# The eigenvalues of Hessian matrices of the complete and complete bipartite graphs

## TL;DR

This paper analyzes the eigenvalues of Hessian matrices for complete and complete bipartite graphs, revealing their Lorentzian metric properties and implications for algebraic structures.

## Contribution

It computes eigenvalues of Hessian matrices for these graphs and demonstrates the Lorentzian nature and algebraic properties related to the strong Lefschetz property.

## Key findings

- Eigenvalues include one positive and multiple negative values.
- The Hessian matrices exhibit Lorentzian metric properties.
- Strong Lefschetz property holds for related Artinian Gorenstein algebras.

## Abstract

In this paper, we consider the Hessian matrices $H_{\Gamma}$ of the complete and complete bipartite graphs, and the special value of $\tilde H_{\Gamma}$ at $x_{i}=1$ for all $x_{i}$. We compute the eigenvalues of $\tilde H_{\Gamma}$. We show that one of them is positive and that the others are negative. In other words, the metric with respect to the symmetric matrix $\tilde H_{\Gamma}$ is Lorentzian. Hence those Hessian $\det (H_{\Gamma})$ are not identically zero. As an application, we show the strong Lefschetz property for the Artinian Gorenstein algebra associated to the graphic matroids of the complete and complete bipartite graphs with at most five vertices.