Mostar index and bounded maximum degree

Abstract

Došlić et al. defined the Mostar index of a graph $G$ as $Mo(G)=\sum _{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. For a graph $G$ of order $n$ and maximum degree at most $\Delta$, we show $Mo(G)\le \frac{\Delta }{2}{n}^2-(1-o(1))c_{\Delta }n\log (\log (n)),$ where $c_{\Delta }>0$ only depends on $\Delta$ and the $o(1)$ term only depends on $n$. Furthermore, for integers $n_0$ and $\Delta$ at least $3$, we show the existence of a $\Delta$-regular graph of order $n$ at least $n_0$ with $Mo(G)\ge \frac{\Delta }{2}{n}^2-c_{\Delta }^{\prime }n\log (n),$ where $c_{\Delta }^{\prime }>0$ only depends on $\Delta$.