‎On Power Graph of Some Finite Rings

Consider a ring R with order p or p2, and let P(R) represent its multi-plicative power graph. For two distinct rings R1 and R2 that possess identity element 1, we define a new structure called the unit semi-cartesian prod- uct of their multiplicative power graphs. This combined structure, denoted as G.H, is constructed by taking the Cartesian product of the vertex sets V (G) × V (H), where G = P(R1) and H = P(R2). The edges in G.H are formed based on specific conditions: for vertices (g, h) and (g, h), an edge exists between them if g = g, g is a vertex in G, and the product hh forms a vertex in H. Our exploration focuses on understanding the characteristics of the multi-plicative power graph resulting from the unit semi-cartesian product P(R1).P(R2), where R1 and R2 represent distinct rings. Additionally, we o˙er insights into the properties of the multiplicative power graphs inherent in rings of order p or p2.