Maximizing the Mostar index for bipartite graphs and split graphs

Došlić et al.~defined the Mostar index of a graph $G$ as $\sum _{uv\in E(G)}\mathrm{\mid }{n}_G(u,v)-n_G(v,u)\mathrm{\mid }$, 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 $v$. Contributing to conjectures posed by Došlić et al., we show that the Mostar index of bipartite graphs of order $n$ is at most $\frac{\sqrt{3}}{18}{n}^3$, and that the Mostar index of split graphs of order $n$ is at most $\frac{4}{27}{n}^3$.