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Paper   IPM / M / 2323
School of Mathematics
  Title:   When a zero-divisor graph is planar or complete r-partite graph
1.  S. Akbari
2.  H. R. Maimani
3.  S. Yassemi
  Status:   Published
  Journal: J. Algebra
  Vol.:  270
  Year:  2003
  Pages:   169-180
  Supported by:  IPM
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) ≠ ∅, 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 |\TZ(R)|=p2, |R|=p3, and R is isomorphic to exactly one of the rings \mathbbZp3, [(\mathbbZp[x,y])/((xy,y2x))],   [(\mathbbZp2[y])/((py,y2ps))], where 1 ≤ s < p.

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