Vol. 5, No. 1, 2012

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Diameter, girth and cut vertices of the graph of equivalence classes of zero-divisors

Blake Allen, Erin Martin, Eric New and Dane Skabelund

Vol. 5 (2012), No. 1, 51–60

We explore the properties of ΓE(R), the graph of equivalence classes of zero-divisors of a commutative Noetherian ring R. We determine the possible combinations of diameter and girth for the zero-divisor graph Γ(R) and the equivalence class graph ΓE(R), and examine properties of cut-vertices of ΓE(R).

zero-divisor graph, diameter, girth, cut vertices
Mathematical Subject Classification 2010
Primary: 13A99
Received: 19 January 2011
Revised: 2 August 2011
Accepted: 18 August 2011
Published: 28 April 2012

Communicated by Scott Chapman
Blake Allen
Department of Mathematics
Utah Valley University
Orem, UT 84058
United States
Erin Martin
Department of Physics and Mathematics
William Jewell College
Liberty, MO 64068
United States
Eric New
Department of Mathematics and Statistics
The College of New Jersey
Ewing, NJ 08628
United States
Dane Skabelund
Department of Mathematics
Brigham Young University
Provo, UT 84602
United States