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 Amendola 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：我们研究了对数似然函数具有唯一复临界点的高斯统计模型，即最大似然度为一的模型。我们利用Amendola等人建立的关于最大似然度为一模型与同调多项式之间的联系。研究了图的生成树生成函数，并证明当图为弦图时该多项式为同调多项式。对于$n \geq 4$的$n$个顶点上的圈图，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：生成树生成函数是同调多项式当且仅当该图为弦图。我们还为这些模型的定义方程提供了代数公式。利用现有结果，我们开展了构建同调多项式新族的计算研究。最后，我们分析了此类多项式的对称行列式表示，并给出了所涉矩阵规模的上界。