Séminaire Lotharingien de Combinatoire, 84B.38 (2020), 12 pp.

Takahiro Nagaoka and Akiko Yazawa

Strict Log-Concavity of the Kirchhoff Polynomial and Its Applications

Abstract. Anari, Gharan, and Vinzant showed that the basis generating functions for all matroids are log-concave. In this paper, we show that Kirchhoff polynomials, i.e. the basis generating functions for simple graphic matroids, are strictly log-concave. Our key observation is that the Kirchhoff polynomial of a complete graph can be seen as the irreducible relative invariant of a certain prehomogeneous vector space. Furthermore, we prove that an algebra associated to a graphic matroid satisfies the strong Lefschetz property and Hodge-Riemann bilinear relation at degree one.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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