Cacti with maximum Kirchhoff index

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.