Discrete Dynamics in Nature and Society

Volume 2016 (2016), Article ID 4682527, 9 pages

http://dx.doi.org/10.1155/2016/4682527

## Further Results on Resistance Distance and Kirchhoff Index in Electric Networks

^{1}Department of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China^{2}Department of Public Courses, Anhui Xinhua University, Hefei 230088, China^{3}Research Center for Complex Systems and Network Science, Department of Mathematics, Southeast University, Nanjing 210096, China

Received 11 September 2015; Revised 31 December 2015; Accepted 6 January 2016

Academic Editor: Juan R. Torregrosa

Copyright © 2016 Qun Liu et al. 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

In electric circuit theory, it is of great interest to compute the effective resistance between any pairs of vertices of a network, as well as the Kirchhoff index. Let be the graph obtained from by inserting a new vertex into every edge of and by joining by edges those pairs of these new vertices which lie on adjacent edges of . The set of such new vertices is denoted by . The -vertex corona of and , denoted by , is the graph obtained from vertex disjoint and copies of by joining the th vertex of to every vertex in the th copy of . The -edge corona of and , denoted by , is the graph obtained from vertex disjoint and copies of by joining the th vertex of to every vertex in the th copy of . The objective of the present work is to obtain the resistance distance and Kirchhoff index for composite networks such as -vertex corona and -edge corona networks.

#### 1. Introduction

For decades we want to know what a graph looks like. We want to reveal the principles of the networks behaviour covered by their complex topology and dynamics. We want to learn about how the network structure evolves over time and how it affects the properties of the dynamic processes on it. Klein and Randić [1] introduced a new distance function named resistance distance based on electric network theory; the resistance distance between vertices and , denoted by , is defined to be the effective electrical resistance between them if each edge of is replaced by a unit resistor. Resistance distance is an important character of a graph, which can imply many of its dynamic properties. The Kirchhoff index of is the sum of resistance distances between all pairs of vertices of . The Kirchhoff index of a network denotes the mean first-passage time measuring the efficiency of random walks; see [1]. The resistance distance and the Kirchhoff index attracted extensive attention due to its wide applications in complex network, physics, chemistry, and others. For more information on resistance distance and Kirchhoff index of graphs, the readers are referred to [2–22] and the references therein. In view of the applications of the resistance distance and the Kirchhoff index, it is of great interest to calculate this parameter for composite networks and find possible relations between the resistance distance and Kirchhoff indexes of the original networks and those of their composite networks; see, for instance, [3].

Let be a graph with vertex set and edge set . Let be the degree of vertex in and let be the diagonal matrix with all vertex degrees of as its diagonal entries. For a graph , let and denote the adjacency matrix and vertex-edge incidence matrix of , respectively. The matrix is called the Laplacian matrix of , where is the diagonal matrix of vertex degrees of . We use to denote the eigenvalues of . Let be the line graph of .

The -inverse of is a matrix such that . If is singular, then it has infinite many -inverses [6]. We use to denote any -inverse of a matrix , and let denote the -entry of . For a square matrix , the group inverse of , denoted by , is the unique matrix such that , , and . It is known that exists if and only if [6, 9]. If is real symmetric, then exists and is a symmetric -inverse of . Actually, is equal to the Moore-Penrose inverse of since is symmetric [9]. It is known that resistance distances in a connected graph can be obtained from any -inverse of [7, 8].

For a graph , let be the graph obtained from by inserting a new vertex into every edge of and by joining by edges those pairs of these new vertices which lie on adjacent edges of . Let be the set of newly added vertices; that is, . Next, we define two new operations as follows.

Let and be two vertex-disjoint graphs.

*Definition 1. *The -vertex corona of and , denoted by , is the graph obtained from vertex disjoint and copies of by joining the th vertex of to every vertex in the th copy of .

*Definition 2. *The -edge corona of and , denoted by , is the graph obtained from vertex disjoint and copies of by joining the th vertex of to every vertex in the th copy of .

Note that if has vertices and edges for , then has vertices and has vertices.

Let and denote a path and cycle with vertices, respectively. From these definitions, Figure 1 shows the graphs and .