Resistance distance in connected balanced digraphs

TL;DR

This paper proves that in certain connected balanced directed graphs, the resistance distance is always less than or equal to the shortest path distance, generalizing previous results and confirming a longstanding conjecture.

Contribution

It introduces a new class of connected balanced digraphs where resistance distance is bounded by shortest path distance, extending earlier findings and simplifying proofs.

Findings

Resistance distance is less than or equal to shortest path distance in the studied class.

The result generalizes previous work on directed cacti.

Provides an affirmative answer to a well-known conjecture.

Abstract

Let $D = (V, E)$ be a strongly connected and balanced digraph with vertex set $V$ and arc set $E.$ The classical distance $d_{ij}^D$ from $i$ to $j$ in $D$ is the length of a shortest directed path from $i$ to $j$ in $D.$ Let $L$ be the Laplacian matrix of $D$ and $ L^{\dagger} = ( l_{ij}^{\dagger} )$ be the Moore-Penrose inverse of $L.$ The resistance distance from $i$ to $j$ is then defined by $r_{ij}^D := l_{ii}^{\dagger } + l_{jj}^{\dagger } - 2 l_{ij}^{\dagger }.$ Let $\{ D_1, D_2, ...., D_k \}$ be a sequence of strongly connected balanced digraphs with $D_i \cap D_j$ having at most one vertex in common for all $i \neq j$ and with $r_{ij}^{D_t} \leq d_{ij}^{D_t} \ \forall \ t = 1 \ \mathrm{to} \ k.$ Let $\mathcal{C}$ be a collection of connected, balanced digraphs, each member of which is a finite union of the form $D_1 \cup D_2 \cup ....\cup D_k$ where each $D_i$ is a connected and balanced digraph with $D_{i} \cap ( D_1 \cup D_2 \cup ....\cup D_{i-1} )$ being a single vertex, for all $i,$ $1 < i \leq k.$ In this paper, we show that for any digraph $D$ in $\mathcal{C}$, $r_{ij}^D \leq d_{ij}^D \ (*)$. This is established by partitioning the Laplacian matrix of $D$. This generalizes the main result in [3]. As a corollary, we deduce a simpler proof of the result in [3], namely, that for any directed cactus $D$, the inequality (*) holds. Our results provide an affirmative answer to a well known interesting conjecture ( cf : Conjecture 1.3 ).