Totally nonnegative matrices

An m-by-n matrix A is called totally nonnegative (resp. totally positive) if the determinant of every square submatrix (i.e., minor) of A is nonnegative (resp. positive). The class of totally nonnegative matrices has been studied considerably, and this class arises in a variety of applications such as differential equations, statistics, mathematical biology, approximation theory, integral equations and combinatorics. The main purpose of this thesis is to investigate several aspects of totally nonnegative matrices such as spectral problems, determinantal inequalities, factorizations and entry-wise products. It is well-known that the eigenvalues of a totally nonnegative matrix are nonnegative. However, there are many open problems about what other properties exist for the eigenvalues of such matrices. In this thesis we extend classical results concerning the eigenvalues of a totally nonnegative matrix and prove that the positive eigenvalues of an irreducible totally nonnegative matrix are distinct. We also demonstrate various new relationships between the sizes and the number of Jordan blocks corresponding to the zero eigenvalue of an irreducible totally nonnegative matrix. These relationships are a necessary first step to characterizing all possible Jordan canonical forms of totally nonnegative matrices. Another notion investigated is determinantal inequalities among principal minors of totally nonnegative matrices. A characterization of all inequalities that hold among products of principal minors of totally nonnegative matrices up to at most 5 indices is proved, along with general conditions which guarantee when the product of two principal minors is less than another product of two principal minors. A third component of this thesis is a study of entry-wise products of totally nonnegative matrices. In particular, we consider such topics as: closure under this product, questions related to zero/non-zero patterns, and determinantal inequalities associated with this special product. Finally, a survey of classical results and recent developments, including: commonalities and differences among totally nonnegative matrices and other positivity classes of matrices; perturbations and factorizations of totally nonnegative matrices, are discussed

On the Null Space Structure Associated with Trees and Cycles

In this work, we study the structure of the null spaces of matrices associated with graphs. Our primary tool is utilizing Schur complements based on certain collections of independent vertices. This idea is applied in the case of trees, and seems to represent a unifying theory within the context of the support of the null space. We extend this idea and apply it to describe the null vectors and corresponding nullities of certain symmetric matrices associated with cycle

On Perron complements of totally nonnegative matrices

AbstractAn n×n matrix is called totally nonnegative if every minor of A is nonnegative. The problem of interest is to describe the Perron complement of a principal submatrix of an irreducible totally nonnegative matrix. We show that the Perron complement of a totally nonnegative matrix is totally nonnegative only if the complementary index set is based on consecutive indices. We also demonstrate a quotient formula for Perron complements analogous to the so-called quotient formula for Schur complements, and verify an ordering between the Perron complement and Schur complement of totally nonnegative matrices, when the Perron complement is totally nonnegative

Inequalities for totally nonnegative matrices: Gantmacher--Krein, Karlin, and Laplace

A real linear combination of products of minors which is nonnegative over all totally nonnegative (TN) matrices is called a determinantal inequality for these matrices. It is referred to as multiplicative when it compares two collections of products of minors and additive otherwise. Set theoretic operations preserving the class of TN matrices naturally translate into operations preserving determinantal inequalities in this class. We introduce set row/column operations that act directly on all determinantal inequalities for TN matrices, and yield further inequalities for these matrices. These operations assist in revealing novel additive inequalities for TN matrices embedded in the classical identities due to Laplace $[$Mem$.$ Acad$.$ Sciences Paris $1772]$ and Karlin $(1968).$ In particular, for any square TN matrix $A,$ these derived inequalities generalize -- to every i^{\mbox{th}} row of $A$ and j^{\mbox{th}} column of ${\rm adj} A$ -- the classical Gantmacher--Krein fluctuating inequalities $(1941)$ for $i=j=1.$ Furthermore, our row/column operations reveal additional undiscovered fluctuating inequalities for TN matrices. The introduced set row/column operations naturally birth an algorithm that can detect certain determinantal expressions that do not form an inequality for TN matrices. However, the algorithm completely characterizes the multiplicative inequalities comparing products of pairs of minors. Moreover, the underlying row/column operations add that these inequalities are offshoots of certain ''complementary/higher'' ones. These novel results seem very natural, and in addition thoroughly describe and enrich the classification of these multiplicative inequalities due to Fallat--Gekhtman--Johnson $[$Adv$.$ Appl$.$ Math$.$ $2003]$ and later Skandera $[$J$.$ Algebraic Comb$.$ $2004].$Comment: 2 figures, 39 pages, and minor adjustments in the expositio

On acyclic and unicyclic graphs whose minimum rank equals the diameter

AbstractThe 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

The minimum rank of universal adjacency matrices

In this paper we introduce a new parameter for a graph called the {\it minimum universal rank}. This parameter is similar to the minimum rank of a graph. For a graph $G$ the minimum universal rank of $G$ is the minimum rank over all matrices of the form $$U(\alpha, \beta, \gamma, \delta) = \alpha A + \beta I + \gamma J + \delta D$$ where $A$ is the adjacency matrix of $G$, $J$ is the all ones matrix and $D$ is the matrix with the degrees of the vertices in the main diagonal, and $\alpha\neq 0, \beta, \gamma, \delta$ are scalars. Bounds for general graphs based on known graph parameters are given, as is a formula for the minimum universal rank for regular graphs based on the multiplicity of the eigenvalues of $A$. The exact value of the minimum universal rank of some families of graphs are determined, including complete graphs, complete bipartite graph, paths and cycles. Bounds on the minimum universal rank of a graph obtained by deleting a single vertex are established. It is shown that the minimum universal rank is not monotone on induced subgraphs, but bounds based on certain induced subgraphs, including bounds on the union of two graphs, are given. Finally we characterize all graphs with minimum universal rank equal to 0 and to 1

Eigenvalue location for nonnegative and Z-matrices

AbstractLet Lk0 denote the class of n × n Z-matrices A = tl − B with B ⩾ 0 and ϱk(B) ⩽ t < ϱk + 1(B), where ϱk(B) denotes the maximum spectral radius of k × k principal submatrices of B. Bounds are determined on the number of eigenvalues with positive real parts for A ϵ Lk0, where k satisfies, ⌊n2⌋ ⩽ k ⩽ n − 1. For these classes, when k = n − 1 and n − 2, wedges are identified that contain only the unqiue negative eigenvalue of A. These results lead to new eigenvalue location regions for nonnegative matrices

On the graph complement conjecture for minimum rank

AbstractThe minimum rank of a graph has been an interesting and well studied parameter investigated by many researchers over the past decade or so. One of the many unresolved questions on this topic is the so-called graph complement conjecture, which grew out of a workshop in 2006. This conjecture asks for an upper bound on the sum of the minimum rank of a graph and the minimum rank of its complement, and may be classified as a Nordhaus–Gaddum type problem involving the graph parameter minimum rank. The conjectured bound is the order of the graph plus two. Other variants of the graph complement conjecture are introduced here for the minimum semidefinite rank and the Colin de Verdière type parameter ν. We show that if the ν-graph complement conjecture is true for two graphs then it is true for the join of these graphs. Related results for the graph complement conjecture (and the positive semidefinite version) for joins of graphs are also established. We also report on the use of recent results on partial k-trees to establish the graph complement conjecture for graphs of low minimum rank