Mostar index and bounded maximum degree

Abstract

Došlić et al. defined the Mostar index of a graph $G$ as $Mo\left(G\right)=\sum _{uv\in E\left(G\right)}|n_G(u,v)-n_G(v,u)|$, 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 $Mo\left(G\right)\le \frac{\Delta }{2}{n}^2-(1-o\left(1\right))c_{\Delta }nlog(log\left(n\right)),$ where $c_{\Delta }>0$ only depends on $\Delta$ and the $o\left(1\right)$ 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 $Mo\left(G\right)\ge \frac{\Delta }{2}{n}^2-c_{\Delta }^{\prime }nlog\left(n\right),$ where $c_{\Delta }^{\prime }>0$ only depends on $\Delta$.