Digraphs associated with finite rings

Abstract

Let A be a finite commutative ring with unity (ring for short). Define a mapping φ A² → A² by (a, b) → (a + b, ab). One can interpret this mapping as a finite directed graph (digraph) G = G(A) with vertices A² and arrows defined by φ. The main idea is to connect ring properties of A to graph properties of G. Particularly interesting are rings A = Z/nZ. Their graphs should reflect number-theoretic properties of integers. The first few graphs G<inf>n</inf> = G(Z/nZ) 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.