Abstract
Graph product plays a key role in many applications of graph theory because many large graphs can be constructed from small graphs by using graph products. Here, we discuss two of the most frequent graph-theoretical products. Let and be two graphs. The Cartesian product of any two graphs and is a graph whose vertex set is and if either and or and . The tensor product of and is a graph whose vertex set is and if and . The strong product of any two graphs and is a graph whose vertex set is defined by and edge set is defined by . The resistance distance among two vertices and of a graph is determined as the effective resistance among the two vertices when a unit resistor replaces each edge of . Let and denote a path and a cycle of order , respectively. In this paper, the generalized inverse of Laplacian matrix for the graphs and was procured, based on which the resistance distances of any two vertices in and can be acquired. Also, we give some examples as applications, which elucidated the effectiveness of the suggested method.
1. Introduction
Graph products [1] became an interesting area of research, and different types of products have been worked out in graph theory and other fields. The Cartesian product of any two graphs and is a graph whose vertex set is and two vertices and are adjacent in if and only if either and is adjacent to in , or and is adjacent to in . The tensor product of and is a graph whose vertex set is the Cartesian product of and distinct vertices and are adjacent in if is adjacent to and is adjacent to . The strong product of graph and is a graph whose vertex set is and distinct vertices and are adjacent in if either and is adjacent to , or and is adjacent to , or is adjacent to and is adjacent to . It is the union of Cartesian product and tensor product. Sabidussi first proposed it in 1960 [2]. Let and be the path and the cycle graphs of order , respectively. From the definition of tensor and strong product of graphs, the graphs and are depicted in Figure 1. The graph depicted in Figure 1(b) is also called a King’s graph which is a strong product of two path graphs.
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The resistance distance is a function of the distance in graphs, as suggested by Klein and Randi [3]. The resistance distance between any two vertices of a simple connected graph, , is equal to the resistance between two equivalent points on an electrical network, constructed in such a way as , of each of the edge to replace a load resistance of 1 ohm. It is symbolized by , where . The computation of resistance is relevant to a wide range of applications ranging from random walks [4], opinion formation [5], classical transport in disordered media [6], robustness of coupled oscillators network [7–9], first-passage processes [10], identifying the influential spreader node in a network [11], lattice Greens functions [12, 13], and resistance distance [3, 14] to graph theory [13, 15, 16]. At present, the resistance distance is a very suitable tool and internal graphic measurement to express the wave-like or fluid-like communication between two vertices [17]. It is also well studied in chemical and mathematical literature [3, 18–23].
Many kinds of formulae were attained for calculating the resistance distance, i.e., probabilistic formulae [4, 24], algebraic formulae [25–31], combinatorial formula [28], and so forth. Resistance distances have been procured for certain types of graphs, i.e., wheels and fans [18], cyclic graphs [32], some fullerene graphs [33], Cayley graphs [34], regular graphs [35, 36], pseudodistance regular graphs [37], and so forth. In recent years, the resistance distance of some graphic operations has been calculated (see [20, 38–41]).
In the present paper, we investigated the generalized inverse of Laplacian matrix for the graphs and , based on which two-vertex resistances in and can be procured.
We ordered the paper in the following way. Section 2 covers some preliminary knowledge, i.e., basic definitions and necessary lemmas. In Section 3, we prove our main results, i.e., the generalized inverse of Laplacian matrix for tensor and strong product networks and . In Section 4, as an application, we present a few examples. The final remarks are given in Section 5.
2. Preliminaries and Lemmas
Let be a simple graph, and the vertex and edge sets of are symbolized by and , respectively. The adjacency matrix of a graph is an matrix, whose element is one when there is an edge among vertex and vertex and zero when there is no edge between vertex and vertex . Let be diagonal matrix with diagonal entries . For a graph , let be a Laplacian matrix of order . The incidence matrix of a graph is an matrix, where and are numbers of vertices and edges, respectively, such that is 1 if the vertex and edge are incident and 0 otherwise. The identity matrix is an square matrix with 1s on main diagonal and 0s elsewhere.
