Symmetrically colored Gaussian graphical models with toric vanishing ideal

A colored Gaussian graphical model is a linear concentration model in which equalities among the concentrations are specified by a coloring of an underlying graph. The model is called RCOP if this coloring is given by the edge and vertex orbits of a subgroup of the automorphism group of the graph. We show that RCOP Gaussian graphical models on block graphs are toric in the space of covariance matrices and we describe Markov bases for them. To this end, we learn more about the combinatorial structure of these models and their connection with Jordan algebras.