Bicyclic graphs with extremal degree resistance distance

TL;DR

This paper investigates bicyclic graphs to identify those with the minimum and maximum degree resistance distance, providing a complete characterization of extremal cases based on resistance distance properties.

Contribution

It determines and characterizes the bicyclic graphs with extremal degree resistance distance, filling a gap in understanding resistance metrics in such graph classes.

Findings

Identified bicyclic graphs with minimum degree resistance distance.

Identified bicyclic graphs with maximum degree resistance distance.

Provided a complete characterization of extremal bicyclic graphs.

Abstract

Let $r(u,v)$ be the resistance distance between two vertices $u, v$ of a simple graph $G$, which is the effective resistance between the vertices in the corresponding electrical network constructed from $G$ by replacing each edge of $G$ with a unit resistor. The degree resistance distance of a simple graph $G$ is defined as ${D_R}(G) = \sum\limits_{\{u,v\} \subseteq V(G)} {[d(u) + d(v)]r(u,v)},$ where $d(u)$ is the degree of the vertex $u$. In this paper, the bicyclic graphs with extremal degree resistance distance are strong-minded. We first determine the $n$-vertex bicyclic graphs having precisely two cycles with minimum and maximum degree resistance distance. We then completely characterize the bicyclic graphs with extremal degree resistance distance.