Affiliation:
1. College of Mathematics and System Sciences Xinjiang University Urumqi Xinjiang People's Republic of China
Abstract
AbstractIn 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.
Funder
National Natural Science Foundation of China