Minimum Rank, Maximum Nullity, and Zero Forcing Number of Graphs

This chapter represents an overview of research related to a notion of the “rank of a graph" and the dual concept known as the “nullity of a graph," from the point of view of associating a fixed collection of symmetric or Hermitian matrices to a given graph. This topic and related questions have enjoyed a fairly large history within discrete mathematics, and have become very popular among linear algebraists recently, partly based on its connection to certain inverse eigenvalue problems, but also because of the many interesting applications (e.g., to communication complexity in computer science and to control of quantum systems in mathematical physics) and implications to several facets of graph theory. This chapter is divided into eight parts, beginning in Section 1 with what we feel is the standard minimum rank problem concerning symmetric matrices over the real numbers associated with a simple graph. We continue with important variants of the standard minimum rank problem and related parameters, including the concept of minimum rank over other fields (Section 2), the positive semidefinite minimum rank of a graph (Section 3), graph coloring parameters known as zero forcing numbers (Section 4) and the more classical and celebrated parameters due to Y. Colin de Verdière (Section 5). Section 6 contains more advanced topics relevant to the previous five sections, and Section 7 discusses two well-known conjectures related to minimum rank. Whereas the first seven sections concern primarily symmetric matrices and the diagonal of the matrix is free, in Section 8 we discuss minimum rank problems with no symmetry assumption but the diagonal constrained. NB: The topics discussed in this chapter are in an active research area and the facts presented here represent the state of knowledge as of the writing of this chapter in 2012. Disciplines Algebra | Applied Mathematics | Discrete Mathematics and Combinatorics Comments This is an Accepted Manuscript of the book chapter Fallat, Shaun M., and Leslie Hogben. "Minimum Rank, Maximum Nullity, and Zero Forcing Number of Graphs." In Handbook of Linear Algebra, Second Edition, edited by Leslie Hogben. Boca Raton, Florida : CRC Press/Taylor & Francis Group, 2014: 46-1 to 46-36. Posted with permission. This book chapter is available at Iowa State University Digital Repository: https://lib.dr.iastate.edu/math_pubs/212 Minimum Rank, Maximum Nullity, and Zero Forcing Number of Graphs Shaun M. Fallat∗ Leslie Hogben† This is the content of Chapter 46 of Handbook of Linear Algebra 2nd ed. copyright 2014 by CRC Press. Reformatted but not updated 5/22/19. This chapter represents an overview of research related to a notion of the “rank of a graph” and the dual concept known as the “nullity of a graph,” from the point of view of associating a fixed collection of symmetric or Hermitian matrices to a given graph. This topic and related questions have enjoyed a fairly large history within discrete mathematics, and have become very popular among linear algebraists recently, partly based on its connection to certain inverse eigenvalue problems, but also because of the many interesting applications (e.g., to communication complexity in computer science and to control of quantum systems in mathematical physics) and implications to several facets of graph theory. This chapter is divided into eight parts, beginning in Section 1 with what we feel is the standard minimum rank problem concerning symmetric matrices over the real numbers associated with a simple graph. We continue with important variants of the standard minimum rank problem and related parameters, including the concept of minimum rank over other fields (Section 2), the positive semidefinite minimum rank of a graph (Section 3), graph coloring parameters known as zero forcing numbers (Section 4) and the more classical and celebrated parameters due to Y. Colin de Verdière (Section 5). Section 6 contains more advanced topics relevant to the previous five sections, and Section 7 discusses two well-known conjectures related to minimum rank. Whereas the first seven sections concern primarily symmetric matrices and the diagonal of the matrix is free, in Section 8 we discuss minimum rank problems with no symmetry assumption but the diagonal constrained. NB: The topics discussed in this chapter are in an active research area and the facts presented here represent the state of knowledge as of the writing of this chapter in 2012. 1 Standard Minimum Rank of Graphs For a given simple graph G, we associate, in a natural way, a collection of matrices whose entries reflect adjacency in G. We are interested in quantities such as the smallest possible rank and the largest possible nullity over all matrices in this collection. Of particular ∗Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada, shaun.fallat@uregina.ca. †Department of Mathematics, Iowa State University, Ames, IA 50011, USA, and American Institute of Mathematics, 600 E. Brokaw Road, San Jose, CA 95112, USA, hogben@aimath.org.