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.

在本文中，我们考虑了海森矩阵\（H _ {\伽玛} \）的完整和完全二部图，以及特殊值\（{\波浪号{H}} _ {\伽玛} \）在\（所有\（x_ {i} \）的x_ {i} = 1 \）。我们计算\（{\ tilde {H}} _ {\ Gamma} \）的特征值。我们显示其中一个是积极的，而其他则是消极的。换句话说，关于对称矩阵\（\ tilde {H} _ {\ Gamma} \）的度量是洛伦兹式的。因此，那些黑森州\（\ det（H _ {\ Gamma}）\）不完全相同为零。作为一个应用程序，我们展示了Artinian Gorenstein代数的强Lefschetz属性，该代数与具有最多五个顶点的完整和完整二部图的图形拟阵有关。