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

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 $G_1\diamond G_2$, 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 $G_1\diamond G_2$ are investigated, based on which the resistance distances of any two vertices in $G_1\circ G_2$ and $G_1\diamond G_2$ can be obtained. Moreover, some examples as applications are presented, which illustrate the correction and efficiency of the proposed method.