Geodesic
Digraphs Associated With Finite Rings
Publications de l'Institut Mathématique, _N_S_92 (2012) no. 106, p. 35 .

Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

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.
Classification : 11T99 05C90
Keywords: finite rings, finite graphs, symmetric polynomials 
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     author = {Aleksandar T. Lipkovski},
     title = {Digraphs {Associated} {With} {Finite} {Rings}},
     journal = {Publications de l'Institut Math\'ematique},
     pages = {35 },
     publisher = {mathdoc},
     volume = {_N_S_92},
     number = {106},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_2012_N_S_92_106_a2/}
}
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Aleksandar T. Lipkovski. Digraphs Associated With Finite Rings. Publications de l'Institut Mathématique, _N_S_92 (2012) no. 106, p. 35 . http://geodesic.mathdoc.fr/item/PIM_2012_N_S_92_106_a2/