The Extremal Graphs for (Sum) Balaban Index of Spiro and Polyphenyl Hexagonal Chains

As highly discriminant distancebased topological indices, the Balaban index and the sumBalaban index of a graph $G$ are defined as $J(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)D_{G}(v)}}$ and $SJ(G)=frac{m}{mu+1}sumlimits_{uvin E} frac{1}{sqrt{D_{G}(u)+D_{G}(v)}}$, respectively, where $D_{G}(u)=sumlimits_{vin V}d(u,v)$ is the distance sum of vertex $u$ in $G$, $m$ is the number of edges and $mu$ is the cyclomatic number of $G$. They are useful distancebased descriptor in chemometrics. In this paper, we focus on the extremal graphs of spiro and polyphenyl hexagonal chains with respect to the Balaban index and the sumBalaban index.