Computing the Mostar Index in Networks with Applications to Molecular Graphs

Recently, a bond-additive topological descriptor, named as the Mostar index, has been introduced as a measure of peripherality in networks. For a connected graph $G$, the Mostar index is defined as $Mo(G) = sum_{e=uv in E(G)} |n_u(e) - n_v(e)|$, where for an edge $e=uv$ we denote by $n_u(e)$ the number of vertices of $G$ that are closer to $u$ than to $v$ and by $n_v(e)$ the number of vertices of $G$ that are closer to $v$ than to $u$. In the present paper, we prove that the Mostar index of a weighted graph can be computed in terms of Mostar indices of weighted quotient graphs. Inspired by this result, several generalizations to other versions of the Mostar index already appeared in the literature. Furthermore, we apply the obtained method to benzenoid systems, tree-like polyphenyl systems, and to a fullerene patch. Closed-form formulas for two families of these molecular graphs are also deduced.