Wayne Barrett

• Wayne Barrett

The parameter $q(G)$ of an $n$-vertex graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. We show that all $G$ with $e(\overline{G}) = |E(\overline{G})| \leq \lfloor n/2 \rfloor -1$ have $q(G)=2$. We conjecture that any $G$ with $e(\overline{G}) \leq n-3$ satisfies $q(G) = 2$. We show that...

For an $n \times n$ matrix $A$, let $q(A)$ be the number of distinct eigenvalues of $A$. If $G$ is a connected graph on $n$ vertices, let $\mathcal{S}(G)$ be the set of all real symmetric $n \times n$ matrices $A=[a_{ij}]$ such that for $i\neq j$, $a_{ij}=0$ if and only if $\{i,j\}$ is not an edge of $G$. Let $q(G)={\rm min}\{q(A)\,:\,A \in \mathca...

In this article, we extend the notion of the Laplacian spread to simple directed graphs (digraphs) using the restricted numerical range. First, we provide Laplacian spread values for several families of digraphs. Then, we prove sharp upper bounds on the Laplacian spread for all polygonal and balanced digraphs. In particular, we show that the validi...

The parameter $q(G)$ of a graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. It is shown that the minimum number of edges necessary for a connected graph $G$ to have $q(G)=2$ is $2n-4$ if $n$ is even, and $2n-3$ if $n$ is odd. In addition, a characterization of graphs for which equality i...

We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number of vertices of high eccentricity in a graph. We prove that this lower bound implies a strengthening of the Laplacian Spread Conjecture. We discuss further conjectures, also strengthening the Laplacian Spread Conjecture, that include a conjecture for si...

We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number of vertices of high eccentricity in a graph. We prove that this lower bound implies a strengthening of the Laplacian Spread Conjecture. We discuss further conjectures, also strengthening the Laplacian Spread Conjecture, that include a conjecture for si...

In [3], [4] the authors found formulas for the number of spanning 2-forests (equivalently for the resistance distance) between any pair of vertices in a “straight linear 2-tree” and in a “linear 2-tree with a single bend.” The main result of this paper generalizes these formulas to linear 2-trees with any number of bends. We also give alternative a...

A spanning 2-forest separating vertices u and v of an undirected connected graph is a spanning forest with 2 components such that u and v are in distinct components. Aside from their combinatorial significance, spanning 2-forests have an important application to the calculation of resistance distance or effective resistance. The resistance distance...

A spanning 2-forest separating vertices $u$ and $v$ of an undirected connected graph is a spanning forest with 2 components such that $u$ and $v$ are in distinct components. Aside from their combinatorial significance, spanning 2-forests have an important application to the calculation of resistance distance or effective resistance. The resistance...

In this note we consider the bent linear 2-tree and provide an explicit formula for the resistance distance $r_{G_n}(1,n)$ between the first and last vertices of the graph. We call the graph $G_n$ with vertex set $V(G_n) = \{ 1, 2, \ldots, n\}$ and $\{i,j\} \in E(G_n)$ if and only if $0<|i-j| \leq 2$ a straight linear 2-tree. We define the graph $H...

We consider the graph $G_n$ with vertex set $V(G_n) = \{ 1, 2, \ldots, n\}$ and $\{i,j\} \in E(G_n)$ if and only if $0<|i-j| \leq 2$. We call $G_n$ the straight linear 2-tree on $n$ vertices. Using $\Delta$--Y transformations and identities for the Fibonacci and Lucas numbers we obtain explicit formulae for the resistance distance $r_{G_n}(i,j)$ be...

We consider the graph $G_n$ with vertex set $V(G_n) = \{ 1, 2, \ldots, n\}$ and $\{i,j\} \in E(G_n)$ if and only if $0<|i-j| \leq 2$. We call $G_n$ the straight linear 2-tree on $n$ vertices. Using $\Delta$--Y transformations and identities for the Fibonacci and Lucas numbers we obtain explicit formulae for the resistance distance $r_{G_n}(i,j)$ be...

The inverse eigenvalue problem of a given graph $G$ is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in $G$. Barrett et al. introduced the Strong Spectral Property (SSP) and the Strong Multiplicity Property (SMP) in [8]. In that paper it was shown that if a graph has a matrix...

The inverse eigenvalue problem of a given graph $G$ is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in $G$. Barrett et al. introduced the Strong Spectral Property (SSP) and the Strong Multiplicity Property (SMP) in [8]. In that paper it was shown that if a graph has a matrix...

For a given graph G and an associated class of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G, the collection of all possible spectra for such matrices is considered. Building on the pioneering work of Colin de Verdiere in connection with the Strong Arnold Property, two extensions are devised that target a b...

