TY - JOUR
A2 - Koukouvinos, Christos
AU - Arav, Marina
AU - Hall, Frank
AU - Li, Zhongshan
AU - Rao, Bhaskara
PY - 2009
DA - 2009/06/02
TI - ZPC Matrices and Zero Cycles
SP - 520923
VL - 2009
AB - 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=1k−1Zi)|≤1 for all k (2≤k≤m) 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.
SN - 1687-9163
UR - https://doi.org/10.1155/2009/520923
DO - 10.1155/2009/520923
JF - International Journal of Combinatorics
PB - Hindawi Publishing Corporation
KW -
ER -