Discrete Dynamics in Nature and Society
Volume 2016 (2016), Article ID 4682527, 9 pages
http://dx.doi.org/10.1155/2016/4682527

Research Article

Further Results on Resistance Distance and Kirchhoff Index in Electric Networks

Qun Liu,¹ Jia-Bao Liu,² and Jinde Cao³

¹Department of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
²Department of Public Courses, Anhui Xinhua University, Hefei 230088, China
³Research 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 .

Figure 1: and .

Bu et al. investigated resistance distance in subdivision-vertex join and subdivision-edge join of graphs [4]. Liu et al. [5] gave the resistance distance and Kirchhoff index of -vertex join and -edge join of two graphs. Motivated by these works, in this paper we will work on two different composite networks: -vertex corona and -edge corona networks. In fact, we will follow the techniques used in [4] but in a slightly different method in order to obtain the effective resistances and the Kirchhoff indexes of -vertex corona and -edge corona networks in terms of the same parameters on the factors.

2. Preliminaries

In this section we list some lemmas as underlying but necessary preliminaries, which will be used in the proofs of our main results.

Lemma 3 (see [7, 9]). Let be a connected graph. Then

Let denote the column vector of dimension with all the entries equal to one. We will often use to denote an all-ones column vector if the dimension can be read from the context.

Lemma 4 (see [4]). For any graph, one has

Lemma 5 (see [17]). Let

be a nonsingular matrix. If and are nonsingular, then

where

For a square matrix , let denote the trace of .

Lemma 6 (see [10]). Let be a connected graph on vertices. Then

Lemma 7. Let

be the Laplacian matrix of a connected graph. If is nonsingular, then

is a symmetric -inverse of , where .

Proof. Since is symmetric, exists and is symmetric. Since

we know that

is a symmetric -inverse of .

Remarks 1. The above result is similar to Lemma in [10]; this is another form of Lemma , but in the process of computing the resistance distance of graphs and , we use this formula to be more superior than Lemma in [10].

3. Resistance Distance in -Vertex Coronae and -Edge Coronae of Two Graphs

We first give explicit resistance distance for the arbitrary two-vertex resistance distance in .

Theorem 8. Let be an -regular graph on vertices and edges, and let be an arbitrary graph on vertices and edges. Then the following holds:

(i)For any , one has

(ii)For any , one has

(iii)For any , let be edges incident to in . For any , , one has

(iv)For any ( ), let   (resp., ) be edges incident to   (resp., ) in , one has

(v)For any , let be edges incident to in . For any , , one has

(vi)For any , ,

Proof. Let and be the adjacency matrix of and incidence matrix of , respectively. Then the Laplacian matrix of is as follows:

Let , , , and

Note that . Let . By Lemma 5, we have

Now we are ready to calculate -inverse of .
First we begin with the calculation about .
Let ; then

By Lemma 7, we have .
Next according to Lemma 7, we calculate and :

We are ready to compute .
Let , ; then

Based on Lemmas 5 and 7, the following matrix

is a symmetric -inverse of , where and .
For any , by Lemma 3 and (21), we have

For any , by Lemma 3 and (21), we have

For any , let be edges incident to in . For any , , by Lemma 3 and (21), we have

For any ), let (resp., ) be edges incident to (resp., ) in . By Lemma 3 and (21), we have

For any , let be edges incident to in . For any , , by Lemma 3 and (21), we have

For any , , by Lemma 3 and (21), we have

Next it will give the formulae for resistance distance of graph when is a regular graph as follows.

Theorem 9. Let be an -regular graph with vertices, and let be an arbitrary graph with vertices. Then the following holds:

(i)For any , one has

(ii)For any , one has

(iii)For any , let be edges incident to in . For any , one has

(iv)For any   ( ), let   (resp., ) be edges incident to   (resp., ) in ; one has

(v)For any , let be edges incident to in . For any , , one has

(vi)For any , , one has

Proof. Let and be the adjacency matrix of and incidence matrix of , respectively. Then the Laplacian matrix of is as follows:

Now we are ready to calculate a -inverse of .
First we are ready to calculate .
Let , , then

By Lemma 7, we have .
Next we compute and :

We are ready to compute :

Based on Lemmas 5 and 7, the following matrix

is a symmetric -inverse of , where and .
For any , by Lemma 3 and (38), we have

For any , by Lemma 3 and (38), we have

For any , let be edges incident to in . For any , by Lemma 3 and (38), we have

For any   , let (resp., ) be edges incident to (resp., ) in . By Lemma 3 and (38), we have

For any , let be edges incident to in . For any , , by Lemma 3 and (38), we have

For any , , by Lemma 3 and (38), we have

4. Kirchhoff Index in -Vertex Corona and -Edge Corona of Two Graphs

Theorem 10. Let be an -regular graph on vertices and edges, and let be an arbitrary graph on vertices and edges. Then the following holds:

Proof. Let be the symmetric -inverse of . Then

Note that the eigenvalues of are . Then

So

Next, we calculate . Since, by Lemma 4, , then

where . Since , then

Let ; then

So

Lemma 6 implies that

Then plugging and into the equation above, we obtain the required result.