The automorphisms of a graph can be used to characterize the graph's symmetries. Here we show that if a graph has any kind of symmetry, it is possible to decompose the graph's (weighted) adjacency matrix $A$ into a number of smaller matrices with respect to any one of its automorphisms. The collective eigenvalues of these resulting matrices are the...

We investigate connections between the symmetries (automorphisms) of a graph and its spectral properties. Whenever a graph has a symmetry, i.e. a nontrivial automorphism $\phi$, it is possible to use $\phi$ to decompose any matrix $M\in\mathbb{C}^{n \times n}$ appropriately associated with the graph. The result of this decomposition is a number of...

This paper examines 0–1 symmetric matrices for which the inverse of the matrix has all entries of the form ±α for some α and constant diagonal. Several constructions of such matrices are given as well as a strong connection to matrices that are invertible and have all principal submatrices of order singular.

Given an n×nn×n matrix, its principal rank characteristic sequence is a sequence of length n+1n+1 of 0s and 1s where, for k=0,1,…,nk=0,1,…,n, a 1 in the kth position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix. The principal rank characteristic sequences for symmetric matrices over va...

Let G be a simple undirected graph on n vertices and let 8(G) be the class of real symmetric n x n matrices whose nonzero off-diagonal entries correspond to the edges of G. Given 2n 1 real numbers Ai >t > A2 > /t2 > " " " > An > ttn i > An, and a vertex v of G, the question is addressed of whether or not there exists A E 8(G) with eigenvalues Ai,.....

Barrett et al. asked in W. Barrett et al. Minimum rank of edge subdivisions of graphs. Electronic Journal of Linear Algebra, 18:530-563, 2009.], whether the maximum nullity is equal to the zero forcing number for all complete subdivision graphs. We prove that this equality holds. Furthermore, we compute the value of M(F, G) = Z(G) by introducing th...

We establish the bounds 4/3 <= b(v) <= b(xi) <= root 2, where b(v) and b(xi) are the Nordhaus-Gaddum sum upper bound multipliers, i.e., v(G) + v((G) over bar) <= b(v)vertical bar G vertical bar and xi(G) + xi((G) over bar) <= b(xi)vertical bar G vertical bar for all graphs G, and v and xi are Colin de Verdiere type graph parameters. The Nordhaus-Ga...

Let G be a simple undirected graph on n vertices and let S(G) be the class of real symmetric n×n matrices whose nonzero off-diagonal entries correspond exactly to the edges of G. Given 2n-1 real numbers λ 1 ≥μ 1 ≥λ 2 ≥μ 2 ≥⋯≥λ n-1 ≥μ n-1 ≥λ n , and a vertex v of G, the question is addressed of whether or not there exists A∈S(G) with eigenvalues λ 1...

Let F be a field, let G be an undirected graph on n vertices, and let SF(G)SF(G) be the class of all F-valued symmetric n×nn×n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For each graph G, there is an associated minimum rank class MRF(G)MRF(G) consisting of all matrices A∈SF(G)A∈SF(G)...

Tree‐width, and variants that restrict the allowable tree decompositions, play an important role in the study of graph algorithms and have application to computer science. The zero forcing number is used to study the maximum nullity/minimum rank of the family of symmetric matrices described by a graph. We establish relationships between these param...

Let F be a field, let G be a simple graph on n vertices, and let SF (G) be the class of all F-valued symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For each graph G, there is an associated minimum rank class, MRF (G) consisting of all matrices A ∈ SF (G) with rank A = mrF...

We consider the problem of computing inertia sets for graphs. By using tools for combining the inertia sets of smaller graphs we can reduce this problem to understanding the inertia sets for three-connected graphs that are not joins. We term such graphs atoms and give the inertia sets for all atoms on at most seven vertices. This can be used to com...

The 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...

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=[ai,j] with ai,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 a...

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. Let mr(G) denote the minimum rank of all matrices in S(G), and mr +(G) the minimum rank of all positive semidefinite matrices in S(G). All graphs G w...

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...

Let G be an undirected graph on n vertices and let S(G) be the set of all real sym-metric 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), a question which was previously answered when G is a tr...

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 te...

Let F be a field, let G be an undirected graph on n vertices, and let S(F, G) be the set of all F -valued symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The minimum rank of G over F is defined to be mr(F, G) = min{rank A| A 2 S(F, G)}. The problem of finding the minimum ra...

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 r...

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n x 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...

Our main result is a sharp bound for the number of vertices in a minimal forbidden subgraph for the graphs having minimum rank at most 3 over the finite field of order 2. We also list all 62 such minimal forbidden subgraphs. We conclude by exploring how some of these results over the finite field of order 2 extend to arbitrary fields and demonstrat...

