The eigenvalues of the Hessian matrices of the generating functions for trees with $k$ components

TL;DR

This paper investigates the eigenvalues of the Hessian matrices of generating functions related to truncated graphic matroids, demonstrating non-vanishing Hessians with exactly one positive eigenvalue for complete and bipartite graphs, and applying results to algebraic properties.

Contribution

It computes the eigenvalues of the Hessian matrices of basis generating functions for truncated graphic matroids of complete and bipartite graphs, revealing their spectral properties and algebraic implications.

Findings

Hessian matrices of the generating functions do not vanish.

Hessian matrices have exactly one positive eigenvalue.

Results imply the strong Lefschetz property for associated Gorenstein algebras.

Abstract

Let us consider a truncated matroid $M_{\Gamma}^{r}$ of rank $r$ of a graphic matroid of a graph $\Gamma$. The basis for $M_{\Gamma}^{r}$ is the set of the forests with $r$ edges in $\Gamma$. We consider this basis generating function and compute its Hessian. In this paper, we show that the Hessian of the basis generating function of the truncated matroid of the graphic matroid of the complete or complete bipartite graph does not vanish by calculating the eigenvalues of the Hessian matrix. Moreover, we show that the Hessian matrix of the basis generating function of the truncated matroid of the graphic matroid of the complete or complete bipartite graph has exactly one positive eigenvalue. As an application, we show the strong Lefschetz property for the Artinian Gorenstein algebra associated to the truncated matroid.