Mathematics Interdisciplinary Research (Aug 2023)
On Power Graph of Some Finite Rings
Abstract
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 $p^2$.
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