Let F be a finite field, G =( V, E) be an undirected graph on n vertices, and let S(F, G) be the set of all symmetric n × n matrices over F whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G.L et mr(F, G) be the minimum rank of all matrices in S(F, G). If F is a finite field with pt elements, p � =2 ,...

Let F be a field, G =( V, E) be an undirected graph on n vertices, and let S(F, G) be the set of all symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For example, if G is a path, S(F, G )co nsists of the symmetric irreducible tridiagonal matrices. Let mr(F, G) be the minimum...

Let R be the real numbers and R-n the vector space of all column vectors of length n. Let l(n) be the convex set of all real correlation matrices of size n. If V is a subspace of R-n of dimension k, we consider the face F-V of l(n) consisting of all A is an element of l(n) such that V subset of N(A), i.e., AV = 0. If F-V is nonempty, we say that V...

The positive semidefinite (PSD) completion problem is concerned with determining the set of PSD completions of a partial matrix. Previous work has focused on determining whether or not a given partial PSD matrix has a PSD completion, by examining characteristics of the graph of the matrix. Our aim is to move beyond the existence question to that of...

A function f from the symmetric group Sn into R is called a class function if it is constant on each conjugacy class. Let df be the generalized matrix function associated with f, mapping the n-by-n Hermitian matrices to R. For example, if f(σ)=sgn(σ), then df(A)=detA. Let Kn(Kn(R)) denote the closed convex cone of those f for which df(A)⩾0 for all...

A function f from the symmetric group Sn into R is called a class function if f(σ) = f(τ) whenever τ is conjugate to σ. Let df be thegeneralized matrix function associated with f, mapping the n-by-n positive semidefinite Hermitian matricesto R. For example, if f(σ) = sgn(σ), then df(A) = det A. We consider the cone Kn of those f for which df(A) ≥ 0...

Among various matrix completion problems that have been considered in recent years, the positive definite completion problem seems to have received the most attention. Indeed, in addition to being a problem of great interest, it is related to various applications as well as other completion problems. It may also be viewed as a fundamental problem i...

The positive semidefinite (PSD) completion problem is concerned with determining the set of PSD completions of a partial matrix. Previous work has focused on determining whether or not a given partial PSD matrix has a PSD completion, by examining characteristics of the graph of the matrix. Our aim is to move beyond the existence question to that of...

. We consider the question of whether a real partial positive definite matrix (in which the specified off-diagonal entries consist of a full n cycle) has a positive definite completion. This lies in contrast to the previously studied chordal case. We give two solutions. In one, we describe about n 2 independent conditions on angles associated with...

Several inequalities relating the rank of a positive semidefinite matrix with the ranks of various principal submatrices are presented. These inequalities are analogous to known determinantal inequalities for positive definite matrices, such as Fischer's inequality, Koteljanskii's inequality, and extensions of these associated with chordal graphs.

We consider the problem of identifying all determinantal inequalities valid on all positive definite matrices. This is fundamentally a combinatorial problem about relations between collections of index sets. We describe some general structure of this problem and give sufficient and necessary conditions that coincide for collections of no more than...

Given a complex n-by-n matrix A, and α{1,2,…,n} denote by A[α] the principal submatrix of A lying in the rows and columns indicated by α. For a partition k=(k 1,…,kp ) of n k 1≥ċ≥kp , Σki =n, into p nonnegative integers, we define the symmetrized Fischer product If A is positive definite, each product in the sum is an upper bound for detA by Fische...

Let M n (F) denote the set of n-by-n matrices over the field F. We consider the following question: Among matrices A ∈ M n (F) with rank A = k < n, how many diagonal entries of A must be changed, at worst, in order to guarantee that the rank of A is increased. Our initial motivation arose from an error pointed out in [BOvdD], but we also view this...

Define n × n matrices Dn = (dij) andCn = (cij) by dij = 1 ifi|j, 0 otherwise, and Cn = (0, 1, 1,…,1)T(1, 0, 0,…,0). Let An = Dn + Cn. The matrix is of number-theoretic significance because det An = M(n) is Mertens' function. We give a simplified derivation of the characteristic polynomial of An, give precise asymptotic estimates for its two “large”...

An n × n real matrix T ∈ Mk if T=D+A where D is diagonal and rank.A= k. For 0 ≤ k ≤ n − 1 and A diagonally symmetrizable, we prove that all but k eigenvalues of T lie in the closed interval between the minimum and maximum diagonal entry of D, but show that no such result holds for general A. This answers an open problem posed by Furth and Sierksma....

Let A be a partial positive definite Hermitian matrix, and let G(A) be the undirected graph of the specified off-diagonal entries of A. When G(A) is a chordal graph, we present an explicit determinantal formula for the (unique) determinant-maximizing positive definite completion (maximum-entropy completion) of A. In order to do this, we first give...

