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The graph polynomial for the number of independent sets of size $k$ in ageneral undirected graph is shown to be equal to an elementary symmetricpolynomial of the vertex monomials, which are determined by the edges incidentat the vertices. The edge variables that comprise the vertex monomials areshown to be nilpotent of degree two. The index of nilpotency of the algebragenerated by the graph's vertex monomials is the order of a maximum independentset of the graph. Graph polynomials for the number of cliques and vertex coversare derived from the graph polynomial for independent sets. The graphpolynomial for the number of bipartite cuts is given as a coefficient of amultivariate Laurent polynomial.

Graph Polynomials for the Numbers of Independent Sets and Bipartite Cuts for Undirected Graphs