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 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 with and , the maximum Wiener index is achieved by the one in which the union of the paths between and forms a directed cycle of length , where and are the vertex of degree 3. We confirm the validity of the conjecture.