On the Balaban Index of Chain Graphs

Abstract

The Balaban index and sum-Balaban index of a connected (molecular) graph G are defined as $$\begin{aligned} J(G)&=\frac{m}{\mu +1} \sum _{uv\in E(G)}\frac{1}{\sqrt{\sigma _{G}(u)\sigma _{G}(v)}}~ \text{ and }\\ SJ(G)&=\frac{m}{\mu +1} \sum _{uv\in E(G)}\frac{1}{\sqrt{\sigma _{G}(u)+\sigma _{G}(v)}}, \end{aligned}$$ respectively, where m is the number of edges, $$\mu $$ is the cyclomatic number, $$\sigma _G(u)$$ is the sum of distances between vertex u and all other vertices of G. In this paper, we establish that $$\begin{aligned} K\left( DS(n-3,\,1)\right)>K\left( DS(n-4,\,2)\right)>\cdots >K \left( DS\left( \left\lceil \frac{n}{2}\right\rceil -1,\, \left\lfloor \frac{n}{2}\right\rfloor -1\right) \right) \end{aligned}$$ $$(K=J,\,SJ),$$ where $$DS(p,\,q)$$ is a double star on $$n\,(=p+q+2,\,p\ge q)$$ vertices. As an application, we determine the extremal graphs of the Balaban index and the sum-Balaban index in the class of chain graphs G on n vertices, where G is a tree or a unicyclic graph. Finally, we give an open problem on Balaban (sum-Balaban) index of connected chain graphs.