Vol. 4, No. 1, 2011

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Five-point zero-divisor graphs determined by equivalence classes

Florida Levidiotis and Sandra Spiroff

Vol. 4 (2011), No. 1, 53–64
Abstract

We study condensed zero-divisor graphs (those whose vertices are equivalence classes of zero-divisors of a ring R) having exactly five vertices. In particular, we determine which graphs with exactly five vertices can be realized as the condensed zero-divisor graph of a ring. We provide the rings for the graphs which are possible, and prove that the rest of graphs can not be realized via any commutative ring. There are 34 graphs in total which contain exactly five vertices.

Keywords
condensed zero-divisor graphs, equivalence classes of zero-divisors
Mathematical Subject Classification 2000
Primary: 13A99
Secondary: 05C99
Milestones
Received: 17 June 2010
Revised: 11 February 2011
Accepted: 23 February 2011
Published: 22 September 2011

Communicated by Scott Chapman
Authors
Florida Levidiotis
Department of Mathematics and Statistics
Loyola University Chicago
26 E Pearson Street, #2305
Chicago, IL 60611
United States
Sandra Spiroff
Department of Mathematics
University of Mississippi
Hume Hall 305
P.O. Box 1848
University, MS 38677-1848
United States
http://home.olemiss.edu/depts/mathematics/spiroff.htm