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 .
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, thenwhere
For a square matrix , let denote the trace of .
Lemma 6 (see [10]). Let be a connected graph on vertices. Then
Lemma 7. Letbe the Laplacian matrix of a connected graph. If is nonsingular, thenis a symmetric -inverse of , where .
Proof. Since is symmetric, exists and is symmetric. Sincewe know thatis 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 , , , andNote that . Let . By Lemma 5, we haveNow we are ready to calculate -inverse of .
First we begin with the calculation about .
Let ; thenBy Lemma 7, we have .
Next according to Lemma 7, we calculate and :We are ready to compute .
Let , ; thenBased on Lemmas 5 and 7, the following matrixis 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 , , thenBy Lemma 7, we have .
Next we compute and :We are ready to compute :Based on Lemmas 5 and 7, the following matrixis 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 SoNext, we calculate . Since, by Lemma 4, , thenwhere . Since , thenLet ; thenSoLemma 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.