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 nu and nv over all edges e = uv of G, where nu(e) and nv(e) 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.