On maximum Wiener index of directed grids

TL;DR

This paper investigates the maximum Wiener index of directed grid graphs, disproving a conjecture that a certain orientation maximizes it, by showing a comb-like orientation yields a larger Wiener index.

Contribution

The paper challenges a previous conjecture by demonstrating that a novel comb-like orientation of directed grids achieves a higher Wiener index.

Findings

A comb-like orientation outperforms the conjectured optimal orientation in maximizing Wiener index.

The maximum Wiener index for directed grids is achieved by a non-intuitive orientation.

The study provides new insights into the structure of directed graphs with extremal Wiener indices.

Abstract

This paper is devoted to Wiener index of directed graphs, more precisely of directed grids. The grid $G_{m,n}$ is the Cartesian product $P_m\Box P_n$ of paths on $m$ and $n$ vertices, and in a particular case when $m=2$, it is a called the ladder graph $L_n$. Kraner \v{S}umenjak et al. proved that the maximum Wiener index of a digraph, which is obtained by orienting the edges of $L_n$, is obtained when all layers isomorphic to one factor are directed paths directed in the same way except one (corresponding to an endvertex of the other factor) which is a directed path directed in the opposite way. Then they conjectured that the natural generalization of this orientation to $G_{m,n}$ will attain the maximum Wiener index among all orientations of $G_{m,n}$. In this paper we disprove the conjecture by showing that a comb-like orientation of $G_{m,n}$ has significiantly bigger Wiener index.