Resistance matrices of balanced directed graphs

Abstract

Let $G$ be a strongly connected and balanced directed graph. The Laplacian matrix of $G$ is then the matrix (not necessarily symmetric) $L:=D-A$, where $A$ is the adjacency matrix of $G$ and $D$ is the diagonal matrix such that the row sums and the column sums of $L$ are equal to zero. Let $L^\dag=[l^{\dag}_{ij}]$ be the Moore-Penrose inverse of $L$. We define the resistance between any two vertices $i$ and $j$ of $G$ by $r_{ij}:=l^{\dag}_{ii}+l^{\dag}_{jj}-2l^{\dag}_{ij}$. In this paper, we derive some interesting properties of the resistance and the corresponding resistance matrix $[r_{ij}]$.