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On the cayley graph of a commutative ring with respect to its zero-divisors

Aalipour, G ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1080/00927872.2015.1027359
  3. Publisher: Taylor and Francis Inc
  4. Abstract:
  5. Let R be a commutative ring with unity and R+ and Z*(R) be the additive group and the set of all nonzero zero-divisors of R, respectively. We denote by ℂ𝔸𝕐(R) the Cayley graph Cay(R+, Z*(R)). In this article, we study ℂ𝔸𝕐(R). Among other results, it is shown that for every zero-dimensional nonlocal ring R, 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 ℂ𝔸𝕐(R). As a result, ℂ𝔸𝕐(R) gives an algebraic construction for vertex transitive graphs of maximum connectivity. In addition, we characterize all zero-dimensional semilocal rings R whose ℂ𝔸𝕐(R) is perfect. We also study Reg(ℂ𝔸𝕐(R)) the induced subgraph on the regular elements of R. This graph gives a family of vertex transitive graphs as well. We show that if R is a Noetherian ring and Reg(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(ℂ𝔸𝕐(R)) are determined
  6. Keywords:
  7. Chromatic number ; Clique number ; Connectivity ; Minimal prime ideal ; Zero-dimensional ring ; Zero-divisor
  8. Source: Communications in Algebra ; Volume 44, Issue 4 , 2016 , Pages 1443-1459 ; 00927872 (ISSN)
  9. URL: http://www.tandfonline.com/doi/abs/10.1080/00927872.2015.1027359