Bounding the Mostar index

TL;DR

This paper improves the upper bounds on the Mostar index of graphs, a measure related to vertex distances, and introduces a new parameter that provides tighter bounds, with implications for graph theory conjectures.

Contribution

The authors establish a sharper upper bound on the parameter $Mo^igstar(G)$, improving previous results and demonstrating its near-optimality, while also deriving bounds based on maximum degree.

Findings

Improved upper bound: $Mo^igstar(G) \\leq (\frac{2}{\sqrt{3}}-1)n^3$.

Bound depending on maximum degree: $Mo^igstar(G)$ expressed in terms of $\Delta$ and $n$.

Results are tight up to lower order terms.

Abstract

Do\v{s}li\'{c} et al. defined the Mostar index of a graph $G$ as $Mo(G)=\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, 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 $v$. They conjectured that $Mo(G)\leq 0.\overline{148}n^3$ for every graph $G$ of order $n$. As a natural upper bound on the Mostar index, Geneson and Tsai implicitly consider the parameter $Mo^\star(G)=\sum\limits_{uv\in E(G)}\big(n-\min\{ d_G(u),d_G(v)\}\big)$. For a graph $G$ of order $n$, they show that $Mo^\star(G)\leq \frac{5}{24}(1+o(1))n^3$. We improve this bound to $Mo^\star(G)\leq \left(\frac{2}{\sqrt{3}}-1\right)n^3$, which is best possible up to terms of lower order. Furthermore, we show that $Mo^\star(G)\leq \left(2\left(\frac{\Delta}{n}\right)^2+\left(\frac{\Delta}{n}\right)-2\left(\frac{\Delta}{n}\right)\sqrt{\left(\frac{\Delta}{n}\right)^2+\left(\frac{\Delta}{n}\right)}\right)n^3$ provided that $G$ has maximum degree $\Delta$.