Sharper lower and upper bounds for the gaussian rank of a graph

An unanswered question in graphical models is "what is the smallest number of observations required to guarantee the existence of the maximum likelihood estimator of the covariance matrix for a graphical gaussian model? " It is clear that the smallest required number depends on the graph representing the model. In this paper we call the smallest required number the gaussian rank of the graph and prove that it is strictly between the subgraph connectivity number and the graph degeneracy number. These lower and upper bounds are in general sharper than the best lower and upper bounds known in the literature and, furthermore, are computable in polynomial time.