p q k + p 2 q 2 6 p q k + 4 p k + 4 q k 2 3 k 2 2 .

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 p q and k < p , let G be a bipartite graph with partite sets U and V having | U | = p and | V | = q , and | E ( G ) | = p q k . If min v h V { deg v h } p 1 2 , then
λ 1 ( G ) p q k + p 2 q 2 6 p q k + 4 p k + 4 q k 2 3 k 2 2
(1)

with equality if and only if G G ( p , q ; k , 1 ; 1 ) .

Proof Let Z = ( x 1 , x 2 , , x p , y 1 , y 2 , , y q ) T be an eigenvector of A ( G ) corresponding to an eigenvalue λ 1 ( G ) . For the sets U and V, let x i = max 1 h p x h and y j = max 1 h q y h , respectively. Also, let us suppose that v 1 is the vertex having minimum degree in V. Then we have
p 1 2 min v h V { deg v h } = deg v 1 = d 1 (say).
Now,
A ( G ) Z = λ 1 ( G ) Z .
(2)
Considering (2), we get
λ 1 ( G ) x i ( q 1 ) y j + y 1 for  u i U
(3)
and
λ 1 ( G ) y 1 d 1 x i for  v 1 V .
(4)
However, from (3) and (4), we clearly obtain
λ 1 2 ( G ) y 1 d 1 [ ( q 1 ) y j + y 1 ] ,
which can be written shortly as
( λ 1 2 ( G ) d 1 ) y 1 ( q 1 ) d 1 y j .
(5)
Since v 1 is the vertex with the minimum degree d 1 in V and the total number of edges in bipartite graph G is p q k , we have
h = 1 p λ 1 ( G ) x h ( p q k d 1 ) y j + d 1 y 1 .
(6)
For v j V , from (2) we get
λ 1 ( G ) y j = u h : u h v j E x h .
In other words, by (6),
λ 1 2 ( G ) y j = u h : u h v j E λ 1 ( G ) x h h = 1 p λ 1 ( G ) x h ( p q k d 1 ) y j + d 1 y 1 ,
that is,
( λ 1 2 ( G ) p q + k + d 1 ) y j d 1 y 1 .
(7)
From (5) and (7), we get
λ 1 4 ( G ) ( p q k ) λ 1 2 ( G ) + d 1 ( p q k q d 1 ) 0 ,
that is,
λ 1 ( G ) p q k + p 2 q 2 2 p q k + k 2 4 p q d 1 + 4 k d 1 + 4 q d 1 2 2 .
(8)
Let us consider a function
f ( x ) = 4 q x 2 + 4 k x 4 p q x , where  x p 1 2 .
Then
f ( x ) = 4 q ( p k q 2 x ) < 0 , as  x p 1 2  and  k < p q .

Thus f ( x ) is a decreasing function on 1 x p 1 2 . Since p k d 1 p 1 2 , 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 d 1 = p k . 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 G G ( p , q ; k , 1 ; 1 ) .

Conversely, by Lemma 1, one can easily see that the equality holds in (1) for the graph G ( p , q ; k , 1 ; 1 ) . □

Remark 1 In Theorem 1, we proved Conjecture 1 for the range min v h V { deg v h } p 1 2 . However, this conjecture is still open for the range p 1 2 < min v h V { deg v h } < p .

Declarations

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.

Authors’ Affiliations

(1)
Department of Mathematics, Sungkyunkwan University
(2)
Department of Mathematics, Faculty of Arts and Science, Uludag University
(3)
Department of Mathematics, Faculty of Science, Selcuk University

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© Das et al.; licensee Springer 2013

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