Mostar index and bounded maximum degree

Došlić et al. defined the Mostar index of a graph G as Mo(G)=∑uv∈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 Δ, we show [Formula presented] where c_Δ>0 only depends on Δ and the o(1) term only depends on n. Furthermore, for integers n₀ and Δ at least 3, we show the existence of a Δ-regular graph of order n at least n₀ with [Formula presented] where c_Δ^′>0 only depends on Δ.