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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Vertex and edge spread of zero forcing number,maximum nullity,and minimum rank of a 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 $i\ne j$) is nonzero whenever $\{i\text{,}j\}$ is an edge in G and is zero otherwise; maximum nullity is taken over the same set of matrices. The zero forcing number is the minimum size of a zero forcing set of vertices and bounds the maximum nullity from above. The spread of a graph parameter at a vertex v or edge e of G is the difference between the value of the parameter on G and on $G-v$ or $G-e$. Rank spread (at a vertex) was introduced in [4]. This paper introduces vertex spread of the zero forcing number and edge spreads for minimum rank/maximum nullity and zero forcing number. Properties of the spreads are established and used to determine values of the minimum rank/maximum nullity and zero forcing number for various types of grids with a vertex or edge deleted.     

Isospectral flows that preserve matrix structure

The matrix A = (a_ij) ∈ S_n is said to lie on a strict undirected graph G if a_ij = 0 (i ≠ j) whenever (i, j) is not in E(G). If S is skew-symmetric, the isospectral flow maintains the spectrum of A. We consider isospectral flows that maintain a matrix A(t) on a given graph G. We review known results for a graph G that is a (generalised) path, and construct isospectral flows for a (generalised) ring, and a star, and show how a flow may be constructed for a general graph. The analysis may be applied to the isospectral problem for a lumped-mass finite element model of an undamped vibrating system. In that context, it is important that the flow maintain other properties such as irreducibility or positivity, and we discuss whether they are maintained.     

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.     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

The inertia set of the join of graphs

Let G=(V,E) be a graph with V={1,2,…,n}. Denote by S(G) the set of all real symmetric n×n matrices A=[a_i,j] with a_i,j≠0, i≠j if and only if ij is an edge of G. Denote by I^↗(G) the set of all pairs (p,q) of natural numbers such that there exists a matrix A∈S(G) with at most p positive and q negative eigenvalues. We show that if G is the join of G₁ and G₂, then I^↗(G)?{(1,1)}=I^↗(G₁∨K₁)∩I^↗(G₂∨K₁)?{(1,1)}. Further, we show that if G is a graph with s isolated vertices, then , where denotes the graph obtained from G be removing all isolated vertices, and we give a combinatorial characterization of graphs G with (1,1)∈I^↗(G). We use these results to determine I^↗(G) for every complete multipartite graph G.     

Expected values of parameters associated with the minimum rank of a graph

We investigate the expected value of various graph parameters associated with the minimum rank of a graph, including minimum rank/maximum nullity and related Colin de Verdière-type parameters. Let G(v,p) denote the usual Erd?s-Rényi random graph on v vertices with edge probability p. We obtain bounds for the expected value of the random variables mr(G(v,p)), M(G(v,p)), ν(G(v,p)) and ξ(G(v,p)), which yield bounds on the average values of these parameters over all labeled graphs of order v.     

Forest-Like Permutations

Given a permutation , construct a graph G _π on the vertex set {1, 2,..., n} by joining i to j if (i) i < j and π(i) < π(j) and (ii) there is no k such that i < k < j and π(i) < π(k) < π(j). We say that π is forest-like if G _π is a forest. We first characterize forest-like permutations in terms of pattern avoidance, and then by a certain linear map being onto. Thanks to recent results of Woo and Yong, these show that forest-like permutations characterize Schubert varieties which are locally factorial. Thus forest-like permutations generalize smooth permutations (corresponding to smooth Schubert varieties). We compute the generating function of forest-like permutations. As in the smooth case, it turns out to be algebraic. We then adapt our method to count permutations for which G _π is a tree, or a path, and recover the known generating function of smooth permutations. Received March 27, 2006     

Some intersection theorems

LetL(A) be the set of submatrices of anm×n matrixA. ThenL(A) is a ranked poset with respect to the inclusion, and the poset rank of a submatrix is the sum of the number of rows and columns minus 1, the rank of the empty matrix is zero. We attack the question: What is the maximum number of submatrices such that any two of them have intersection of rank at leastt? We have a solution fort=1,2 using the followoing theorem of independent interest. Letm(n,i,j,k) = max(|F|;|G|), where maximum is taken for all possible pairs of families of subsets of ann-element set such thatF isi-intersecting,G isj-intersecting andF ansd,G are cross-k-intersecting. Then fori≤j≤k, m(n,i,j,k) is attained ifF is a maximali-intersecting family containing subsets of size at leastn/2, andG is a maximal2k?i-intersecting family. Furthermore, we discuss and Erd?s-Ko-Rado-type question forL(A), as well.     

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.     

Least squares solutions to AX = B for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation

A matrix A∈R^n×n is called a bisymmetric matrix if its elements a_i,j satisfy the properties a_i,j=a_j,i and a_i,j=a_n-j+1,n-i+1 for 1?i,j?n. This paper considers least squares solutions to the matrix equation AX=B for A under a central principal submatrix constraint and the optimal approximation. A central principal submatrix is a submatrix obtained by deleting the same number of rows and columns in edges of a given matrix. We first discuss the specified structure of bisymmetric matrices and their central principal submatrices. Then we give some necessary and sufficient conditions for the solvability of the least squares problem, and derive the general representation of the solutions. Moreover, we also obtain the expression of the solution to the corresponding optimal approximation 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.