On maximum Wiener index of directed grids
Keywords:Wiener index, topological graph theory, graph distance
This paper is devoted to Wiener index of directed graphs, more precisely of directed grids. The grid Gm, n is the Cartesian product Pm□Pn of paths on m and n vertices, and in a particular case when m = 2, it is a called the ladder graph Ln. Kraner Šumenjak et al. in 2021 proved that the maximum Wiener index of a digraph, which is obtained by orienting the edges of Ln, 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 Gm, n will attain the maximum Wiener index among all orientations of Gm, n. In this paper we disprove the conjecture by showing that a comb-like orientation of Gm, n has significiantly bigger Wiener index.