Abstract
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 Menon are related to clique polynomials. This establishes a connection between the clique polynomial and decompositions of finite sets, symmetric groups and analysis.
Highlights
The graphs considered here are finite and without loops or multiple edges
We define a clique in G to be a subgraph of G which is a complete graph
We show that a number-theoretic function defined by Menon [3] is really the clique polynomial of a complete graph, when weights are assigned in a particular manner
Summary
We derive explicit formulae for the general and simple clique polynomials of the complete graph and for their generating functions. We show that a number-theoretic function defined by Menon [3] is really the clique polynomial of a complete graph, when weights are assigned in a particular manner. Instead of cliques, we take the components of a cover of G to be members of a general family F of connected graphs, the resulting polynomial is called the. We will not give an independent derivation of the generating function for the clique polynomial of the complete graph. We use a result given in Farrell [I] (Theorem I) on the generating function for the general F-polynomial of the complete graph.
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