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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 707954, 5 pages
http://dx.doi.org/10.1155/2013/707954
Research Article

The Distance Matrices of Some Graphs Related to Wheel Graphs

School of Mathematics and Information Science, Yantai University, Yantai, Shandong 264005, China

Received 25 November 2012; Revised 29 May 2013; Accepted 30 May 2013

Academic Editor: Maurizio Porfiri

Copyright © 2013 Xiaoling Zhang and Chengyuan Song. 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.

Abstract

Let denote the distance matrix of a connected graph . The inertia of is the triple of integers (), where , , and denote the number of positive, 0, and negative eigenvalues of , respectively. In this paper, we mainly study the inertia of distance matrices of some graphs related to wheel graphs and give a construction for graphs whose distance matrices have exactly one positive eigenvalue.

1. Introduction

A simple graph consists of , a nonempty set of vertices, and , a set of unordered pairs of distinct elements of called edges. All graphs considered here are simple and connected. Let be a simple connected graph with vertex set and edge set . The distance between two vertices , is denoted by and is defined as the length of the shortest path between and in . The distance matrix of is denoted by and is defined by . Since is a symmetric matrix, its inertia is the triple of integers , where , , and denote the number of positive, , and negative eigenvalues of , respectively.

The distance matrix of a graph has numerous applications to chemistry [1]. It contains information on various walks and self-avoiding walks of chemical graphs. Moreover, the distance matrix is not only immensely useful in the computation of topological indices such as the Wiener index [1] but also useful in the computation of thermodynamic properties such as pressure and temperature virial coefficients [2]. The distance matrix of a graph contains more structural information compared to a simple adjacency matrix. Consequently, it seems to be a more powerful structure discriminator than the adjacency matrix. In some cases, it can differentiate isospectral graphs although there are nonisomorphic trees with the same distance polynomials [3]. In addition to such applications in chemical sciences, distance matrices find applications in music theory, ornithology [4], molecular biology [5], psychology [4], archeology [6], sociology [7], and so forth. For more information, we can see [1] which is an excellent recent review on the topic and various uses of distance matrices.

Since the distance matrix of a general graph is a complicated matrix, it is very difficult to compute its eigenvalues. People focus on studying the inertia of the distance matrices of some graphs. Unfortunately, up to now, only few graphs are known to have exactly one positive -eigenvalue, such as trees [8], connected unicyclic graphs [9], the polyacenes, honeycomb and square lattices [10], complete bipartite graphs [11], , and iterated line graphs of some regular graphs [12], and cacti [13]. This inspires us to find more graphs whose distance matrices have exactly one positive eigenvalue.

The wheel graph of vertices is a graph that contains a cycle of length plus a vertex (sometimes called the hub) not in the cycle such that is connected to every other vertex. In this paper, we first study the inertia of the distance matrices in wheel graphs if one or more edges are removed from the graph, and then, with the help of the structural characteristics of wheel graphs, we give a construction for graphs whose distance matrices have exactly one positive eigenvalue.

2. Preliminaries

We first give some lemmas that will be used in the main results.

Lemma 1 (see [14]). Let be a Hermitian matrix with eigenvalues and one of its principal submatrices. Let have eigenvalues . Then the inequalities hold.

For a square matrix, let denote the sum of cofactors of . Form the matrix by subtracting the first row from all other rows then the first column from all other columns and let denote the principle submatrix obtained from by deleting the first row and first column.

Lemma 2 (see [15]). .
A cut vertex is a vertex the removal of which would disconnect the remaining graph; a block of a graph is defined to be a maximal subgraph having no cut vertices.

Lemma 3 (see [15]). If is a strongly connected directed graph with blocks , then

Lemma 4. Let Then

Proof. Let Comparing to , we get the following: Expanding the determinant according to the last column and then the last line, we get the following incursion: that is, Since , , and , from the above incursion, we get the following: So, we have the following: This completes the proof.

3. Main Results

In the following, we always assume that , where is the hub of .

Theorem 5. Let . Then where .

Proof. Without loss of generality, we may assume that . Let
Then Expanding the determinant according to the second line, we get the following incursion: where and are defined as in Lemma 4.
By Lemma 4, we get the following: Since , , , and , according to the above incursion, we get the following: where . This completes the proof.

Corollary 6. Let . Then

Proof. We will prove the result by induction on .
If , is obviously true.
Suppose that the result is true for ; that is, , , .
Since is a principle submatrix of , by Lemma 1, the eigenvalues of interlace the eigenvalues of . By Theorem 5, . So, has one negative eigenvalue more than . According to the induction hypothesis, we get , , and . This completes the proof.

Theorem 7. One has where .

Proof. Consider the following Expanding the above determinant according to the second line, we get the following: where is defined as in Theorem 5.
By Theorem 5, when , we get the following:
This completes the proof.

Similar to Corollary 6, we can get the following corollary.

Corollary 8. (i)  If is even, , , .
(ii)  If is odd, , , .
Denote by the graph obtained from by deleting the vertex and all the edges adjacent to ; that is, . Let be any subset of with . In the following, we always denote by the graph obtained from by deleting all the edges in .

Theorem 9. One has , , .

Proof. Denote the components of by . Let denote the graph that contains plus the vertex such that is connected to every other vertex, . Then each is isomorphism to or , where is an edge of . By Lemma 2 and some direct calculations, we get the following: It is easy to check that is also true when is isomorphism to .
In the following, we will prove the theorem by introduction on .
For , , where is an edge of , by Corollary 6, we get the result.
Suppose the result is true for .
For , let . Then by the induction hypothesis, , , and , which implies that where is a positive integer.
Since by Lemma 3,
Then where , if is even and , if is odd.
In this case, similar to Corollary 6, we can easily get , , and .
Up to now, we have proved the result.

Let denote the graph formed by only identifying the vertex of with the vertex of , where and are arbitrary vertices of and , respectively.

Lemma 10 (see [13]). Let denote the Cartesian product of connected graphs and , where and . Then we have(i); (ii); (iii).

Theorem 11. Let and be the hubs of and , respectively. Suppose and are any subsets of and with , , respectively. Then, the distance matrix of the graph has exactly one positive eigenvalue.

Proof. Since and are the hubs of and , respectively, must be isomorphism to some , where is the hub of and is any subset of with . By Theorem 9 and Lemma 10, we get the result.

Given an arbitrary integer , for , let be the hub of and any subset of . Suppose .

Theorem 12. For an arbitrary integer , the distance matrix of the graph has exactly one positive eigenvalue.

Proof. We will prove the conclusion by induction on .
If , by Theorem 9, the conclusion is true.
Suppose the conclusion is true for . For convenience, let . Then . By Lemma 10, we have the following: Since , we get . By the induction hypothesis, we get . This completes the proof.

Remark 13. Let and be any two graphs with the same form as in Theorem 12. Making Cartesian product of graphs and , by Lemma 10 and Theorem 12, we get a series of graphs whose distance matrices have exactly one positive eigenvalue.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work was supported by NSFC (11126256) and NSF of Shandong Province of China (ZR2012AQ022).

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