Disproof of a conjecture on the minimum Wiener index of signed trees

TL;DR

This paper disproves a conjecture that the path with alternating signs minimizes the Wiener index among signed trees, by providing counterexamples for trees with at least 30 vertices.

Contribution

The paper constructs counterexamples to a previous conjecture, showing that the minimal Wiener index in signed trees is not always achieved by the alternating path.

Findings

Counterexamples for n ≥ 30

Conjecture is false for large trees

Minimal Wiener index not always on alternating paths

Abstract

The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices. Sam Spiro [The Wiener index of signed graphs, Appl. Math. Comput., 416(2022)126755] recently introduced the Wiener index for a signed graph and conjectured that the path $P_n$ with alternating signs has the minimum Wiener index among all signed trees with $n$ vertices. By constructing an infinite family of counterexamples, we prove that the conjecture is false whenever $n$ is at least 30.