On ABS index of unicyclic graphs with fixed diameter

Abstract

  Given a simple connected graph $G=(V(G),E(G))$, the atom-bond sum-connectivity ($ABS$) index is defined as $$ABS(G)=\sum_{u v \in E(G)} \sqrt{\frac{d(u)+d(v)-2}{d(u)+d(v)}}=\sum_{u v \in E(G)}\sqrt{1-\frac{2}{d(u)+d(v)}},$$ where $d(u)$ and $d(v)$ are the degrees of $u,v\in V(G)$, respectively.

  In this paper, let $\mathcal{U}_{n,\alpha }$ be a set of all unicyclic graphs of order $n$ with diameter $\alpha$. Firstly, we present the minimum $ABS$ index of $G\in \mathcal{U}_{n,\alpha }$ with $ alpha \ge2$. We also determine the maximum $ABS$ index of $G\in \mathcal{U}_{n,\alpha }$ with $\alpha \ge4$. Finally, the corresponding extremal graphs with the sharp upper and lower bounds have been characterized, respectively.