Robert Grone

Positive Definite Completions of Partial Hermitian Matrices

Abstract The question of which partial Hermitian matrices (some entries specified, some free) may be completed to positive definite matrices is addressed. It is shown that if the diagonal entries are specified and principal minors, composed of specified entries, are positive, then, if the undirected graph of the specified entries is chordal, a positive definite completion necessarily exists. Furthermore, if this graph is not chordal, then examples exist without positive definite completions. In case a positive definite completion exists, there is a unique matrix, in the class of all positive definite completions, whose determinant is maximal, and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial Hermitian matrix. Additional observations regarding positive definite completions are made.

The Laplacian spectrum of a graph

Let G be a graph. The Laplacian matrix

The Laplacian Spectrum of a Graph II

L(G) = D(G) - A(G)

Ordering trees by algebraic connectivity

is the difference of the diagonal matrix of vertex degrees and the 0-1 adjacency matrix. Various aspects of the spectrum of

A Matrix Analysis Approach to Higher-Order Approximations for Divergence and Gradients Satisfying a Global Conservation Law

L(G)

Extremal correlation matrices

are investigated. Particular attention is given to multiplicities of integer eigenvalues and to the effect on the spectrum of various modifications of G.

A bound for the complexity of a simple graph

Let G be a graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then L(G) = D(G) - A(G) is the Laplacian matrix of G. The first section of this paper is devoted to properties of Laplacian integral graphs, those for which the Laplacian spectrum consists entirely of integers. The second section relates the degree sequence and the Laplacian spectrum through majorization. The third section introduces the notion of a d-cluster, using it to bound the multiplicity of d in the spectrum of L(G).

Large eigenvalues of the laplacian

LetG be a graph onn vertices. Denote byL(G) the difference between the diagonal matrix of vertex degrees and the adjacency matrix. It is not hard to see thatL(G) is positive semidefinite symmetric and that its second smallest eigenvalue,a(G) > 0, if and only ifG is connected. This observation led M. Fiedler to calla(G) thealgebraic connectivity ofG. Given two trees,T1 andT2, the authors explore a graph theoretic interpretation for the difference betweena(T1) anda(T2).

Eigenvalues and the degree sequences of graphs

One-dimensional, second-order finite-difference approximations of the derivative are constructed which satisfy a global conservation law. Creating a second-order approximation away from the boundary is simple, but obtaining appropriate behavior near the boundary is difficult, even in one dimension on a uniform grid. In this article we exhibit techniques that allow the construction of discrete versions of the divergence and gradient operator that have high-order approximations at the boundary. We construct such discretizations in the one-dimensional situation which have fourth-order approximation both on the boundary and in the interior. The precision of the high-order mimetic schemes in this article is as high as possible at the boundary points (with respect to the bandwidth parameter). This guarantees an overall high order of accuracy. Furthermore, the method described for the calculation of the approximations uses matrix analysis to streamline the various mimetic conditions. This contributes to a marked clarity with respect to earlier approaches. This is a crucial preliminary step in creating higher-order approximations of the divergence and gradient for nonuniform grids in higher dimensions.

Characterizations of sign patterns of inverse-positive matrices

Abstract Let R n denote the convex, compact set of all real n -by- n positive semidefinite matrices with main-diagonal entries equal to 1. We examine the extreme points of R n focusing mainly on their rank. the principal result is that R n contains extreme points of rank k if and only if k ( k +1)⩽2 n .