Let ϵ1 = (1, 0,…, 0) and ϵ = (1, 1,…, 1) be complex vectors of length n, and let Dn = (dij), with and 0 otherwise, be an n × n complex matrix. Then the sizes of the Jordan blocks of the matrix ϵTϵ1 + Dn corresponding to the eigenvalue 1 are , where {n} is the greatest odd integer less than or equal to n.

Define n× nmatricesDn = (dij) and Cn = (cij) by dij = 1 if i∣j, 0 otherwise and Cn = (0, 1, 1,…, 1)T(1, 0, 0,…, 0). Let An = Dn + Cn. We use the directed graph of An −In to obtain the characteristic polynomial of An. Then we show that all but [log2n]+1 of the eigenvalues of An are equal to 1 and that ϱ(An) is asymptotically equal to √n as n → ∞.

For a nonsingular n-by-n matrix $A = [a_{ij} ]$, let $\alpha \subseteq \{ 1,2, \cdots ,n\} $ and let $A[\alpha ]$ denote the principal submatrix of A lying in the rows and columns indicated by $\alpha $. We determine the combinatorial circumstances under which the $(i,j)$ entry of the Schur complement $(A^{ - 1} [\alpha ])^{ - 1} $ equals $a_{ij} $...

In this paper, equality of ranks of particular submatrices in a matrix and its inverse is shown. These results are applied briefly to local duality and functional structure. Examples from production economics are used to illustrate the significance of the results.

In an earlier paper, formulae for det A as a ratio of products of principal minors of A were exhibited, for any given symmetric zero-pattern of A−1. These formulae may be presented in terms of a spanning tree of the intersection graph of certain index sets associated with the zero pattern of A−1. However, just as the determinant of a diagonal and o...

All possible graph-theoretic generalizations of a certain sort for the Hadamard-Fischer determinantal inequalities are determined. These involve ratios of products of principal minors which dominate the determinant. Furthermore, the cases of equality in these inequalities are characterized, and equality is possible for every set of values which can...

For any given set S of n distinct positive numbers, we construct a symmetric n-by-n (strictly) totally positive matrix whose spectrum is S. Thus, in order to be the spectrum of an n-by-n totally positive matrix, it is necessary and sufficient that n numbers be positive and distinct.

Given a Toeplitz matrix T with banded inverse [i.e., (T−1)ij=0 for j−i>p], we show that the elements of T can be expressed in terms of the roots of a polynomial. Then, using properties we have previously established, we generalize this result appropriately to allow singular T and show that the converse also holds. Finally, we give a sufficient cond...

The determinant of a matrix is expressed in terms of certain of its principal minors by a formula which can be “read off” from the graph of the inverse of the matrix. The only information used is the zero pattern of the inverse, and each zero pattern yields one or more corresponding formulae for the determinant.

We establish a correspondence between the vanishing of a certain set of minors of a matrix A and the vanishing of a related set of minors of A×1. In particular, inverses of banded matrices are characterized. We then use our results to find patterns for Toeplitz matrices with banded inverses. Finally we give an interesting determinant formula for in...

Tridiagonal or Jacobi matrices arise in many diverse branches of mathematics and have been studied extensively. However, there is little written about the inverses of such matrices. In this paper we characterize those matrices with nonzero diagonal elements whose inverses are tridiagonal. The arguments given are elementaryand show that matrices wit...

1. In their paper “Permutation Problems and Special Functions,” Askey and Ismail [ 1 ] give the following striking identity. Consider three boxes containing j, k, m distinguishable balls, and consider all possible rearrangements of these balls such that each box still has the same number of balls; i.e., j end up in the first, k in the second, m in...

In this paper we use the properties of the covariance matrix of a Gaussian Markovian family to give a probabilistic proof of a theorem about inverses of tridiagonal matrices.

In this paper we use the properties of the covariance matrix of a Gaussian Markovian family to give a probabilistic proof of a theorem about inverses of tridiagonal matrices.

A one-parameter family of partition functions is considered which for zero value of the parameter reduces to the spherical model of a ferromagnet. The model for > 0 is closer to the usual discrete lattice spin model of a ferromagnet than is the spherical model. The first four terms in of the limiting value of the partition function are calculated a...

A one parameter family of models is considered which for zero value of the parameter alpha reduces to the familiar spherical model. The models for alpha > 0 have a symmetry close to that of an Ising model. A heuristic discussion of phase transitions in these models leads to the conjecture that, if the spherical model (alpha = 0) exhibits a phase tr...

July 1996, volume 122, number 584 (fourth of 5 numbers) Incluye bibliografía