One would think that by analogy with the matching-generating polynomial, independence polynomial, etc., a path polynomial whose coefficients are the numbers of paths of length k would be defined. While no such polynomial seems not to have been defined in the literature, they are defined in this work. The path polynomial, perhaps defined here for the first time, is therefore the polynomial P_G(x) = sum_(k = 1)^(n - 1) p_k x^k whose coefficients p_k give the number of simple paths of length k present in a graph G on n nodes.

cycle graph | graph path | Hamiltonian path