Resistance distances in generalized join graphs

Abstract

Let $H$ be a graph with vertex set $V\left(H\right)=\{v_1,v_2,\dots ,v_k\}$. The generalized join graph $H[G_1,G_2,\dots ,G_k]$ is obtained from $H$ by replacing each vertex $v_i$ with a graph $G_i$ and joining each vertex in $G_i$ with each vertex in $G_j$ provided $v_i{v}_j\in E\left(H\right)$. If every $G_i$ is an independent set of $n_i$ vertices, then we write $H[G_1,G_2,\dots ,G_k]$ as $H[n_1,n_2,\dots ,n_k]$, which is called the blow-up of $H$. In this paper we introduce the local complement transformation in electrical networks and obtain an electrically equivalent graph of $H[n_1,n_2,\dots ,n_k]$. As their applications, we obtain formulae for resistance distances of some vertex-weighted graphs and give a unified technique to compute resistance distances in $H[G_1,G_2,\dots ,G_k]$ when every $G_i$ for $1\le i\le k$ is a graph consisting of a matching and isolated vertices, which results in closed formulae for resistance distances of $H[G_1,G_2,\dots ,G_k]$ when $H$ are some given graphs.