Publications de l'Institut Mathematique 2012 Volume 92, Issue 106, Pages: 35-41
https://doi.org/10.2298/PIM1206035L
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Digraphs associated with finite rings
Lipkovski Aleksandar T. (Faculty of Mathematics, Belgrade)
Let A be a finite commutative ring with unity (ring for short). Define a
mapping φ : A2 → A2 by (a, b) 7→ (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
Gn = 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.
Keywords: finite rings, finite graphs, symmetric polynomials
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