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

In this paper, we consider the Hessian matrices $H_{\mathrm{\Gamma }}$ of the complete and complete bipartite graphs, and the special value of ${\tilde{H}}_{\mathrm{\Gamma }}$ at $x_i=1$ for all $x_i$. We compute the eigenvalues of ${\tilde{H}}_{\mathrm{\Gamma }}$. 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 ${\tilde{H}}_{\mathrm{\Gamma }}$ is Lorentzian. Hence those Hessian $\det (H_{\mathrm{\Gamma }})$ 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.