On Z_k−vertex-magic labeling of prime graph PG(Z_n)

Let G = (V(G),E(G)) be a graph, (A,+) be an Abelian group with identity 0_A, and (R, +, ·) be a ring. The A-vertex-magic labeling of G is a mapping from V (G) to A− {0_A} such that the total labels of every adjacent vertex with u are equal for every u in V (G). The prime graph over R, denoted by PG(R), is a graph with V (PG(R)) = R such that uv is an edge if and only if uRv = {0_R} or vRu = {0_R}, for every vertex u ≠ v. In this article, we discuss the Z_k-vertex-magic labeling of the prime graph over the ring Z_n. We study some literature to develop the properties of Z_k-vertex-magic labeling of PG(R). We investigate some classes of prime graphs over ring Z_n for n = p, n = p², and n = pq, with p ≠ q primes.