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.     

Lower bounds for minimum semidefinite rank from orthogonal removal and chordal supergraphs

The minimum semidefinite rank (msr) of a graph is the minimum rank among positive semidefinite matrices with the given graph. The OS-number is a useful lower bound for msr, which arises by considering ordered vertex sets with some connectivity properties. In this paper, we develop two new interpretations of the OS-number. We first show that OS-number is also equal to the maximum number of vertices which can be orthogonally removed from a graph under certain nondegeneracy conditions. Our second interpretation of the OS-number is as the maximum possible rank of chordal supergraphs who exhibit a notion of connectivity we call isolation-preserving. These interpretations not only give insight into the OS-number, but also allow us to prove some new results. For example we show that msr(G) = |G| ? 2 if and only if OS(G) = |Gzsfnc ? 2.     

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.     

On the minimum semidefinite rank of a simple graph

The minimum semidefinite rank (msr) of a graph is defined to be the minimum rank among all positive semidefinite matrices whose zero/nonzero pattern corresponds to that graph. We recall some known facts and present new results, including results concerning the effects of vertex or edge removal from a graph on msr.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

The elliptic matrix completion problem

Completions of partial elliptic matrices are studied. Given an undirected graph G, it is shown that every partial elliptic matrix with graph G can be completed to an elliptic matrix if and only if the maximal cliques of G are pairwise disjoint. Further, given a partial elliptic matrix A with undirected graph G, it is proved that if G is chordal and each specified principal submatrix defined by a pair of intersecting maximal cliques is nonsingular, then A can be completed to an elliptic matrix. Conversely, if G is nonchordal or if the regularity condition is relaxed, it is shown that there exist partial elliptic matrices which are not completable to an elliptic matrix. In the process we obtain several results concerning chordal graphs that may be of independent interest.     

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.     

On the Sparsity Order of a Graph and Its Deficiency in Chordality

Given a graph G on n nodes, let denote the cone consisting of the positive semidefinite matrices (with real or complex entries) having a zero entry at every off-diagonal position corresponding to a non edge of G. Then, the sparsity order of G is defined as the maximum rank of a matrix lying on an extreme ray of the cone . It is known that the graphs with sparsity order 1 are the chordal graphs and a characterization of the graphs with sparsity order 2 is conjectured in [1] in the real case. We show in this paper the validity of this conjecture. Moreover, we characterize the graphs with sparsity order 2 in the complex case and we give a decomposition result for the graphs with sparsity order in both real and complex cases. As an application, these graphs can be recognized in polynomial time. We also indicate how an inequality from [17] relating the sparsity order of a graph and its minimum fill-in can be derived from a result concerning the dimension of the faces of the cone . Received August 31, 1998/Revised April 26, 2000     

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.     

The minimum semidefinite rank of a triangle-free graph

We employ a result of Moshe Rosenfeld to show that the minimum semidefinite rank of a triangle-free graph with no isolated vertex must be at least half the number of its vertices. We define a Rosenfeld graph to be such a graph that achieves equality in this bound, and we explore the structure of these special graphs. Their structure turns out to be intimately connected with the zero-nonzero patterns of the unitary matrices. Finally, we suggest an exploration of the connection between the girth of a graph and its minimum semidefinite rank, and provide a conjecture in this direction.     

Unitarily achievable zero patterns and traces of words in A and A

Our primary objective is to identify a natural and substantial problem about unitary similarity on arbitrary complex matrices: which 0-patterns may be achieved for any given n-by-n complex matrix via some unitary similarity of it. To this end, certain restrictions on “achievable” 0-patterns are mentioned, both positional and, more important, on the maximum number of achievable 0’s. Prior results fitting this general question are mentioned, as well as the “first” unresolved pattern (for 3-by-3 matrices!). In the process a recent question is answered.A closely related additional objective is to mention the best known bound for the maximum length of words necessary for the application of Specht’s theorem about which pairs of complex matrices are unitarily similar, which seems not widely known to matrix theorists. In the process, we mention the number of words necessary for small size matrices.     

Converse to the Parter-Wiener theorem: The case of non-trees

Through a succession of results, it is known that if the graph of an Hermitian matrix A is a tree and if for some index j, λ∈σ(A)∩σ(A(j)), then there is an index i such that the multiplicity of λ in σ(A(i)) is one more than that in A. We exhibit a converse to this result by showing that it is generally true only for trees. In particular, it is shown that the minimum rank of a positive semidefinite matrix with a given graph G is ?n-2 when G is not a tree. This raises the question of how the minimum rank of a positive semidefinite matrix depends upon the graph in general.     

A note on universally optimal matrices and field independence of the minimum rank of a graph

For a simple graph G on n vertices, the minimum rank of G over a field F, written as mr^F(G), is defined to be the smallest possible rank among all n×n symmetric matrices over F whose (i,j)th entry (for i≠j) is nonzero whenever {i,j} is an edge in G and is zero otherwise. A symmetric integer matrix A such that every off-diagonal entry is 0, 1, or -1 is called a universally optimal matrix if, for all fields F, the rank of A over F is the minimum rank of the graph of A over F. Recently, Dealba et al. [L.M. Dealba, J. Grout, L. Hogben, R. Mikkelson, K. Rasmussen, Universally optimal matrices and field independence of the minimum rank of a graph, Electron. J. Linear Algebra 18 (2009) 403-419] initiated the study of universally optimal matrices and established field independence or dependence of minimum rank for some families of graphs. In the present paper, more results on universally optimal matrices and field independence or dependence of the minimum rank of a graph are presented, and some results of Dealba et al. [5] are improved.     

The inverse inertia problem for graphs: Cut vertices, trees, and a counterexample

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in terms of a new family of graph parameters, the maximal disconnection numbers of a graph. We also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs. Finally, we give an example of a graph that is not inertia-balanced, which settles an open problem from the October 2006 AIM Workshop on Spectra of Families of Matrices described by Graphs, Digraphs and Sign Patterns. We also determine some restrictions on the inertia set of any graph.