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Zero forcing parameters and minimum rank problems |
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| Authors: | Francesco Barioli H Tracy Hall Leslie Hogben Bryan Shader Hein van der Holst |
| Institution: | a Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga TN 37403, USA b Department of Mathematics, Brigham Young University, Provo UT 84602, USA c Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada d Department of Mathematics, Iowa State University, Ames, IA 50011, USA e American Institute of Mathematics, 360 Portage Ave, Palo Alto, CA 94306, USA f Department of Mathematics, University of Wyoming, Laramie, WY 82071, USA g Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada V8W 3R4 h School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA |
| Abstract: | The zero forcing number Z(G), which is the minimum number of vertices in a zero forcing set of a graph G, is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by G. It is shown that for a connected graph of order at least two, no vertex is in every zero forcing set. The positive semidefinite zero forcing number Z+(G) is introduced, and shown to be equal to |G|-OS(G), where OS(G) is the recently defined ordered set number that is a lower bound for minimum positive semidefinite rank. The positive semidefinite zero forcing number is applied to the computation of positive semidefinite minimum rank of certain graphs. An example of a graph for which the real positive symmetric semidefinite minimum rank is greater than the complex Hermitian positive semidefinite minimum rank is presented. |
| Keywords: | 05C50 15A03 15A18 15B57 |
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