Resistance distances in corona and neighborhood corona graphs with Laplacian generalized inverse approach

Jia-Bao Liu; Xiang-Feng Pan; Fu-Tao Hu

paper

Resistance distances in corona and neighborhood corona graphs with Laplacian generalized inverse approach

arXiv:1503.07842

Abstract

Let $G_{1}$ and $G_{2}$ be two graphs on disjoint sets of $n_{1}$ and $n_{2}$ vertices, respectively. The corona of graphs $G_{1}$ and $G_{2}$ , denoted by $G_{1} \circ G_{2}$ , is the graph formed from one copy of $G_{1}$ and $n_{1}$ copies of $G_{2}$ where the $i$ -th vertex of $G_{1}$ is adjacent to every vertex in the $i$ -th copy of $G_{2}$ . The neighborhood corona of $G_{1}$ and $G_{2}$ , denoted by $G_{1} ⋄ G_{2}$ , is the graph obtained by taking one copy of $G_{1}$ and $n_{1}$ copies of $G_{2}$ and joining every neighbor of the $i$ -th vertex of $G_{1}$ to every vertex in the $i$ -th copy of $G_{2}$ by a new edge. In this paper, the Laplacian generalized inverse for the graphs $G_{1} \circ G_{2}$ and $G_{1} ⋄ G_{2}$ are investigated, based on which the resistance distances of any two vertices in $G_{1} \circ G_{2}$ and $G_{1} ⋄ G_{2}$ can be obtained. Moreover, some examples as applications are presented, which illustrate the correction and efficiency of the proposed method.