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

On maximum Wiener index of directed grids

“…Let D m,n be the orientation of G m,n with all P m -layers oriented up except the last P m -layer which is oriented down, and all P n -layers oriented to the left except the first P n -layer which is oriented to the right, see Figure 3. The following conjecture was stated in [17]. The conjecture naturally generalizes the result for m = 2, but in this paper we show that it is not true if m ≥ 3.…”

“…In [17], the authors did not evaluate the Wiener index of D m,n . In order to be able to compare the Wiener indices of C m,n and D m,n , we prove the following statement.…”

“…If m = 2, the grid is called the ladder graph L n . Kraner Šumenjak et al [17] proved that the maximum Wiener index of a digraph whose underlying graph is L n is (8n 3 + 3n 2 − 5n + 6)/3. Moreover, the optimal orientation of L n is attained for orientation presented in Figure 1.…”

“…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 .…”

On maximum Wiener index of directed grids

“…If m = 2, the grid is called the ladder graph L n . Kraner Šumenjak et al [26] proved that the maximum Wiener index of a digraph whose underlying graph is L n is (8n 3 + 3n 2 − 5n + 6)/3. Moreover, the optimal orientation of L n is attained for orientation presented in Figure 1.…”