Zero Divisor Graph of a Poset with Respect to an Ideal Authors Vinayak Joshi Department of Mathematics University of Pune Article

First Online: 10 May 2011 Received: 05 August 2009 Accepted: 31 March 2011 DOI :
10.1007/s11083-011-9216-2

Cite this article as: Joshi, V. Order (2012) 29: 499. doi:10.1007/s11083-011-9216-2
Abstract In this paper, we introduce the zero divisor graph G _{I} (P ) of a poset P (with 0) with respect to an ideal I . It is shown that G _{I} (P ) is connected with its diameter ≤3, and if G _{I} (P ) contains a cycle, then the core K of G _{I} (P ) is a union of 3-cycles and 4-cycles. Further, the chromatic number and clique number of G _{I} (P ) are shown to be equal. This proves a form of Beck’s conjecture for posets with 0. The complete bipartite zero divisor graphs are characterized.

Keywords Zero divisor graph Clique number Chromatic number Annihilator Semi-ideal Ideal Prime ideal Semiprime ideal Dedicated to Professor N. K. Thakare on his 71st birthday.

This research is supported by Board of College and University Development, University of Pune, via the project SC-66.

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