Prime Graph over Cartesian Product over Rings and Its Complement

Graph theory is a branch of algebra that growing rapidly both in concept and application studies. This graph can be used chemistry, transportation, cryptographic problems, coding theory, design communication network, etc. There currently bridge between graphs algebra, especially an algebraic structures, namely algebra. One researchs on formed by prime ring elements or called over R. The commutative R (PG(R))) construction with set vertices V(PG(R))=R two x y are adjacent if satisfy xRy={0}, for x≠y. Girth the shortest cycle length contains PG(R) written gr(PG(R)). Order denoted |V(PG(R))| size |E(PG(R))|. In this paper, we discussed cartesian product rings Z_m×Z_n its complement. We focused only m=p_1, n=p_2 n=〖p_2〗^2, where p_1 p_2 numbers. Then, obtained some properties related to order size, degree, girth. also observe examples. Moreover, found correction statement (Pawar & Joshi, 2019) about complement gave counter example that.