The Maximum Balaban Index (Sum-Balaban Index) of Unicyclic Graphs

Abstract:

      The Balaban index of a connected graph $G$ is defined as $$J(G)=\frac{|E(G)| }{\mu +1}\sum_{e=uv\in E(G)}\frac{1}{{^{\sqrt{D_{G}(u)D_{G}(v)}}} },$$ and the Sum-Balaban index is defined as $$SJ(G)=\frac{|E(G)| }{\mu 1}\sum_{e=uv\in E(G)}\frac{1}{{^{\sqrt{D_{G}(u) D_{G}(v)}}} }, $$ where $D_{G}(u)=\sum_{w\in V(G)}d_{G}(u,w),$ and $\mu$ is the cyclomatic number of $G$. In this paper, the unicyclic graphs with the maximum Balaban index and the maximum Sum-Balaban index among all unicyclic graphs on $n$ vertices are characterized, respectively.