  We characterize the equivalence and the weak equivalence of Cayley graphs for
a finite group $\C{A}$. Using these characterizations, we find degree
distribution polynomials for weak equivalence of some graphs including 1)
circulant graphs of prime power order, 2) circulant graphs of order $4p$, 3)
circulant graphs of square free order and 4) Cayley graphs of order $p$ or
$2p$. As an application, we find an enumeration formula for the number of weak
equivalence classes of circulant graphs of prime power order, order $4p$ and
square free order and Cayley graphs of order $p$ or $2p$.
