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 \(J(G) = \frac{m}{\mu+1} \sum_{uv \in E(G)} ({D_uD_v})^{-\frac{1}{2}}\) 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.