Further results regarding the degree resistance distance of cacti

TL;DR

This paper investigates the degree resistance distance in cacti graphs, corrects previous errors, and identifies the second and third minimal values along with their extremal structures.

Contribution

It corrects prior inaccuracies and determines the second and third minimal degree resistance distances in connected cacti graphs, characterizing the extremal graphs.

Findings

Corrected previous errors in degree resistance distance calculations.

Identified second-minimum and third-minimum degree resistance distances.

Characterized extremal graphs achieving these distances.

Abstract

A graph $G$ is called a cactus if each block of $G$ is either an edge or a cycle. Denote by $Cact(n;t)$ the set of connected cacti possessing $n$ vertices and $t$ cycles. In this paper, we show that there are some errors in [J. Du, G. Su, J. Tu, I. Gutman, The degree resistance distance of cacti, Discrete Appl. Math. 188 (2015) 16-24.], and we present some results which correct their mistakes. We also give the second-minimum and third-minimum degree resistance distances among graphs in $Cact(n;t)$, and characterize the corresponding extremal graphs as well.