On the Mostar index of trees and product graphs

Abstract

If $G$ is a graph, and if for $e=uv\in E(G)$ the number of vertices closer to $u$ than to $v$ is denoted by $n_u$, then ${\rm Mo}(G) = \sum_{uv\in E(G)} |n_u-n_v|$ is the Mostar index of $G$. In this paper, the Mostar index is studied on trees and graph products. Lower and upper bounds are given on the difference between the Mostar indices of a tree and a tree obtained by contraction one of its edges and the  corresponding extremal trees are characterized. An upper bound on the Mostar index for the class of all trees but the stars is proved. Extremal results are also determined on the $(k+1)$-th largest/smallest Mostar index. The index is also studied on Cartesian and corona products.