Resistance matrices of balanced directed graphs

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 $L^†=[l^†_{ij}]$ be the Moore-Penrose inverse of $L$. We define the resistance between any two vertices $i$ and $j$ of $G$ by $r_{ij}:=l^†_{ii}+l^†_{jj}-2l^†_{ij}$. In this paper, we derive some interesting properties of the resistance and the corresponding resistance matrix $[r_{ij}]$.

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