The maximum degree resistance distance of cacti

TL;DR

This paper characterizes the cacti graphs with the maximum degree resistance distance among all cacti with a given number of vertices and cycles, extending previous unicyclic graph results.

Contribution

It provides a complete characterization of extremal cacti graphs with maximum degree resistance distance, generalizing earlier work on unicyclic graphs.

Findings

Identifies extremal cacti with maximum degree resistance distance.

Extends previous results from unicyclic to cactus graphs.

Provides a theoretical framework for analyzing resistance distances in complex graphs.

Abstract

Various topological indices, based on the distances between the vertices of a graph, are widely used in theoretical chemistry. The degree resistance distance of a 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 $u,$ and $R(u, v)$ the resistance distance between the vertices $u$ and $v.$ A graph $G$ is called a cactus if each block of $G$ is either an edge or a cycle. In this paper, we completely characterize the extremal cacti having the maximum degree resistance distance among all cacti with $n$ vertices and $t$ cycles, and extend some results of a recent paper [J. Tu, J. Du, G. Su, The unicyclic graphs with maximum degree resistance distance, Appl. Math. Comput. 268 (2015) 859-864].