On the spectral radius of bipartite graphs which are nearly complete

Abstract

For $p,q,r,s,t\in {\mathbb{Z}}^{+}$ with $rt\le p$ and $st\le q$, let $G=G(p,q;r,s;t)$ be the bipartite graph with partite sets $U=\{u_1,\dots ,u_p\}$ and $V=\{v_1,\dots ,v_q\}$ such that any two edges $u_i$ and $v_j$ are not adjacent if and only if there exists a positive integer k with $1\le k\le t$ such that $(k-1)r+1\le i\le kr$ and $(k-1)s+1\le j\le ks$. Under these circumstances, Chen et al. (Linear Algebra Appl. 432:606-614, 2010) presented the following conjecture:

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

$${\lambda }_1(G)\le {\lambda }_1(G(p,q;k,1;1))=\sqrt{\frac{pq-k+\sqrt{p^2{q}^2-6pqk+4pk+4q{k}^2-3k^2}}{2}}.$$

In this paper, we prove this conjecture for the range ${min}_{v_h\in V}\{deg{v}_h\}\le \lfloor \frac{p-1}{2}\rfloor$.

MSC:05C05, 05C50.

1 Introduction

Let G be a (simple) graph with the vertex and edge sets given by $V(G)=\{v_1,v_2,\dots ,v_n\}$ and $E(G)=\{v_i{v}_j\mid v_i\text{ and }{v}_j\text{ are adjacent}\}$, respectively. The adjacency matrix of G on n vertices is an $n\times n$ matrix $A(G)$ whose entries $a_{ij}$ are given by

$$a_{ij}=\{\begin{array}{cc}1;\hfill & \text{if }{v}_i{v}_j\in E(G),\hfill \\ 0;\hfill & \text{otherwise}.\hfill \end{array}$$

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:

$${\lambda }_1(G)\ge {\lambda }_2(G)\ge \cdots \ge {\lambda }_{n-1}(G)\ge {\lambda }_n(G).$$

The largest eigenvalue ${\lambda }_1(G)$ 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 $G=(U\cup V,E)$ is such a bipartite graph, where $U=\{u_1,u_2,\dots ,u_p\}$, $V=\{v_1,v_2,\dots ,v_q\}$ are two sets of vertices and E is the set of edges defined as a subset of $U\times V$. As a usual notation, the degrees of vertices $u_i\in U$ and $v_j\in V$ will be denoted by $deg{u}_i$ and $deg{v}_j$, respectively. For the integers $p,q,r,s,t\in {\mathbb{Z}}^{+}$ satisfying $rt\le p$ and $st\le q$, let us denote the bipartite graph G by $G(p,q;r,s;t)$ with the above partite sets U and V such that $u_i\in U$ and $v_j\in V$ are not adjacent if and only if there exists a $k\in {\mathbb{Z}}^{+}$ with $1\le k\le t$ such that $(k-1)r+1\le i\le kr$ and $(k-1)s+1\le j\le ks$.

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 $|U|=p$, $|V|=q$ and $p\le q$, in the same reference, the authors also gave an affirmative answer to the conjecture [[11], Conjecture 1.2] by taking $|E(G)|=pq-2$ into account of a bipartite graph. Furthermore, refining the same conjecture for the number of edges is at least $pq-p+1$, there still exists the following conjecture.

Conjecture 1 [3]

For positive integers p, q and k satisfying $p\le q$ and $k<p$, let G be a bipartite graph with partite sets U and V having $|U|=p$ and $|V|=q$, and $|E(G)|=pq-k$. Then

$$\lambda (G)\le \lambda (G(p,q;k,1;1))=\sqrt{\frac{pq-k+\sqrt{p^2{q}^2-6pqk+4pk+4q{k}^2-3k^2}}{2}}.$$

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 ${min}_{v_h\in V}\{deg{v}_h\}\le \lfloor \frac{p-1}{2}\rfloor$.

