The resistance distance of a dual number weighted graph

TL;DR

The paper develops a framework for dual number weighted graphs to study perturbations of electrical networks. By establishing the existence and explicit forms of $L_{w}^{\{1\}}$ and $L_{w}^{\dag}$, it derives exact formulas for resistance distance $R_{ij}(G^{w})$ and Kirchhoff index $Kf(G^{w})$ under perturbations, expressed through the original Laplacian and the perturbation $\widehat{L}$. It also provides sharp perturbation bounds for $R_{ij}$ and $Kf$ in terms of the spectrum of $L$ and $\widehat{L}$, and specializes to edge-wise perturbations to quantify their impact on network efficiencies. These results enable precise sensitivity analysis of network resistance measures under infinitesimal dual-number perturbations, with implications for robustness and network design in perturbed environments.

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.

Theorems & Definitions (20)

• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof