Minimum rank problems

A graph describes the zero-nonzero pattern of a family of matrices, with the type of graph (undirected or directed, simple or allowing loops) determining what type of matrices (symmetric or not necessarily symmetric, diagonal entries free or constrained) are described by the graph. The minimum rank problem of the graph is to determine the minimum among the ranks of the matrices in this family; the determination of maximum nullity is equivalent. This problem has been solved for simple trees [P.M. Nylen, Minimum-rank matrices with prescribed graph, Linear Algebra Appl. 248 (1996) 303-316, C.R. Johnson, A. Leal Duarte, The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree, Linear and Multilinear Algebra 46 (1999) 139-144], trees allowing loops [L.M. DeAlba, T.L. Hardy, I.R. Hentzel, L. Hogben, A. Wangsness. Minimum rank and maximum eigenvalue multiplicity of symmetric tree sign patterns, Linear Algebra Appl. 418 (2006) 389-415], and directed trees allowing loops [F. Barioli, S. Fallat, D. Hershkowitz, H.T. Hall, L. Hogben, H. van der Holst, B. Shader, On the minimum rank of not necessarily symmetric matrices: a preliminary study, Electron. J. Linear Algebra 18 (2000) 126-145]. We survey these results from a unified perspective and solve the minimum rank problem for simple directed trees.     

On acyclic and unicyclic graphs whose minimum rank equals the diameter

The minimum rank of a graph G is defined as the smallest possible rank over all symmetric matrices governed by G. It is well known that the minimum rank of a connected graph is at least the diameter of that graph. In this paper, we investigate the graphs for which equality holds between minimum rank and diameter, and completely describe the acyclic and unicyclic graphs for which this equality holds.     

A characterization of graphs with rank 4

The rank of a graph G is defined to be the rank of its adjacency matrix. In this paper, we consider the following problem: What is the structure of a connected graph with rank 4? This question has not yet been fully answered in the literature, and only some partial results are known. In this paper we resolve this question by completely characterizing graphs G whose adjacency matrix has rank 4.     

Unitary matrix digraphs and minimum semidefinite rank

For an undirected simple graph G, the minimum rank among all positive semidefinite matrices with graph G is called the minimum semidefinite rank (msr) of G. In this paper, we show that the msr of a given graph may be determined from the msr of a related bipartite graph. Finding the msr of a given bipartite graph is then shown to be equivalent to determining which digraphs encode the zero/nonzero pattern of a unitary matrix. We provide an algorithm to construct unitary matrices with a certain pattern, and use previous results to give a lower bound for the msr of certain bipartite graphs.     

Minimum rank of skew-symmetric matrices described by a graph

The minimum (symmetric) rank of a simple graph G over a field F is the smallest possible rank among all symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. The problem of determining minimum (symmetric) rank has been studied extensively. We define the minimum skew rank of a simple graph G to be the smallest possible rank among all skew-symmetric matrices over F whose ijth entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We apply techniques from the minimum (symmetric) rank problem and from skew-symmetric matrices to obtain results about the minimum skew rank problem.     

Lower Bounds in Minimum Rank Problems

The minimum rank of a graph is the smallest possible rank among all real symmetric matrices with the given graph. The minimum semidefinite rank of a graph is the minimum rank among Hermitian positive semidefinite matrices with the given graph. We explore connections between OS-sets and a lower bound for minimum rank related to zero forcing sets as well as exhibit graphs for which the difference between the minimum semidefinite rank and these lower bounds can be arbitrarily large.     

An upper bound for the minimum rank of a graph

For a graph G of order n, the minimum rank of G is defined to be the smallest possible rank over all real symmetric n×n matrices A whose (i,j)th entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. We prove an upper bound for minimum rank in terms of minimum degree of a vertex is valid for many graphs, including all bipartite graphs, and conjecture this bound is true over for all graphs, and prove a related bound for all zero-nonzero patterns of (not necessarily symmetric) matrices. Most of the results are valid for matrices over any infinite field, but need not be true for matrices over finite fields.     

Matroid Matching Via Mixed Skew-Symmetric Matrices

Tutte associates a V by V skew-symmetric matrix T, having indeterminate entries, with a graph G=(V,E). This matrix, called the Tutte matrix, has rank exactly twice the size of a maximum cardinality matching of G. Thus, to find the size of a maximum matching it suffices to compute the rank of T. We consider the more general problem of computing the rank of T + K where K is a real V by V skew-symmetric matrix. This modest generalization of the matching problem contains the linear matroid matching problem and, more generally, the linear delta-matroid parity problem. We present a tight upper bound on the rank of T + K by decomposing T + K into a sum of matrices whose ranks are easy to compute.Part of this research was done while the authors visited the Fields Institute in Toronto, Canada. The research was partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada.     

Zero forcing sets and the minimum rank of graphs

The 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. This paper introduces a new graph parameter, Z(G), that is the minimum size of a zero forcing set of vertices and uses it to bound the minimum rank for numerous families of graphs, often enabling computation of the minimum rank.     

The Laplacian of a Graph as a Density Matrix: A Basic Combinatorial Approach to Separability of Mixed States

We study entanglement properties of mixed density matrices obtained from combinatorial Laplacians. This is done by introducing the notion of the density matrix of a graph. We characterize the graphs with pure density matrices and show that the density matrix of a graph can be always written as a uniform mixture of pure density matrices of graphs. We consider the von Neumann entropy of these matrices and we characterize the graphs for which the minimum and maximum values are attained. We then discuss the problem of separability by pointing out that separability of density matrices of graphs does not always depend on the labelling of the vertices. We consider graphs with a tensor product structure and simple cases for which combinatorial properties are linked to the entanglement of the state. We calculate the concurrence of all graphs on four vertices representing entangled states. It turns out that for these graphs the value of the concurrence is exactly fractional. Received July 28, 2004     

Zero forcing parameters and minimum rank problems

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.     

Bicyclic graphs for which the least eigenvalue is minimum

The spread of a graph is defined to be the difference between the greatest eigenvalue and the least eigenvalue of the adjacency matrix of the graph. In this paper we determine the unique graph with minimum least eigenvalue among all connected bicyclic graphs of order n. Also, we determine the unique graph with maximum spread in this class for each n?28.     

The minimum rank problem: A counterexample

We provide a counterexample to a recent conjecture that the minimum rank over the reals of every sign pattern matrix can be realized by a rational matrix. We use one of the equivalences of the conjecture and some results from projective geometry. As a consequence of the counterexample we show that there is a graph for which the minimum rank of the graph over the reals is strictly smaller than the minimum rank of the graph over the rationals. We also make some comments on the minimum rank of sign pattern matrices over different subfields of R.     

Minimum rank of outerplanar graphs

The problem of finding the minimum rank over all symmetric matrices corresponding to a given graph has grown in interest recently. It is well known that the minimum rank of any graph is bounded above by the clique cover number, the minimum number of cliques needed to cover all edges of the graph. We generalize the idea of the clique cover number by defining the rank sum of a cover to be the sum of the minimum ranks of the graphs in the cover. Using this idea we obtain a combinatorial solution to the minimum rank problem for an outerplanar graph. As a consequence the minimum rank of an outerplanar graph is field independent and all outerplanar graphs have a universally optimal matrix. We also consider implications of the main result to the inverse inertia problem.     

Cycles of Nonzero Elements in Low Rank Matrices

Dedicated to the memory of Paul Erdős We consider the problem of finding some structure in the zero-nonzero pattern of a low rank matrix. This problem has strong motivation from theoretical computer science. Firstly, the well-known problem on rigidity of matrices, proposed by Valiant as a means to prove lower bounds on some algebraic circuits, is of this type. Secondly, several problems in communication complexity are also of this type. The special case of this problem, where one considers positive semidefinite matrices, is equivalent to the question of arrangements of vectors in euclidean space so that some condition on orthogonality holds. The latter question has been considered by several authors in combinatorics [1, 4]. Furthermore, we can think of this problem as a kind of Ramsey problem, where we study the tradeoff between the rank of the adjacency matrix and, say, the size of a largest complete subgraph. In this paper we show that for an real matrix with nonzero elements on the main diagonal, if the rank is o(n), the graph of the nonzero elements of the matrix contains certain cycles. We get more information for positive semidefinite matrices. Received September 9, 1999 RID="*" ID="*" Partially supported by grant A1019901 of the Academy of Sciences of the Czech Republic and by a cooperative research grant INT-9600919/ME-103 from the NSF (USA) and the MŠMT (Czech Republic).     

Techniques for determining the minimum rank of a small graph

The 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. Minimum rank is a difficult parameter to compute. However, there are now a number of known reduction techniques and bounds that can be programmed on a computer; we have developed a program using the open-source mathematics software Sage to implement several techniques. We have also established several additional strategies for computation of minimum rank. These techniques have been used to determine the minimum ranks of all graphs of order 7.     

Linearly independent vertices and minimum semidefinite rank

We study the minimum semidefinite rank of a graph using vector representations of the graph and of certain subgraphs. We present a sufficient condition for when the vectors corresponding to a set of vertices of a graph must be linearly independent in any vector representation of that graph, and conjecture that the resulting graph invariant is equal to minimum semidefinite rank. Rotation of vector representations by a unitary matrix allows us to find the minimum semidefinite rank of the join of two graphs. We also improve upon previous results concerning the effect on minimum semidefinite rank of the removal of a vertex.     

Classes of graphs with minimum skew rank 4

The minimum skew rank of a simple graph G   is the smallest possible rank among all real skew-symmetric matrices whose (i,j)$(i,j)$-entry is nonzero if and only if the edge joining i and j is in G. It is known that a graph has minimum skew rank 2 if and only if it consists of a complete multipartite graph and some isolated vertices. Some necessary conditions for a graph to have minimum skew rank 4 are established, and several families of graphs with minimum skew rank 4 are given. Linear algebraic techniques are developed to show that complements of trees and certain outerplanar graphs have minimum skew rank 4.     

On unimodular graphs

We study graphs whose adjacency matrices have determinant equal to 1 or −1, and characterize certain subclasses of these graphs. Graphs whose adjacency matrices are totally unimodular are also characterized. For bipartite graphs having a unique perfect matching, we provide a formula for the inverse of the corresponding adjacency matrix, and address the problem of when that inverse is diagonally similar to a nonnegative matrix. Special attention is paid to the case that such a graph is unicyclic.     

Maximum generic nullity of a graph

For a graph G of order n, the maximum nullity of G is defined to be the largest possible nullity over all real symmetric n×n matrices A whose (i,j)th entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Maximum nullity and the related parameter minimum rank of the same set of matrices have been studied extensively. A new parameter, maximum generic nullity, is introduced. Maximum generic nullity provides insight into the structure of the null-space of a matrix realizing maximum nullity of a graph. It is shown that maximum generic nullity is bounded above by edge connectivity and below by vertex connectivity. Results on random graphs are used to show that as n goes to infinity almost all graphs have equal maximum generic nullity, vertex connectivity, edge connectivity, and minimum degree.