On cacti with large Mostar index

Abstract

The Mostar index of a graph $G$ is defined as the sum of absolute values of the  differences between $n_u$ and $n_v$ over all edges $uv$ of $G$,  where $n_u$ and $n_v$ are respectively,  the number of vertices of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of vertices of $G$ lying closer to vertex $v$ than to vertex $u$.  A cactus is a graph in which any two cycles have at most one common vertex.  In this paper, we determine all the $n$-vertex cacti with the largest Mostar index, and we give a sharp upper bound of the Mostar index for cacti of order $n$ with $k$ cycles, and characterize all the cacti that achieve this  bound.