On Sum-Balaban Index

Abstract

The sum-Balaban index of a connected graph $G$ is defined as

$$J_e(G)=\frac{m}{\mu +1}\sum _{uv\in E(G)}{(D_u+D_v)}^{-\frac{1}{2}},$$

where $D_u$ is the sum of distances between vertex $u$ and all other vertices, $\mu$ is the cyclomatic number, $E(G)$ is the edge set, and $m=|E(G)|$. We establish various upper and lower bounds for the sum-Balaban index, and determine the trees with the largest, second-largest, and third-largest as well as the smallest, second-smallest, and third-smallest sum-Balaban indices among the $n$-vertex trees for $n\ge 6$.