‎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, denotedas g.h, is constructed by taking the cartesian product of the vertex setsv (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 formsa 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 intothe properties of the multiplicative power graphs inherent in rings of orderp or p2.