the extremal graphs for (sum-) balaban index of spiro and polyphenyl hexagonal chains

As highly discriminant distance-based topological indices, the balaban index and the sum-balaban 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 distance-based 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 sum-balaban index.