On Minimum Generalized ABC Index of Graphs

Abstract

The generalized $ABC$ index of a graph $G$, denoted by $ABC_{\alpha}$, is defined as the sum of the terms $[(d(v)+d(u)-2)/d(v)d(u)]^\alpha$ over all pairs of adjacent vertices, where $d(u)$ is the degree of the vertex $u$ and $\alpha$ is a real number.
In this paper, we prove that for $\alpha\leq-1$, the balanced double broom is the unique tree that minimizes $ABC_\alpha$ among trees of order $n$ with diameter $d$, and trees of order $n$ with $k$ pendent vertices.