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 2 H. The Cayley sum 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 2 H. Let S be the set of homogeneous elements and Z(R) be the set of zero-divisors of R. In this paper, we study Cay+(R; Z(R)) (total graph) and Cay(R; S). In particular, if R is an Artinian graded ring, we show that Cay(R; S) is isomorphic to a Hamming graph and conversely any Hamming graph is isomorphic to a subgraph of Cay(R; S) for some fnite graded ring R.
