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The regular graph of a commutative ring

Akbari, S ; Sharif University of Technology | 2013

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  1. Type of Document: Article
  2. DOI: 10.1007/s10998-013-7039-1
  3. Publisher: 2013
  4. Abstract:
  5. Let R be a commutative ring, let Z(R) be the set of all zero-divisors of R and Reg(R) = RZ(R). The regular graph of R, denoted by G(R), is a graph with all elements of Reg(R) as the vertices, and two distinct vertices x, y ∈ Reg(R) are adjacent if and only if x+y ∈ Z(R). In this paper we show that if R is a commutative Noetherian ring and 2 ∈ Z(R), then the chromatic number and the clique number of G(R) are the same and they are 2n, where n is the minimum number of prime ideals whose union is Z(R). Also, we prove that all trees that can occur as the regular graph of a ring have at most two vertices
  6. Keywords:
  7. Chromatic number ; Clique number ; Noetherian ring ; Regular graph ; Zero-divisors
  8. Source: Periodica Mathematica Hungarica ; Volume 67, Issue 2 , 2013 , Pages 211-220 ; 00315303 (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S1571065313002965