Orientations of graphs with maximum Wiener index

Abstract

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 $T$ 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 ${\Theta }_{a,b,c}$ with $a\ge b\ge c$ and $b>1$, the maximum Wiener index is achieved by the one in which the union of the paths between $u_1$ and $u_2$ forms a directed cycle of length $a+b+2$, where $u_1$ and $u_2$ are the vertex of degree 3. We confirm the validity of the conjecture.