On the Cayley graph of a commutative ring with respect to its zero-divisors
| Autor: | Saieed Akbari, Ghodratollah Aalipour |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2013 |
| Předmět: |
Discrete mathematics
Vertex (graph theory) Finite ring Noetherian ring Algebra and Number Theory Mathematics::Combinatorics Cayley graph Mathematics::Commutative Algebra 010102 general mathematics Induced subgraph 0102 computer and information sciences Commutative ring Commutative Algebra (math.AC) Mathematics - Commutative Algebra 01 natural sciences Combinatorics 010201 computation theory & mathematics FOS: Mathematics Mathematics - Combinatorics Combinatorics (math.CO) 0101 mathematics Connectivity Zero divisor 05C15 05C25 05C40 05C69 16N40 Mathematics |
| Popis: | Let $R$ be a commutative ring with unity and $R^{+}$ be $Z^*(R)$ be the additive group and the set of all non-zero zero-divisors of $R$, respectively. We denote by $\mathbb{CAY}(R)$ the Cayley graph $Cay(R^+,Z^*(R))$. In this paper, we study $\mathbb{CAY}(R)$. Among other results, it is shown that for every zero-dimensional non-local ring $R$, $\mathbb{CAY}(R)$ is a connected graph of diameter 2. Moreover, for a finite ring $R$, we obtain the vertex connectivity and the edge connectivity of $\mathbb{CAY}(R)$. We investigate rings $R$ with perfect $\mathbb{CAY}(R)$ as well. We also study $Reg(\mathbb{CAY}(R))$ the induced subgraph on the regular elements of $R$. This graph gives a family of vertex transitive graphs. We show that if $R$ is a Noetherian ring and $Reg(\mathbb{CAY}(R))$ has no infinite clique, then $R$ is finite. Furthermore, for every finite ring $R$, the clique number and the chromatic number of $Reg(\mathbb{CAY}(R))$ are determined. 21 pages, 1 figure |
| Databáze: | OpenAIRE |
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