We study the Gaussian statistical models whose log-likelihood function has a unique complex critical point, i.e., has maximum likelihood degree one. We exploit the connection developed by Am\'endola et. al. between the models having maximum likelihood degree one and homaloidal polynomials. We study the spanning tree generating function of a graph and show this polynomial is homaloidal when the graph is chordal. When the graph is a cycle on $n$ vertices, $n \geq 4$, we prove the polynomial is not homaloidal, and show that the maximum likelihood degree of the resulting model is the $n$th Eulerian number. These results support our conjecture that the spanning tree generating function is a homaloidal polynomial if and only if the graph is chordal. We also provide an algebraic formulation for the defining equations of these models. Using existing results, we provide a computational study on constructing new families of homaloidal polynomials. In the end, we analyze the symmetric determinantal representation of such polynomials and provide an upper bound on the size of the matrices involved.

翻译：我们研究对数似然函数具有唯一复临界点（即最大似然度为一）的高斯统计模型。本文利用Améndola等人提出的模型与同调多项式之间的关联，探讨了图的生成树生成函数，证明当图为弦图时该多项式为同调多项式。当图为$n$个顶点（$n \geq 4$）的环时，我们证明该多项式不是同调的，并表明所得模型的最大似然度为第$n$个欧拉数。这些结果支持我们的猜想：生成树生成函数为同调多项式当且仅当图为弦图。我们还给出了这些模型定义方程的代数形式。基于现有结论，我们通过计算研究构建了同调多项式的新族。最后，分析了该类多项式的对称行列式表示，并给出了相关矩阵规模的上界。