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arXiv · 2303.14810

A Note on Polynomial Certificates for Walk Inequalities

Nadja Willenborg·Sven Kosub
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Abstract

Let $w_m(G)$ denote the total number of walks of length $m$ in an undirected graph $G$. Spectral decomposition shows that $(w_m(G))_{m\ge 0}$ is a moment sequence of a finite positive measure. We use exchangeability of its product measures to turn global nonnegativity of polynomial symmetrizations into universal inequalities for the number of walks. For exponent vectors $\alpha,\beta$ in distinct permutation orbits, $\mathrm{Sym}(x^\beta-x^\alpha)$ is globally nonnegative exactly when $\beta$ is coordinatewise even and majorizes $\alpha$. This finite criterion includes several classical inequalities as special cases; univariate polynomials and squared alternants also yield linear and Hankel determinant inequalities.

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Nadja Willenborg, Sven Kosub. 2023-03-26. A Note on Polynomial Certificates for Walk Inequalities. https://arxiv.org/abs/2303.14810

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