I was just pondering some things about graph theory and I was hoping someone could weigh in or direct me to a good resource.
The first is maximal clique polynomials, which would be a polynomial where the coefficient on x^n is the number of maximal cliques of size n in the graph. I'm not sure if it exists and goes by a different name, or if it just hasn't been seriously studied yet, but I can't seem to find a paper on them. I have seen research on clique polynomials (same idea but with coefficients counting cliques, not maximal cliques). I would love to read a paper about maximal clique polynomials if it exists.
Secondly, is there a term for cliques that are contained by more than one maximal clique, and does this distinction turn out to be useful anywhere? For example, if we take two complete graphs on 4 vertices, ABCD and A'B'C'D', and merge vertices A and A' into one vertex A, then 1-clique A is a subgraph of two maximal cliques: ABCD, and AB'C'D'. Is characterizing cliques into those that are contained by only one maximal clique and those that are contained by more than one maximal clique interesting at all? I have the feeling it might be if we're analyzing maximal clique polynomials, but I'm not sure.