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Some properties of a cayley graph of a commutative ring

Aalipour, G ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1080/00927872.2012.745866
  3. Abstract:
  4. Let R be a commutative ring with unity and R+, U(R), and Z*(R) be the additive group, the set of unit elements, and the set of all nonzero zero-divisors of R, respectively. We denote by ℂAY(R) and GR, the Cayley graph Cay(R+, Z*(R)) and the unitary Cayley graph Cay(R+, U(R)), respectively. For an Artinian ring R, Akhtar et al. (2009) studied GR. In this article, we study ℂAY(R) and determine the clique number, chromatic number, edge chromatic number, domination number, and the girth of ℂAY(R). We also characterize all rings R whose ℂAY(R) is planar. Moreover, we determine all finite rings R whose ℂAY(R) is strongly regular. We prove that ℂAY(R) is strongly regular if and only if it is edge transitive. As a consequence, we characterize all finite rings R for which GR is a strongly regular graph
  5. Keywords:
  6. Chromatic number ; Clique number ; Edge transitive graph ; Strongly regular graph ; Zero-divisor
  7. Source: Communications in Algebra ; Vol. 42, issue. 4 , Dec , 2014 , pp. 1582-1593 ; ISSN: 00927872
  8. URL: http://www.tandfonline.com/doi/abs/10.1080/00927872.2012.745866#.VdrJibW5JIE