On the Atom-Bond Connectivity Index of Cacti


Hawei Dong, Xiaoxia Wu




The Atom-Bond Connectivity (ABC) index of a connected graph $G$ is defined as $\operatorname{ABC}(G)=\sum_{uv\in E(G)}\sqrt{\frac{d(u)+d(v)-2}{d(u)d(v)}}$ where $d(u)$ is the degree of vertex $u$ in $G$. A connected graph $G$ is called a cactus if any two of its cycles have at most one common vertex. Denote by $\Cal G^0(n,r)$ the set of cacti with $n$ vertices and $r$ cycles and $\Cal G^1(n,p)$ the set of cacti with $n$ vertices and $p$ pendent vertices. In this paper, we give sharp bounds of the ABC index of cacti among $\Cal G^0(n,r)$ and $\Cal G^1(n,p)$ respectively, and characterize the corresponding extremal cacti.