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    Cycle Polynomial

    Definition

    One would think that by analogy with the matching-generating polynomial, independence polynomial, etc., a cycle polynomial whose coefficients are the numbers of cycles of length k would be defined. While no such polynomial seems not to have been defined in the literature (instead, "cycle polynomials" commonly instead refers to a polynomial corresponding to cycle indices of permutation groups), they are defined in this work. The cycle polynomial, perhaps defined here for the first time, is therefore the polynomial C_G(x) = sum_(k = 3)^n c_k x^k whose coefficients c_k give the number of simple cycles present in a graph G on n nodes.

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