On Zero Divisor Graphs

By J. Coykendall, Sean Sather-Wagstaff, Laura Sheppardson, Sandra Spiroff. 2012

97 citations recorded by the source corpus.

We survey the research conducted on zero divisor graphs, with a focus on zero divisor graphs determined by equivalence classes of zero divisors of a commutative ring R. In particular, we consider the problem of classifying star graphs with any finite number of vertices. We study the pathology of a zero divisor graph in terms of cliques, we investigate when the clique and chromatic numbers are equal, and we show that the girth of a Noetherian ring, if finite, is 3. We also introduce a graph for modules that is useful for studying zero divisor graphs of trivial extensions.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1e05b62d56c2f288e206407ef37cb51083e0e2d1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Ayman Badawi
2. David F. Anderson
3. John D. LaGrange
4. F. Soheilnia
5. Ahmad Yousefian Darani

In Geometry and Topology

1. Ayman Badawi
2. David F. Anderson
3. John D. LaGrange

Citations in the corpus

• Commutative Boolean monoids, reduced rings, and the compressed zero-divisor graph (2012, background): “As defined in [20,4], two distinct vertices v and w of a graph Γ are called complements in Γ , written v ⊥ w, if v and w are adjacent, and no vertex of Γ is adjacent to both v and w.”
• Von Neumann Regular and Related Elements in Commutative Rings (2012, background): “We next show that some of the results from [9] carry over to Γ(Idem(R)) and Γ(vnr(R)), but first we recall some definitions.”