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

Chuanming Wang; Songlin Guo; Wei Wang

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

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arxiv 2208.01984 v1 pith:6LFTBQMQ submitted 2022-08-03 math.CO

Songlin Guo , Wei Wang , Chuanming Wang This is my paper · ORCID

classification math.CO

keywords indexwienersignedconjecturegraphminimumtreesvertices

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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.

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