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.
From: Domingos Salazar [view email]
[v1]
Wed, 24 Jun 2026 18:16:36 UTC (15 KB)
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