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

By Saieed Akbari, H. Maimani, Siamak Yassemi. 2003

213 citations recorded by the source corpus.

Abstract 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 | Z ( R )|= p 2 , | R |= p 3 , and R is isomorphic to exactly one of the rings Z p 3 , Z p [x,y] (xy,y 2 −x) , Z p 2 [y] (py,y 2 −ps) , where 1⩽ s p .

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b5be88e8b2857cc1393bd37d87da0979ced6cb15

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Citations in the corpus

On the Genus of the Total Graph of a Commutative Ring (2013, background): “The planarity of zero divisor graphs were studied in [2, 3, 8, 16].”
A Zero Divisor Graph Determined by Equivalence Classes of Zero Divisors (2007, background): “This graph has some advantages over the earlier zero divisor graphs Γ(R) in [4], [1], [2], [3], or subsequent zero divisor graphs determined by an ideal of R in [5], [8].”
Zero-divisor graphs of amalgamated duplication of a ring along an ideal (2007, background): “(b)⇒(c). This follows from [7, Proposition 2]. (c)⇒(d). This follows from [ 2, Theorem 2.4 ].”
On the Dot Product Graph of a Commutative Ring (2015, unknown): “Recently, there has been considerable attention in the literature to associating graphs with algebraic structures (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [17], [18], [19], [20], [21], [23], [24], [25], and [26]).”
Zero-Divisor Graph with Respect to an Ideal (2006, background): “…Iran S. Yassemi Department of Mathematics and Computer Science, Faculty of Science, University of Tehran, Tehran, Iran and School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran LetR be a commutative ring with nonzero identity and let I be an ideal ofR.”
Commutative rings with toroidal zero-divisor graphs (2007, background): “The planar case has been discussed previously in [2], [7] and in [18].”
Planar zero-divisor graphs (2007, background): “When working with polynomial rings, say K[X]/I , we will let x denote the coset X + I . Our goal in this paper is to determine all finite commutative rings whose zero-divisor graphs are planar, thus answering a question of Anderson, Frazier, Lauve, and Livingston [2, Question 5.3] “For which finite ”
Zero-divisor graphs of genus one (2006, background): “4] and give an affirmative answer to a question raised in [1]: Is it true that, for any local ring R of cardinality 32, which is not a field, Γ (R) is not planar? A simple graph G is an ordered pair of disjoint sets (V ,E) such that V = V (G) is the vertex set of G and E = E(G) is the edge set of G.”
On Zero Divisor Graphs (2012, background): “Throughout, and unless otherwise stated, R will be a commutative ring with unity.”
Classification of Rings with Genus One Zero-Divisor Graphs (2008, methodology): “The following theorem is another useful tool and is a generalization of some ideas used in Akbari et al. (2003).”
Rings whose total graphs have genus at most one (2010, background): “[4]: For which finite commutative rings R is Γ(R) planar? A partial answer was given in [1], but the question remained open for local rings of order 32.”