Trees with Given Diameter Minimizing the Augmented Zagreb Index and Maximizing the ABC Index

Abstract

Let $G$ be a simple connected graph with vertex set $V (G)$ and edge set $E(G)$. The augmented Zagreb index of a graph $G$ is defined as

$$AZI(G)=\sum_{uv\in E(G)}\left(\frac{d_ud_v}{d_u+d_v-2}\right)^3,$$

and the atom-bond connectivity index (ABC index for short) of a graph $G$ is defined as$$ABC(G)=\sum_{uv\in E(G)}\sqrt{\frac{d_u+d_v-2}{d_ud_v}},$$

where $d_u$ and $d_v$ denote the degree of vertices $u$ and $v$ in $G$, respectively. In this paper, trees with given diameter minimizing the augmented Zagreb index and maximizing the ABC index are determined, respectively.