On the maximum atom-bond sum-connectivity index of cacti with given parameters

Abstract

The atom-bond sum-connectivity (ABS) index of a graph $G$ is defined as $ABS(G)=\sum_{uv\in E(G)}\sqrt{\frac{d_{G}(u)+d_{G}(v)-2}{d_{G}(u)+d_{G}(v)}}$, where $d_{G}(u)$ represents the degree of vertex $u\in V(G)$. A connected graph $G$ is called a cactus if its any two cycles have no common edge. Let $\mathscr{C}(n,s)$ be the set of cacti of order $n$ with $s$ cycles, $\mathscr{P}(n,t)$ be the set of cacti of order $n$ with $t$ pendent vertices, $\mathscr{T}(2\beta,q)$ be the set of cacti of order $2\beta$ with a perfect matching $M$ and $q$ cycles, respectively. In this paper we determine the sharp upper bounds on the ABS index of the cacti, respectively, among $\mathscr{C}(n,s)$, $\mathscr{P}(n,t)$ and $\mathscr{T}(2\beta,q)$. Moreover, the corresponding cacti are characterized at which the extremal values are attained.