Resistance Distance in Tensor and Strong Product of Path or Cycle Graphs Based on the Generalized Inverse Approach

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 ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ be two graphs. The Cartesian product ${\mathcal{G}}_1\square {\mathcal{G}}_2$ of any two graphs ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is a graph whose vertex set is $V\left({\mathcal{G}}_1\square {\mathcal{G}}_2\right)=V\left({\mathcal{G}}_1\right)\times V\left({\mathcal{G}}_2\right)$ and $\left(a_1,a_2\right)\left(b_1,b_2\right)\in E\left({\mathcal{G}}_1\square {\mathcal{G}}_2\right)$ if either $a_1=b_1$ and $a_2{b}_2\in E\left({\mathcal{G}}_2\right)$ or $a_1{b}_1\in E\left({\mathcal{G}}_1\right)$ and $a_2=b_2$ . The tensor product ${\mathcal{G}}_1\times {\mathcal{G}}_2$ of ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is a graph whose vertex set is $V\left({\mathcal{G}}_1\times {\mathcal{G}}_2\right)=V\left({\mathcal{G}}_1\right)\times V\left({\mathcal{G}}_2\right)$ and $\left(a_1,a_2\right)\left(b_1,b_2\right)\in E\left({\mathcal{G}}_1\times {\mathcal{G}}_2\right)$ if $a_1{b}_1\in E\left({\mathcal{G}}_1\right)$ and $a_2{b}_2\in E\left({\mathcal{G}}_2\right)$ . The strong product ${\mathcal{G}}_1\mathrm{⊠}{\mathcal{G}}_2$ of any two graphs ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$ is a graph whose vertex set is defined by $V\left({\mathcal{G}}_1\mathrm{⊠}{\mathcal{G}}_2\right)=V\left({\mathcal{G}}_1\right)\times V\left({\mathcal{G}}_2\right)$ and edge set is defined by $E\left({\mathcal{G}}_1\mathrm{⊠}{\mathcal{G}}_2\right)=E\left({\mathcal{G}}_1\square {\mathcal{G}}_2\right)\cup E\left({\mathcal{G}}_1\times {\mathcal{G}}_2\right)$ . The resistance distance among two vertices $u$ and $v$ of a graph $\mathcal{G}$ is determined as the effective resistance among the two vertices when a unit resistor replaces each edge of $\mathcal{G}$ . Let $P_n$ and $C_n$ denote a path and a cycle of order $n$ , respectively. In this paper, the generalized inverse of Laplacian matrix for the graphs $P_{n_1}\times C_{n_2}$ and $P_{n_1}\mathrm{⊠}{P}_{n_2}$ was procured, based on which the resistance distances of any two vertices in $P_{n_1}\times C_{n_2}$ and $P_{n_1}\mathrm{⊠}{P}_{n_2}$ can be acquired. Also, we give some examples as applications, which elucidated the effectiveness of the suggested method.