Let S be a subset of an abelian group G. The addition Cayley graph of G induced by S, which is denoted by (S), is an undirected graph with vertex set G, and two vertices g 1 ,g 2 G are joined by an edge if and only if g 1 +g 2 S. Note that if S is finite, then (S) is regular of degree (we assume that each loop contributes 1 to the degree of the corresponding vertex). We study some basic properties of addition Cayley graph and show that (S) is connected if and only if S is not contained in a coset of a proper subgroup of G, with the possible exception of the non – zero coset of a subgroup of index 2. Let e a graph of the finite set V. The (vertex) connectivity of , denoted by ( ), is the smallest number of vertices which are to be removed from V so that the resulting graph is either disconnected or has only one vertex. For subsets A and B of an abelian group, we write A : = . Suppose that H is a subgroup of an abelian group G and , such that 2g + H