Path Polynomial

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 .

A graph is traceable iff the degree of the path polynomial equals . In particular,  gives the number of Hamiltonian paths, so a graph is traceable iff .

Since path counts in a disconnected graph are the sum of path counts in its connected components, the path polynomial is additive over connected components.