Let H be a subset of a commutative graded ring R. The Cayley graph Cay(R,H) is a

graph whose vertex set is R and two vertices a and b are adjacent if and only if a−b ∈ H

Let S be the set of homogeneous elements we study  Cay(R, S)

In particular, if R is an Artinian graded ring, we show that Cay(R, S) is

isomorphic to a Hamming graph an conversely any Hamming graph is

isomorphic to a subgraph of Cay(R, S) for some finite graded ring R