The resistance distance of a dual number weighted graph

TL;DR

This paper introduces a method to compute resistance distances and Kirchhoff indices in dual number weighted graphs, accounting for perturbations in edge weights, and provides explicit formulas and bounds for these measures.

Contribution

It derives explicit formulas and block representations for the generalized inverses of Laplacian matrices in dual number weighted graphs, including perturbation bounds.

Findings

Explicit formulas for resistance distance and Kirchhoff index in dual number weighted graphs.

Block representations of generalized Laplacian inverses.

Perturbation bounds for resistance distance and Kirchhoff index.

Abstract

For a graph $G=(V,E)$, assigning each edge $e\in E$ a weight of a dual number $w(e)=1+\widehat{a}_{e}\varepsilon$, the weighted graph $G^{w}=(V,E,w)$ is called a dual number weighted graph, where $-\widehat{a}_{e}$ can be regarded as the perturbation of the unit resistor on edge $e$ of $G$. For a connected dual number weighted graph $G^{w}$, we give some expressions and block representations of generalized inverses of the Laplacian matrix of $G^{w}$. And using these results, we derive the explicit formulas of the resistance distance and Kirchhoff index of $G^{w}$. We give the perturbation bounds for the resistance distance and Kirchhoff index of $G$. In particular, when only the edge $e=\{i,j\}$ of $G$ is perturbed, we give the perturbation bounds for the Kirchhoff index and resistance distance between vertices $i$ and $j$ of $G$, respectively.