Abstract.
A sign pattern matrix is a matrix whose entries are from the set {+,−,0}. The purpose of this paper is to obtain bounds on the minimum rank of any symmetric sign pattern matrix A whose graph is a tree T (possibly with loops). In the special case when A is nonnegative with positive diagonal and the graph of A is “star-like”, the exact value of the minimum rank of A is obtained. As a result, it is shown that the gap between the symmetric minimal and maximal ranks can be arbitrarily large for a symmetric tree sign pattern A.
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Supported by NSF grant No. DMS-00700
AMS classification: 05C50, 05C05, 15A48
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Chen, G., Hall, F., Li, Z. et al. On Ranks of Matrices Associated with Trees. Graphs and Combinatorics 19, 323–334 (2003). https://doi.org/10.1007/s00373-002-0522-8
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DOI: https://doi.org/10.1007/s00373-002-0522-8