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 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

whose coefficients
give the number of simple paths of length present in a graph on nodes.

Since the smallest possible path is of length 1, path polynomials have polynomial degree at least 1. In particular, , where is the edge count of a graph
.