A proof of a conjecture on maximum Wiener index of oriented ladder graphs

The ladder graph $$L_n$$ is the Cartesian product of a path on n vertices and a complete graph on two vertices. The Wiener index of a digraph is the sum of distances between all ordered pairs of vertices. In Knor et al. (Bounds in chemical graph theory - advances, 2017) the authors conjectured that the maximum Wiener index of a digraph whose underlying graph is $$L_n$$ is $$(8n^3+3n^2-5n+6)/3$$ . In this article we prove the conjecture.