  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.
