Bounding the Mostar index

Abstract

Došlić et al. defined the Mostar index of a graph G as Mo(G)=∑uv∈E(G)|nG(u,v)−nG(v,u)|, where, for an edge uv of G, the term nG(u,v) denotes the number of vertices of G that have a smaller distance in G to u than to v. They conjectured that Mo(G)≤0.148‾n3 for every graph G of order n. As a natural upper bound on the Mostar index, Geneson and Tsai implicitly consider the parameter Mo⋆(G)=∑uv∈E(G)(n−min⁡{dG(u),dG(v)}). For a graph G of order n, they show that Mo⋆(G)≤524(1+o(1))n3.We improve this bound to Mo⋆(G)≤(23−1)n3, which is best possible up to terms of lower order. Furthermore, we show that Mo⋆(G)≤(2(Δn)2+(Δn)−2(Δn)(Δn)2+(Δn))n3 provided that G has maximum degree Δ.