### 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.

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