Graphs with few trivial critical ideals

The critical ideals of a graph are determinantal ideals of the generalized Laplacian matrix associated to a graph. Let Γ≤i denote the set of simple connected graphs with at most i trivial critical ideals. The main goal is to obtain a characterization of the graphs in Γ≤3 with clique number equal to 2, and the graphs in Γ≤3 with clique number equal to 3. This shows that there exists a strong connection between the structural properties of the graph (like the clique number and the stability number) with its critical ideals.