A Note on Resistance Distances of Graphs - Sun, Wensheng; Yang, Yujun (Frontiers in Physics, 2022)
Let $G$ be a connected graph with vertex set $V(G)$. The resistance distance between any two vertices $u,v \in V(G)$ is the net effective resistance between them in the electric network constructed from $G$ by replacing each edge with a unit resistor. Let $S\subset V(G)$ be a set of vertices such that all the vertices in $S$ have the same neighborhood in $G-S$, and let $G[S]$ be the subgraph induced by $S$. In this note, by the \{1\}-inverse of the Laplacian matrix of $G$, formula for resistance distances between vertices in $S$ is obtained. It turns out that resistance distances between vertices in $S$ could be given in terms of elements in the inverse matrix of an auxiliary matrix of the Laplacian matrix of $G[S]$, which derives the reduction principle obtained in [J. Phys. A: Math. Theor. 41 (2008) 445203] by algebraic method.