Table of Contents
ISRN Algebra
Volume 2011 (2011), Article ID 591041, 24 pages
http://dx.doi.org/10.5402/2011/591041
Research Article

Some Properties of the Complement of the Zero-Divisor Graph of a Commutative Ring

Department of Mathematics, Saurashtra University, Rajkot 360 005, India

Received 19 April 2011; Accepted 17 May 2011

Academic Editors: D. Anderson, V. Drensky, and D. Herbera

Copyright © 2011 S. Visweswaran. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let R be a commutative ring with identity admitting at least two nonzero zero-divisors. Let ( Ξ“ ( 𝑅 ) ) 𝑐 denote the complement of the zero-divisor graph Ξ“ ( 𝑅 ) of 𝑅 . It is shown that if ( Ξ“ ( 𝑅 ) ) 𝑐 is connected, then its radius is equal to 2 and we also determine the center of ( Ξ“ ( 𝑅 ) ) 𝑐 . It is proved that if ( Ξ“ ( 𝑅 ) ) 𝑐 is connected, then its girth is equal to 3, and we also discuss about its girth in the case when ( Ξ“ ( 𝑅 ) ) 𝑐 is not connected. We discuss about the cliques in ( Ξ“ ( 𝑅 ) ) 𝑐 .