On the minimum rank of not necessarily symmetric matrices : a preliminary study

F. Barioli, S.M. Fallat, H.T. Hall, D. Hershkowitz, L. Hogben, H. Holst, van der, B.L. Shader

Onderzoeksoutput: Bijdrage aan tijdschriftTijdschriftartikelAcademicpeer review

30 Citaten (Scopus)
132 Downloads (Pure)

Samenvatting

The minimum rank of a directed graph G is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i, j) is an arc in G and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i _= j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem.
Originele taal-2Engels
Pagina's (van-tot)126-145
TijdschriftElectronic Journal of Linear Algebra
Volume18
StatusGepubliceerd - 2009

Vingerafdruk

Duik in de onderzoeksthema's van 'On the minimum rank of not necessarily symmetric matrices : a preliminary study'. Samen vormen ze een unieke vingerafdruk.

Citeer dit