For a finite and nonempty set S of positive integers, the divisor graph G(S) of S has vertex set S, and two distinct vertices i and j are adjacent if and only if i|j or j|i, while the divisor digraph D(S) of S has vertex set S and (i, j) is an arc of D(S) if and only if i|j. A graph G is a divisor graph if there exists a set S of positive integers such that G is isomorphic to G(S). It is shown that for a divisor graph G with a transitive vertex, G x H is a divisor graph if and only if H has no edges. For m, n in N, with m greater than or equal to five there exists a non-divisor graph G, of order m + n, that has m neighborhoods that are divisor graphs and n neighborhoods that are not divisor graphs.