On maximum Wiener index of directed grids

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 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 Šumenjak et al. [17] 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.