On the Balaban Index of Trees

Abstract

Let $G$ be a connected graph with edge set $E(G)$. The Balaban index of $G$ is defined as $J(G)=\frac{m}{\mu +1}\sum _{uv\in E(G)}(D_u{D}_v{)}^{-\frac{1}{2}}$ where $m=|E(G)|$, and $\mu$ is the cyclomatic number of $G$, $D_u$ is the sum of distances between vertex $u$ and all other vertices of $G$. We determine $n$-vertex trees with the first several largest and smallest Balaban indices.