Let and be the two matrices. The Kronecker product is the matrix acquired from by replacing each element by [42]. Let be the matrix acquired by removing the row and column of a matrix of a graph , and matrix is equal to the row of an incidence matrix of a graph . For example, considering a path graph , the Laplacian matrix is ; then, . The incidence matrix is ; then, .
Let be a square matrix. A matrix is called a -inverse of if it satisfies . -inverse of is represented as . A matrix is a group inverse of a matrix if it meets the following conditions [38]:(i)(ii)(iii)
Let symbolize a group inverse of . If is real symmetric, then exists and is a symmetric -inverse of . Actually, is equal to the Moore–Penrose inverse of since is symmetric [38].
The following lemma is used for computing the resistance distance.
Lemma 1 (see [38]). Let be a Laplacian matrix of a simple graph with vertex set . Then,
Lemma 2 (see [38]). For a nonsingular matrix , if and are nonsingular and , thenis the Schur complement of in .
C. Bu, in [38], stated the following expression.
Lemma 3 (see [38]). Let be a Laplacian matrix of a graph and suppose each a column vector of is a zero vector or ; then, the following matrix is a symmetric -inverse of :where .
The following expression, similar to Lemma 3, also holds for the Laplacian matrix of a simple graph. For more details, see [22, 39, 43].
Lemma 4. If the Laplacian matrix of a simple graph is partitioned as and is nonsingular, then the following matrix is a symmetric -inverse of :where .
3. Main Results
3.1. The Laplacian Generalized Inverse for Graph
Theorem 1. Let and be a path graph and a cycle graph with vertices and , respectively. Then, the symmetric -inverse of iswhere
Proof. Let and . Then,is a partition of , where and are adjacent whenever is an edge in and is an edge in . In , vertices are of degree 2 and vertices are of degree 4. Label the vertices of like in Figure 1(a). According to partition (7), the Laplacian matrix of can be written asWe start with the calculation of . For simplicity, letDue to Lemma 4, we haveBy using Lemma 4, the symmetric -inverse of iswhere
3.2. The Laplacian Generalized Inverse for Graph
Theorem 2. Let and be two paths with and vertices, respectively, and let
Then, the symmetric -inverse of iswhere
Proof. Let and . Then,and . The degree of vertices of isLabel the vertices of like in Figure 1(b). According to partition (16), the Laplacian matrix of can be written asWe start with the calculation of . For simplicity, letDue to Lemma 4, we haveBy using Lemma 4, the symmetric -inverse of iswhere
4. Examples to Summarize the Main Results
Here, we discuss few examples to show that two-vertex resistances in graphs and can be procured by the proposed method.
Example 1. The resistance distance matrix for the graph (see Figure 2(a)).
The Laplacian matrix of isFrom Theorem 1, we obtainBy using Lemma 1 and , the resistance distance matrix of iswhere denotes the two-vertex resistance between vertices and .
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Example 2. The resistance distance matrix for the graph (see Figure 2(b)).
The Laplacian matrix of iswhere .
Based on Theorem 2, we obtain thatBy using Lemma 1 and , the resistance distance matrix of where denotes the two-vertex resistance between vertices and .
5. Conclusion
In this paper, we investigated the resistance distance in the tensor product of a path and a cycle as well as the strong product of two paths. First, we obtained the Laplacian matrix of these two kinds of product graphs. After calculation, we acquire the generalized inverse representations of the Laplacian matrices, and then, applying the generalized inverse theory of block matrices, we obtained the two-vertex resistances. Finally, we applied the above method to compute the resistance distance in graphs and . We obtained the resistance distance between two pair of vertices in tensor and strong product of two classes of graphs. However, the resistance distance for some other graph products has not been solved yet. We recommend the readers to compute the resistance distance for other classes of graphs by using different graph products, i.e., zig zag product, modular product, co-normal product and lexicographical product.
Data Availability
No data were used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
Xiang-Feng Pan was supported by the University Natural Science Research Project of Anhui Province under Grant no. KJ2020A0001.