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International Journal of Combinatorics
Volume 2009 (2009), Article ID 520923, 5 pages
http://dx.doi.org/10.1155/2009/520923
Research Article

ZPC Matrices and Zero Cycles

1Department of Mathematics and Statistics, Georgia State University, Atlanta, GA 30303-3083, USA
2Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN 47809, USA

Received 11 December 2008; Accepted 19 March 2009

Academic Editor: Christos Koukouvinos

Copyright © 2009 Marina Arav et al. 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 H be an m×n real matrix and let Zi be the set of column indices of the zero entries of row i of H. Then the conditions |Zk(i=1k1Zi)|1 for all k(2km) are called the (row) Zero Position Conditions (ZPCs). If H satisfies the ZPC, then H is said to be a (row) ZPC matrix. If HT satisfies the ZPC, then H is said to be a column ZPC matrix. The real matrix H is said to have a zero cycle if H has a sequence of at least four zero entries of the form hi1j1,hi1j2,hi2j2,hi2j3,,hikjk,hikj1 in which the consecutive entries alternatively share the same row or column index (but not both), and the last entry has one common index with the first entry. Several connections between the ZPC and the nonexistence of zero cycles are established. In particular, it is proved that a matrix H has no zero cycle if and only if there are permutation matrices P and Q such that PHQ is a row ZPC matrix and a column ZPC matrix.