On atom-bond connectivity index of cacti

Abstract

The atom-bond connectivity (ABC) index of a connected graph G is defined as $ABC(G)=\sum_{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}}$, where
$d(u)$ is the degree of vertex $u$ in $G$. Denote $\mathscr{G}^0(n,r)$ the set of cacti with $n$ vertices and $r$ cycles and $\mathscr{G}^1(n,r)$ the set of cacti with with $n$ vertices and $r$ pendent vertices. A
connected graph $G$ is called a cactus if any two of its cycles have at most one common vertex. In this paper, I give sharp bounds of ABC index of cacti among $\mathscr{G}^0(n,r)$ and $\mathscr{G}^1(n,r)$ respectively :(1) if $G\in \mathscr{G}^0(n,r)$
and $n\geq 5$, then $ABC(G)\leq \frac{3r}{\sqrt{2}}+(n-2r-1)\sqrt{\frac{n-2}{n-1}}$; (2) if $G\in \mathscr{G}^1(n,r)$ and $n\geq 3$, then
$ABC(G)\leq \frac{3(n-r-1) {2\sqrt{2}}+r\sqrt{\frac{n-2}{n-1}}$ when $n-r$ is odd and $ABC(G)\leq \frac{3(n-r-2)}{2\sqrt{2}}+r\sqrt{\frac{n-3}{n-2}}+\frac{1}{\sqrt{2}}$ when $n-r$ is even, and characterize the corresponding extremal cacti.