The maximum Sum-Balaban Index of Trees with Given Diameter

Abstract

The Sum-Balaban index is defined as
\[SJ(G) = \frac{|E(G)|}{\mu+1} \sum\limits_{uv \in E(G)} \frac{1}{\sqrt{D_G(u)+D_G(v)}}\],
where \(\mu\) is the cyclomatic number of \(G\) and \(D_G(u)=\sum_{u\in V(G)}d_G(u,v)\). In this paper, we characterize the tree with the maximum Sum-Balaban index among all trees with \(n\) vertices and diameter \(d\). We also provide a new proof of the result that the star \(S_n\) is the graph which has the maximum Sum-Balaban index among all trees with \(n\) vertices. Furthermore, we propose a problem for further research.