Cacti with maximum Kirchhoff index

TL;DR

This paper characterizes the maximum Kirchhoff index among cacti graphs, which are connected graphs with blocks that are edges or cycles, and identifies the extremal graph achieving this maximum.

Contribution

It provides a complete characterization of the cacti graphs with the maximum Kirchhoff index and identifies the extremal structure.

Findings

Maximum Kirchhoff index for cacti graphs is established.

Extremal cacti graph structure is identified.

Results contribute to understanding resistance distances in graph theory.

Abstract

The concept of resistance distance was first proposed by Klein and Randi\'c. The Kirchhoff index $Kf(G)$ of a graph $G$ is the sum of resistance distance between all pairs of vertices in $G$. A connected graph $G$ is called a cactus if each block of $G$ is either an edge or a cycle. Let $Cat(n;t)$ be the set of connected cacti possessing $n$ vertices and $t$ cycles, where $0\leq t \leq \lfloor\frac{n-1}{2}\rfloor$. In this paper, the maximum kirchhoff index of cacti are characterized, as well as the corresponding extremal graph.