Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. They are widely used throughout natural sciences, computational biology and many other fields. Computing the vanishing ideal of the model gives us an implicit description of the model. In this paper, we resolve two conjectures of Sturmfels and Uhler from \cite{BS n CU}. In particular, we characterize those graphs for which the vanishing ideal of the Gaussian graphical model is generated in degree $1$ and $2$. These turn out to be the Gaussian graphical models whose ideals are toric ideals, and the resulting graphs are the $1$-clique sums of complete graphs.