On the resistance regular graphs

TL;DR

This paper characterizes resistance regular graphs, providing necessary and sufficient conditions, bounds on resistance spectral radius and energy, and computing these metrics for specific resistance regular graphs.

Contribution

It offers a complete characterization of resistance regular graphs and derives bounds and exact values for their resistance spectral properties.

Findings

Characterization of resistance regular graphs

Sharp bounds for resistance spectral radius

Computed resistance spectrum and energy for specific graphs

Abstract

For a connected graph $G$, its resistance distance matrix is denoted by $R(G)$. A graph is called resistance regular if all the row (or column) sums of $R(G)$ are equal. We provide a necessary and sufficient condition for a simple connected graph to be resistance regular. Additionally, we establish sharp bounds for the resistance spectral radius and present various bounds for the resistance energy of $G$. Furthermore, we compute the resistance spectrum and resistance energy of some resistance regular graphs.