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 minimum rank of the join of graphs and decomposable graphs

AbstractFor a given undirected graph G, the minimum rank of G is defined to be the smallest possible rank over all real symmetric matrices A whose (i,j)th entry is nonzero whenever i≠j and {i,j} is an edge in G. In this work we consider joins and unions of graphs, and characterize the minimum rank of such graphs in the case of ‘balanced inertia’. Several consequences are provided for decomposable graphs, also known as cographs

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

On the normalized Laplacian energy and general Randic index R_{-1} of graphs

In this paper, we consider the energy of a simple graph with respect to its normalized Laplacian eigenvalues, which we call the L-energy. Over graphs of order n that contain no isolated vertices, we characterize the graphs with minimal L-energy of 2 and maximal L-energy of 2bn=2c. We provide upper and lower bounds for L-energy based on its general Randic index R-1(G). We highlight known results for R-1(G), most of which assume G is a tree. We extend an upper bound of R-1(G) known for trees to connected graphs. We provide bounds on the L-energy in terms of other parameters, one of which is the energy with respect to the adjacency matrix. Finally, we discuss the maximum change of L-energy and R-1(G) upon edge deletion

On a relationship between the characteristic and matching polynomials of a uniform hypertree

A hypertree is a connected hypergraph without cycles. Further a hypertree is called an $r$-tree if, additionally, it is $r$-uniform. Note that 2-trees are just ordinary trees. A classical result states that for any 2-tree $T$ with characteristic polynomial $\phi_T(\lambda)$ and matching polynomial $\varphi_T(\lambda)$, then $\phi_T(\lambda)=\varphi_T(\lambda).$ More generally, suppose $\mathcal{T}$ is an $r$-tree of size $m$ with $r\geq2$. In this paper, we extend the above classical relationship to $r$-trees and establish that $$\phi_{\mathcal{T}}(\lambda)=\prod_{H \sqsubseteq \mathcal{T}}\varphi_{H}(\lambda)^{a_{H}},$$ where the product is over all connected subgraphs $H$ of $\mathcal{T}$, and the exponent $a_{H}$ of the factor $\varphi_{H}(\lambda)$ can be written as $$a_H=b^{m-e(H)-|\partial(H)|}c^{e(H)}(b-c)^{|\partial(H)|},$$ where $e(H)$ is the size of $H$, $\partial(H)$ is the boundary of $H$, and $b=(r-1)^{r-1}, c=r^{r-2}$. In particular, for $r=2$, the above correspondence reduces to the classical result for ordinary trees. In addition, we resolve a conjecture by Clark-Cooper [{\em Electron. J. Combin.}, 2018] and show that for any subgraph $H$ of an $r$-tree $\mathcal{T}$ with $r\geq3$, $\varphi_H(\lambda)$ divides $\phi_{\mathcal{T}}(\lambda)$, and additionally $\phi_H(\lambda)$ divides $\phi_{\mathcal{T}}(\lambda)$, if either $r\geq 4$ or $H$ is connected when $r=3$. Moreover, a counterexample is given for the case when $H$ is a disconnected subgraph of a 3-tree.Comment: 36 pages, 4 figure