Resistance distance in directed cactus graphs

TL;DR

This paper proves that in directed cactus graphs, the resistance distance between any two vertices is always less than or equal to the classical directed distance, highlighting a key property of resistance metrics in such graphs.

Contribution

The paper establishes the inequality between resistance and classical distances specifically for directed cactus graphs, a class of graphs where this was previously unproven.

Findings

Resistance distance is always less than or equal to classical distance in directed cactus graphs.

The inequality holds for all strongly connected, balanced directed cactus graphs.

Provides a proof for a previously unproven property of resistance distances in this graph class.

Abstract

Let $G=(V,E)$ be a strongly connected and balanced digraph with vertex set $V=\{1,\dotsc,n\}$. The classical distance $d_{ij}$ between any two vertices $i$ and $j$ in $G$ is the minimum length of all the directed paths joining $i$ and $j$. The resistance distance (or, simply the resistance) between any two vertices $i$ and $j$ in $V$ is defined by $r_{ij}:=l_{ii}^{\dag}+l_{jj}^{\dag}-2l_{ij}^{\dag}$, where $l_{pq}^{\dagger}$ is the $(p,q)^{\rm th}$ entry of the Moore-Penrose inverse of $L$ which is the Laplacian matrix of $G$. In practice, the resistance $r_{ij}$ is more significant than the classical distance. One reason for this is, numerical examples show that the resistance distance between $i$ and $j$ is always less than or equal to the classical distance, i.e. $r_{ij} \leq d_{ij}$. However, no proof for this inequality is known. In this paper, we show that this inequality holds for all directed cactus graphs.