Open Access

Bounds of Eigenvalues of -Minor Free Graphs

Journal of Inequalities and Applications20092009:852406

DOI: 10.1155/2009/852406

Received: 17 February 2009

Accepted: 11 May 2009

Published: 14 May 2009

Abstract

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let be the smallest eigenvalue of . In this paper, we have described the -minor free graphs and showed that (A) let be a simple graph with order . If has no -minor, then . (B) Let be a simple connected graph with order . If has no -minor, then , where equality holds if and only if is isomorphic to .

1. Introduction

In this paper, all graphs are finite undirected graphs without loops and multiple edges. Let be a graph with vertices, edges, and minimum degree or . The spectral radius of is the largest eigenvalue of its adjacency matrix. Let be the smallest eigenvalue of . The join is the graph obtained from by joining each vertex of to each vertex of . A graph is said to be a minor of if can be obtained from by deleting edges, contracting edges, and deleting isolated vertices. A graph is -minor free if has no -minor.

Brualdi and Hoffman [1] showed that the spectral radius satisfies , where , with equality if and only if is isomorphic to the disjoint union of the complete graph and isolated vertices. Stanley [2] improved the above result. Hong et al. [3] showed that if is a simple connected graph then with equality if and only if is either a regular graph or a bidegreed graph in which each vertex is of degree either or . Hong [4] showed that if is a -minor free graph then (1) , where equality holds if and only if is isomorphic to ; (2) , where equality holds if and only if is isomorphic to .

In this paper, we have described the -minor free graphs and obtained that

(a)let be a simple graph with order . If has no -minor, then ;

(b)let be a simple connected graph with order . If has no -minor, then , where equality holds if and only if is isomorphic to .

2. -Minor Free Graphs

The intersection of and is the graph with vertex set and edge set . Suppose is a connected graph and be a minimal separating vertex set of . Then we can write , where and are connected and . Now suppose further that is a complete graph. We say that is a -sum of and , denoted by , if . In particular, let denote a sum of and . Moreover, if or (say ) has a separating vertex set which induces a complete graph, then we can write such that and are connected and is a complete subgraph of . We proceed like this until none of the resulting subgraphs has a complete separating subgraph. The graphs are called the simplical summands of . It is easy to show that the subgraphs are independent of the order in which the decomposition is carried out (see [5]).

Theorem 2.1 (see [6], D. W. Hall; K. Wagner).

A graph has no -minor if and only if it can be obtained by -, -, -summing starting from planar graphs and .

A graph is said to be a edge-maximal -minor free graph if has no -minor and has at least an -minor, where is obtained from by joining any two nonadjacent vertices of . A graph is called a maximal planar graph if the planarity will be not held by joining any two nonadjacent vertices of .

Corollary 2.2.

If is an edge maximal -minor free graph then it can be obtained by -summing starting from and edge maximal planar graphs.

Proof.

This follows from Theorem 2.1.

Lemma 2.3.

If and are two maximal planar graphs with order and , respectively, then is not a maximal planar graph.

Proof.

We denote a planar embedding of by still. Since is a maximal planar graph, every face boundary in is a 3-cycle. Hence the outside face boundary in is a 4-cycle, this implies that the graph is not maximal planar.

Further, we have the following results.

Theorem 2.4.

If is an edge-maximal -minor free graph with vertices then , where , is a maximal planar graph with order .In particular,

(1)when , where ;

(2)when , where ;

(3)when , where ;

(4)when is a maximal planar graph.

Proof.

Suppose that the graphs are the simplical summands of , namely . By Corollary 2.2, is either a maximal planar graph or a . By Lemma 2.3, there is at most a maximal planar graph in . Hence we have , where , is a maximal planar graph with order .

Lemma 2.5 (see [7]).

Let be a simple planar bipartite graph with vertices and edges. Then .

Theorem 2.6.

Let be a simple connected bipartite graph with vertices and edges. If has no -minor, then .

Proof.

Let be a simple connected edge-maximal -minor free graph with vertices and edges. Suppose that the graphs are the simplical summands of . Then is either a maximal planar graph or the graph by Corollary 2.2. Further, without loss generality, we may assume that is a spanning subgraph of . Let the graph be the intersection of and . Then for . If then is a subgraph of , implies that . If is a maximal planar graph then is a simple planar bipartite graph, implies that by Lemma 2.5. Next we prove this result by induction on . For , . Now we assume it is true for and prove it for . Let and . Then by the induction hypothesis.

. Hence .

3. Bounds of Eigenvalues of -Minor Free Graphs

Lemma 3.1 (see [3]).

If is a simple connected graph then with equality if and only if is either a regular graph or a bidegreed graph in which each vertex is of degree either or .

Lemma 3.2.

Let be a simple connected graph with vertices and edges. If ,then , where equality holds if and only if and is either a regular graph or a bidegreed graph in which each vertex is of degree either or .

Proof.

Because when and , is a decreasing function of for , this follows from Lemma 3.1.

Lemma 3.3.

Let be a maximal planar graph with order , and let be a graph with vertices and edges.

(1)If and , where , then .

(2)If and , where , then , .

(3)If and , where , then , .

Proof.

Applying the properties of the maximal planar graphs, this follows by calculating.

Lemma 3.4.

Let be a maximal planar graph with order , and let be a graph with vertices.

(1)If and , where , then .

(2)If and , where , then .

(3)If and , where , then .

Proof.

It follows that (1) and (3) are true by Lemma 3.2 and 5(1)( 3). Next we prove that (2) is true too.

Let be a graph obtained from by expanding (in the simplcal summands of ) to , such that can be obtained by -summing , namely, .

This implies that by (1). Also we have , so .

Theorem 3.5.

Let be a simple graph with order . If has no -minor, then .

Proof.

Since when adding an edge in the spectral radius is strict increasing, we consider the edge-maximal -minor free graph only. Next we may assume that is an edge-maximal -minor free graph.

By Theorem 2.4 and Lemma 3.4, when , .

When , .

When , we have by calculating directly, where , is a maximal planar graph with order (see Theorem 2.4).

Therefore when , .

Remark 3.6.

In Theorem 3.5, the equality holds only if , for the others, the upper bounds of are not sharp. We conjecture that the best bound of is still.

Lemma 3.7 (see [7]).

If is a simple connected graph with vertices, then there exists a connected bipartite subgraph of such that with equality holding if and only if .

Lemma 3.8 (see [7]).

If is a connected bipartite graph with vertices and edges, then , where equality holds if and only if is a complete bipartite graph.

Theorem 3.9.

Let be a simple connected graph with vertices. If has no -minor, then , where equality holds if and only if is isomorphic to .

Proof.

This follows from Lemmas 3.7, 3.8 and Theorem 2.6.

Declarations

Acknowledgments

The author wishes to express his thanks to the referee for valuable comments which led to an improved version of the paper. Work supported by NNSF of China (no. 10671074) and NSF of Zhejian Province (no. Y7080364).

Authors’ Affiliations

(1)
Faculty of Science

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Copyright

© Kun-Fu Fang. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.