Theorem 11. Let be an -regular graph with vertices, and let be an arbitrary graph with vertices. Then

Proof. The proof is similar to those of Theorem 10, omitted.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11461020, 11561042), the Youth Foundation of Hexi University in Gansu Province (no. QN2013-07), and the Teaching Reform Project of Hexi University (no. HXXXJY-2014-011). The work of Jia-Bao Liu was partly supported by the Natural Science Foundation for the Higher Education Institutions of Anhui Province of China under Grant no. KJ2015A331.

References

D. J. Klein and M. Randić, “Resistance distance,” Journal of Mathematical Chemistry, vol. 12, no. 1, pp. 81–95, 1993.

D. J. Klein, “Resistance-distance sum rules,” Croatica Chemica Acta, vol. 75, no. 2, pp. 633–649, 2002.

E. Bendito, A. Carmona, and A. M. Encinas, “The Kirchhoff indices of join networks,” Discrete Applied Mathematics, vol. 160, no. 1-2, pp. 24–37, 2012.

C. Bu, B. Yan, X. Zhou, and J. Zhou, “Resistance distance in subdivision-vertex join and subdivision-edge join of graphs,” Linear Algebra and Its Applications, vol. 458, pp. 454–462, 2014.

X. Liu, J. Zhou, and C. Bu, “Resistance distance and Kirchhoff index of R-vertex join and R-edge join of two graphs,” Discrete Applied Mathematics, vol. 187, pp. 130–139, 2015.

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Springer, New York, NY, USA, 2nd edition, 2003.

R. B. Bapat, Graphs and Matrices, Universitext, Springer, London, UK; Hindustan Book Agency, New Delhi, India, 2010.

R. B. Bapat and S. Gupta, “Resistance distance in wheels and fans,” Indian Journal of Pure and Applied Mathematics, vol. 41, no. 1, pp. 1–13, 2010.

C. Bu, L. Sun, J. Zhou, and Y. Wei, “A note on block representations of the group inverse of Laplacian matrices,” Electronic Journal of Linear Algebra, vol. 23, pp. 866–876, 2012.

L. Sun, W. Wang, J. Zhou, and C. Bu, “Some results on resistance distances and resistance matrices,” Linear and Multilinear Algebra, vol. 63, no. 3, pp. 523–533, 2015.

H. Chen and F. Zhang, “Resistance distance and the normalized Laplacian spectrum,” Discrete Applied Mathematics, vol. 155, no. 5, pp. 654–661, 2007.

W. Xiao and I. Gutman, “Relations between resistance and Laplacian matrices and their applications,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 51, pp. 119–127, 2004.

W. Xiao and I. Gutman, “Resistance distance and Laplacian spectrum,” Theoretical Chemistry Accounts, vol. 110, no. 4, pp. 284–289, 2003.

Y. Yang and D. J. Klein, “A recursion formula for resistance distances and its applications,” Discrete Applied Mathematics, vol. 161, no. 16-17, pp. 2702–2715, 2013.

R. B. Bapat, “Resistance matrix of a weighted graph,” MATCH Communications in Mathematical and in Computer, vol. 50, pp. 73–82, 2004.

Y. Yang and D. J. Klein, “Resistance distance-based graph invariants of subdivisions and triangulations of graphs,” Discrete Applied Mathematics, vol. 181, pp. 260–274, 2015.

F. Zhang, The Schur Complement and Its Applications, Springer, New York, NY, USA, 2005.

J.-B. Liu, X.-F. Pan, L. Yu, and D. Li, “Complete characterization of bicyclic graphs with minimal Kirchhoff index,” Discrete Applied Mathematics, vol. 200, pp. 95–107, 2016.

J.-B. Liu, X.-F. Pan, F.-T. Hu, and F.-F. Hu, “Asymptotic Laplacian-energy-like invariant of lattices,” Applied Mathematics and Computation, vol. 253, pp. 205–214, 2015.

J. B. Liu and J. Cao, “The resistance distances of electrical networks based on Laplacian generalized inverse,” Neurocomputing, vol. 167, pp. 306–313, 2015.

J.-B. Liu and X.-F. Pan, “A unified approach to the asymptotic topological indices of various lattices,” Applied Mathematics and Computation, vol. 270, pp. 62–73, 2015.

Q. Liu, J. B. Liu, and J. Cao, “The Laplacian polynomial and Kirchhoff index of graphs based on R-graphs,” Neurocomputing, 2015.