Bounding the Mostar index

Doslic et al. defined the Mostar index of a graph Gas Mo(G) = Sigma(uv is an element of(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. They conjectured that Mo( G) <= 0.(148) over barn(3) for every graph G of order n. As a natural upper bound on the Mostar index, Geneson and Tsai implicitly consider the parameter Mo-star(G) = Sigma(uv is an element of(G)) (n-min{d(G)(u), d(G)(v)}). For a graph G of order n, they show that Mo-star(G) <= 5/24 (1+o(1))n(3) . We improve this bound to Mo-star(G) <= (2 root 3 -1)n(3) which is best possible up to terms of lower order. Furthermore, we show that Mo-star(G)<=(2(Delta/n)(2) + (Delta/n) - 2(Delta/n) root(Delta/n)(2) + (Delta/n))n(3) provided that G has maximum degree Delta. (c) 2023 Elsevier B.V. All rights reserved.