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

摘要

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 Pn 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.