Maximum reciprocal degree resistance distance index of unicyclic graphs

TL;DR

This paper investigates the maximum reciprocal degree resistance distance index in unicyclic graphs, identifying the extremal graph that maximizes this index among all such graphs with a given number of vertices.

Contribution

It characterizes the unicyclic graph with the highest reciprocal degree resistance distance index for any number of vertices, filling a gap in graph theory extremal problems.

Findings

Identifies the extremal unicyclic graph with maximum RDR index.

Provides a characterization of the extremal graph.

Establishes the maximum RDR index among unicyclic graphs.

Abstract

The reciprocal degree resistance distance index of a connected graph $G$ is defined as $RDR(G)=\sum\limits_{\{u,v\}\subseteq V(G)}\frac {d_G(u)+d_G(v)}{r_G(u,v)}$, where $r_G(u,v)$ is the resistance distance between vertices $u$ and $v$ in $G$. Let $\mathscr {U}_n$ denote the set of unicyclic graphs with $n$ vertices. We study the graph with maximum reciprocal degree resistance distance index among all graphs in $\mathscr {U}_n$, and characterize the corresponding extremal graph.