2 Main result

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

Lemma 1 [3]

Let ${\lambda }_1$ be the spectral radius of the bipartite graph $G(p,q;k,1;1)$. Then

$${\lambda }_1=\sqrt{\frac{pq-k+\sqrt{p^2{q}^2-6pqk+4pk+4q{k}^2-3k^2}}{2}}.$$

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 $p\le q$ and $k<p$, let G be a bipartite graph with partite sets U and V having $|U|=p$ and $|V|=q$, and $|E(G)|=pq-k$. If ${min}_{v_h\in V}\{deg{v}_h\}\le \lfloor \frac{p-1}{2}\rfloor$, then

$${\lambda }_1(G)\le \sqrt{\frac{pq-k+\sqrt{p^2{q}^2-6pqk+4pk+4q{k}^2-3k^2}}{2}}$$

(1)

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

Proof Let $\mathbf{Z}={(x_1,x_2,\dots ,x_p,y_1,y_2,\dots ,y_q)}^T$ be an eigenvector of $A(G)$ corresponding to an eigenvalue ${\lambda }_1(G)$. For the sets U and V, let $x_i={max}_{1\le h\le p}{x}_h$ and $y_j={max}_{1\le h\le q}{y}_h$, respectively. Also, let us suppose that $v_1$ is the vertex having minimum degree in V. Then we have

$$\lfloor \frac{p-1}{2}\rfloor \ge \underset{v_h\in V}{min}\{deg{v}_h\}=deg{v}_1=d_1\text{(say).}$$

Now,

$$A(G)\mathbf{Z}={\lambda }_1(G)\mathbf{Z}.$$

(2)

Considering (2), we get

$${\lambda }_1(G)x_i\le (q-1)y_j+y_1\text{for }{u}_i\in U$$

(3)

and

$${\lambda }_1(G)y_1\le d_1{x}_i\text{for }{v}_1\in V.$$

(4)

However, from (3) and (4), we clearly obtain

$${\lambda }_1^2(G)y_1\le d_1[(q-1)y_j+y_1],$$

which can be written shortly as

$$({\lambda }_1^2(G)-d_1)y_1\le (q-1)d_1{y}_j.$$

(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 $pq-k$, we have

$$\sum _{h=1}^p{\lambda }_1(G)x_h\le (pq-k-d_1)y_j+d_1{y}_1.$$

(6)

For $v_j\in V$, from (2) we get

$${\lambda }_1(G)y_j=\sum _{u_h:u_h{v}_j\in E}{x}_h.$$

In other words, by (6),

$${\lambda }_1^2(G)y_j=\sum _{u_h:u_h{v}_j\in E}{\lambda }_1(G)x_h\le \sum _{h=1}^p{\lambda }_1(G)x_h\le (pq-k-d_1)y_j+d_1{y}_1,$$

that is,

$$({\lambda }_1^2(G)-pq+k+d_1)y_j\le d_1{y}_1.$$

(7)

From (5) and (7), we get

$${\lambda }_1^4(G)-(pq-k){\lambda }_1^2(G)+d_1(pq-k-q{d}_1)\le 0,$$

that is,

$${\lambda }_1(G)\le \sqrt{\frac{pq-k+\sqrt{p^2{q}^2-2pqk+k^2-4pq{d}_1+4k{d}_1+4q{d}_1^2}}{2}}.$$

(8)

Let us consider a function

$$f(x)=4q{x}^2+4kx-4pqx,\text{where }x\le \lfloor \frac{p-1}{2}\rfloor .$$

Then

$$f^{\mathrm{\prime }}(x)=-4q(p-\frac{k}{q}-2x)<0,\text{as }x\le \lfloor \frac{p-1}{2}\rfloor \text{ and }k<p\le q.$$

Thus $f(x)$ is a decreasing function on $1\le x\le \lfloor \frac{p-1}{2}\rfloor$. Since $p-k\le d_1\le \lfloor \frac{p-1}{2}\rfloor$ , 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 $G\cong G(p,q;k,1;1)$.

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 ${min}_{v_h\in V}\{deg{v}_h\}\le \lfloor \frac{p-1}{2}\rfloor$. However, this conjecture is still open for the range $\lfloor \frac{p-1}{2}\rfloor <{min}_{v_h\in V}\{deg{v}_h\}<p$.