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When a zero-divisor graph is planar or a complete r-partite graph

Akbari, S ; Sharif University of Technology | 2003

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  1. Type of Document: Article
  2. DOI: 10.1016/S0021-8693(03)00370-3
  3. Publisher: Academic Press Inc , 2003
  4. Abstract:
  5. Let Γ(R) be the zero-divisor graph of a commutative ring R. An interesting question was proposed by Anderson, Frazier, Lauve, and Livingston: For which finite commutative rings R is Γ (R) planar? We give an answer to this question. More precisely, we prove that if R is a local ring with at least 33 elements, and Γ(R) ≠ 0, then Γ(R) is not planar. We use the set of the associated primes to find the minimal length of a cycle in Γ(R). Also, we determine the rings whose zero-divisor graphs are complete r-partite graphs and show that for any ring R and prime number p, p ≥ 3, if Γ(R) is a finite complete p-partite graph, then Z(R) = p2, R = p3, and R is isomorphic to exactly one of the rings ℤp3, ℤp[x,y]/(xy,y2-x), ℤp2 [y]/(py,y2-ps), where 1 ≤ s < p. © 2003 Published by Elsevier Inc
  6. Keywords:
  7. Bipartite graph ; Girth ; Planar graph ; Zero-divisor graph
  8. Source: Journal of Algebra ; Volume 270, Issue 1 , 2003 , Pages 169-180 ; 00218693 (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0021869303003703