Path decompositions of chains and circuits (Q793050)

For each spanning subgraph C of G consisting of paths \(P_ 1,P_ 2,...,P_ k\) let f(i) be the number of vertices in \(P_ i\), and let \(w(C)=\prod^{k}_{i=1}w_{f(i)}\) where \(w_ 1,w_ 2,w_ 3,...\) are indeterminates. The path polynomial of G is \(\sum w(C)\) where the summation is taken over all of the path covers of G. The path polynomials of paths and cycles are derived.