Further Properties of Trees with Minimal Atom-Bond Connectivity Index

Abstract

Let $G=(V,E)$ be a graph the atom-bond connectivity (ABC) index is defined as the sum of weights $((d_u+d_v-\mathrm{2})/d_u{d}_v{)}^{\mathrm{1}/\mathrm{2}}$ over all edges $uv$ of $G$, where $d_u$ denotes the degree of a vertex $u$ of $G$. In this paper, we determined a few structural features of the trees with minimal ABC index also we characterized the trees with $\text{d}\text{i}\text{a}[T]=\mathrm{2}$ and minimal ABC index, where $[T]$ is induced by the vertices of degree greater than 2 in $T$ and $\text{d}\text{i}\text{a}[T]$ is the diameter of $[T]$.