On the third ABC index of trees and unicyclic graphs

Let $G=(V,E)$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. The third atom-bond connectivity index, $AB{C}_3$ index, of $G$ is defined as $AB{C}_3(G)=\sum _{uv\in E(G)}\sqrt{\frac{e(u)+e(v)-2}{e(u)e(v)}}$, where eccentricity $e(u)$ is the largest distance between $u$ and any other vertex of $G$, namely $e(u)=\max \{d(u,v)\mathrm{\mid }v\in V(G)\}$. This work determines the maximal $AB{C}_3$ index of unicyclic graphs with any given girth and trees with any given diameter, and characterizes the corresponding graphs.