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arXiv · 2606.26275

Bounded Treewidth and Complete Monotonicity for Scott-Sokal Spanning-Tree Polynomials

Domingos S. P. Salazar
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Abstract

Scott and Sokal asked for a structural description of the finite graphs \(G\) for which inverse powers \(T_G^{-\beta}\) of the spanning-tree polynomial are completely monotone. We prove the following bounded-treewidth criterion: if \(G\) is a finite connected simple graph with \(\operatorname{tw}(G)\le k\), then \(T_G^{-\beta}\) is completely monotone for every \(\beta>(k-1)/2\). Consequently, every partial \(3\)-tree is covered throughout the first Scott--Sokal open interval \(1<\beta<3/2\), including finite Apollonian networks, \(K_5-e\), and the four-spoke wheel \(W_4\). The proof combines the real Riesz/Wishart integral for determinants, star--mesh elimination of simplicial vertices, and a Gaussian Laplace kernel for the degree-\(d\) star. General bounded-treewidth graphs follow by chordal completion and monotone deletion of completion edges.

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Domingos S. P. Salazar. 2026-06-24. Bounded Treewidth and Complete Monotonicity for Scott-Sokal Spanning-Tree Polynomials. https://arxiv.org/abs/2606.26275

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