Bounds of Eigenvalues of -Minor Free Graphs

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.