Orientations of graphs with maximum Wiener index.

In this paper, we study the Wiener index of the orientation of trees and theta‐graphs. An orientation of a tree is called no‐zig‐zag if there is no subpath in which edges change the orientation twice. Knor, Škrekovski, and Tepeh conjectured that every orientation of a tree T $T$ achieving the maximum Wiener index is no‐zig‐zag. We disprove this conjecture by constructing a counterexample. Knor, Škrekovski, and Tepeh conjectured that among all orientations of the theta‐graph Θa,b,c ${{\rm{\Theta }}}_{a,b,c}$ with a≥b≥c $a\ge b\ge c$ and b>1 $b\gt 1$, the maximum Wiener index is achieved by the one in which the union of the paths between u1 ${u}_{1}$ and u2 ${u}_{2}$ forms a directed cycle of length a+b+2 $a+b+2$, where u1 ${u}_{1}$ and u2 ${u}_{2}$ are the vertex of degree 3. We confirm the validity of the conjecture. [ABSTRACT FROM AUTHOR]