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arXiv · 2605.25088

The size of the spanning-tree spectrum of simple graphs

Vishesh Jain
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Abstract

For a graph $G$, let $τ(G)$ denote the number of spanning trees. We show that for every fixed $0 < c < 1/4$, the number of distinct values of $τ(G)$, as $G$ ranges over simple graphs on $n$ vertices, is at least $\exp(c n \log n)$ for all sufficiently large $n$. This is optimal up to the choice of the constant $c$ and resolves a conjecture of Chan-Kontorovich-Pak regarding a problem of Sedláček from the late 1960s.

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Vishesh Jain. 2026-05-24. The size of the spanning-tree spectrum of simple graphs. https://arxiv.org/abs/2605.25088

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