There is a natural connection between graphs and simplicial complexes, which shows that every graph is just a special simplicial complex.
Though the clique complex is not explicitly mentioned polynomials related to the clique complex (``clique polynomials") are studied in [41] and [59] (these references were pointed out to the author by R. Stanley). The connection between simplicial complexes and ordered sets now (naturally) is made through comparability graphs:
