The resistance distance of a dual number weighted graph

Abstract

For a graph $G=(V,E)$, assigning each edge $e\in E$ a weight of a dual number $w(e)=1+{\hat{a}}_e\epsilon$, the weighted graph $G^w=(V,E,w)$ is called a dual number weighted graph, where $-{\hat{a}}_e$ 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.

24 pages