Summary: The clique polynomial of a graph is defined. An explicit formula is then derived for the clique polynomial of the complete graph. A fundamental theorem and a reduction process is then given for clique polynomials. Basic properties of the polynomial are also given. It is shown that the number theoretic functions defined by P. K. Menon [J. Comb. Inf. Syst. Sci. 1, 112-121 (1976; Zbl 0429.05009)] are related to clique polynomials. This establishes a connection between the clique polynomial and decompositions of finite sets, symmetric groups and analysis.