Title: The Zero-Divisor Graph Associated to a Semigroup
Abstract: Abstract The zero-divisor graph of a commutative semigroup with zero is the graph whose vertices are the nonzero zero-divisors of the semigroup, with two distinct vertices adjacent if the product of the corresponding elements is zero. New criteria to identify zero-divisor graphs are derived using both graph-theoretic and algebraic methods. We find the lowest bound on the number of edges necessary to guarantee a graph is a zero-divisor graph. In addition, the removal or addition of vertices to a zero-divisor graph is investigated by using equivalence relations and quotient sets. We also prove necessary and sufficient conditions for determining when regular graphs and complete graphs with more than two triangles attached are zero-divisor graphs. Lastly, we classify several graph structures that satisfy all known necessary conditions but are not zero-divisor graphs. Key Words: Zero-divisor graphZero-divisor semigroup2000 Mathematics Subject Classification: Primary 20M14Secondary 05C99 ACKNOWLEDGMENTS Our research was supported by the National Science Foundation Grant DMS 05-52594. Our results were developed from work in the Central Michigan University Research Experience for Undergraduates, and we would like to thank Central Michigan University and Sivaram Narayan for their support. We thank Joseph Hernandez and Ryan Kaliszewski for their helpful suggestions during the research process. We also thank Jesse Geneson and Jacob Steinhardt for their advice and comments on the paper. The authors offer sincere thanks to the referee for the careful reading of this paper and helpful comments which greatly improved the style of this article.
Publication Year: 2010
Publication Date: 2010-08-31
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 20
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