On the Cayley graph of a commutative ring with respect to its\n zero-divisors

2013/05/02 by Ghodratollah Aalipour, Aalipour, Ghodratollah, Saieed Akbari +1 · 2 citations
Mathematics · #05C15 #05C25 #05C40 #05C69 #16N40 #Advanced Topics in Algebra #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1305.0601

openalex publication_date 2013/05/02 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

Let R be a commutative ring with unity and R⁺ be Z^*(R) be the\nadditive group and the set of all non-zero zero-divisors of R, respectively.\nWe denote by mathbbCAY(R) the Cayley graph Cay(R⁺,Z^*(R)). In this\npaper, we study mathbbCAY(R). Among other results, it is shown that for\nevery zero-dimensional non-local ring R, mathbbCAY(R) is a connected\ngraph of diameter 2. Moreover, for a finite ring R, we obtain the vertex\nconnectivity and the edge connectivity of mathbbCAY(R). We investigate\nrings R with perfect mathbbCAY(R) as well. We also study\nReg( mathbbCAY(R)) the induced subgraph on the regular elements of R.\nThis graph gives a family of vertex transitive graphs. We show that if R is a\nNoetherian ring and Reg( mathbbCAY(R)) has no infinite clique, then R is\nfinite. Furthermore, for every finite ring R, the clique number and the\nchromatic number of Reg( mathbbCAY(R)) are determined.\n

Citations

• The Zero-Divisor Graph of a Commutative Ring
• The total graph of a commutative ring

Cited by

• Involutory Cayley graphs of polynomial and power series rings over the ring of integers modulo n
• Toroidal involutory Cayley graphs

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