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Date:
16 Aug 2013
The regular graph of a commutative ring
 S. Akbari,
 F. Heydari
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Let R be a commutative ring, let Z(R) be the set of all zerodivisors of R and Reg(R) = R\Z(R). The regular graph of R, denoted by G(R), is a graph with all elements of Reg(R) as the vertices, and two distinct vertices x, y ∈ Reg(R) are adjacent if and only if x+y ∈ Z(R). In this paper we show that if R is a commutative Noetherian ring and 2 ∈ Z(R), then the chromatic number and the clique number of G(R) are the same and they are 2^{ n }, where n is the minimum number of prime ideals whose union is Z(R). Also, we prove that all trees that can occur as the regular graph of a ring have at most two vertices.
Communicated by László Fuchs
 S. Akbari, D. Kiani, F. Mohammadi and S. Moradi, The total graph and regular graph of a commutative ring, J. Pure Appl. Algebra, 213 (2009), 2224–2228. CrossRef
 D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra, 320 (2008), 2706–2719. CrossRef
 M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, AddisonWesley Publishing Company, 1969.
 J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, Elsevier Publishing Company, New York, 1976.
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 R. Y. Sharp, Steps in Commutative Algebra, Second Edition, Cambridge University Press, 2000.
 Title
 The regular graph of a commutative ring
 Journal

Periodica Mathematica Hungarica
Volume 67, Issue 2 , pp 211220
 Cover Date
 20131201
 DOI
 10.1007/s1099801370391
 Print ISSN
 00315303
 Online ISSN
 15882829
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 05C15
 05C25
 05C69
 13A
 13E05
 regular graph
 Noetherian ring
 zerodivisors
 clique number
 chromatic number
 Authors

 S. Akbari ^{(1)}
 F. Heydari ^{(2)}
 Author Affiliations

 1. Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran
 2. Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran