DIGRAPHS ASSOCIATED WITH FINITE RINGS

Aleksandar T. Lipkovski

Abstract: Let $A$ be a finite commutative ring with unity (ring for short). Define a mapping $\varphi:A^2\to A^2$ by $(a,b)\mapsto(a+b,ab)$. One can interpret this mapping as a finite directed graph (digraph) $G=G(A)$ with vertices $A^2$ and arrows defined by $\varphi$. The main idea is to connect ring properties of $A$ to graph properties of $G$. Particularly interesting are rings $A=\mathbb Z/n\mathbb Z$. Their graphs should reflect number-theoretic properties of integers. The first few graphs $G_n=G(\mathbb Z/n\mathbb Z)$ are drawn and their numerical parameters calculated. From this list, some interesting properties concerning degrees of vertices and presence of loops are noticed and proved.

Keywords: finite rings; finite graphs; symmetric polynomials

Classification (MSC2000): 11T99; 05C90

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