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.
