‎On Power Graph of Some Finite Rings

‎Consider a ring $R$ with order $p$ or $p^2$‎, ‎and let $\mathcal{P}(R)$ represent its multiplicative power graph‎. ‎For two distinct rings $R_1$ and $R_2$ that possess identity element 1‎, ‎we define a new structure called the unit semi-cartesian product of their multiplicative power graphs‎. ‎This combined structure‎, ‎denoted as $G.H$‎, ‎is constructed by taking the Cartesian product of the vertex sets $V(G) \times V(H)$‎, ‎where $G = \mathcal{P}(R1)$ and $H = \mathcal{P}(R2)$‎. ‎The edges in $G.H$ are formed based on specific conditions‎: ‎for vertices $(g,h)$ and $(g^\prime,h^\prime)$‎, ‎an edge exists between them if $g = g^\prime$‎, ‎$g$ is a vertex in $G$‎, ‎and the product $hh^\prime$ forms a vertex in $H$‎. ‎Our exploration focuses on understanding the characteristics of the multiplicative power graph resulting from the unit semi-cartesian product $\mathcal{P}(R1).\mathcal{P}(R2)$‎, ‎where $R_1$ and $R_2$ represent distinct rings‎. ‎Additionally‎, ‎we offer insights into the properties of the multiplicative power graphs inherent in rings of order $p$ or $p^2$‎.