Bryan L. Shader

Characteristic vertices of weighted trees via perron values

We consider a weighted tree T with algebraic connectivity μ, and characteristic vertex v. We show that μ and its associated eigenvectors can be described in terms of the Perron value and vector of a nonnegative matrix which can be computed from the branches of T at v. The machinery of Perron-Frobenius theory can then be used to characterize Type I and Type II trees in terms of these Perron values, and to show that if we construct a weighted tree by taking two weighted trees and identifying a vertex of one with a vertex of the other, then any characteristic vertex of the new tree lies on the path joining the characteristic vertices of the two old trees.

Distances in Weighted Trees and Group Inverse of Laplacian Matrices

In this paper we find formulas for group inverses of Laplacians of weighted trees. We then develop a relationship between entries of the group inverse and various distance functions on trees. In particular, we show that the maximal and minimal entries on the diagonal of the group inverse correspond to certain pendant vertices of the tree and to a centroid of the tree, respectively. We also give a characterization for the group inverses of the Laplacian of an unweighted tree to be an M-matrix.

On graphs with equal algebraic and vertex connectivity

Let G be an undirected unweighted graph on n vertices, let L be its Laplacian matrix, and let L#=(l#i,j) be the group inverse of L. It is known that for Z(L#):=(1/2)max1⩽i,j⩽n∑s=1n|li,s#−lj,s#|, the quantity 1/Z(L#) is a lower bound on the algebraic connectivity a(G) of G, while the vertex connectivity of G, v(G), is an upper bound on a(G). We characterize the graphs G for which v(G)=a(G) and subsequently prove that if n⩾v(G)2, then v(G)=a(G) holds if and only if 1/Z(L#)=a(G)=v(G). We close with an example showing that the equality 1/Z(L#)=a(G) does not necessarily imply that 1/Z(L#)=a(G)=v(G).

APPLICATIONS OF PAZ'S INEQUALITY TO PERTURBATION BOUNDS FOR MARKOV CHAINS

Abstract In several papers Meyer, singly and with coauthors, established the usefulness of the group generalized inverse in the study and computations of various aspects of Markov chains. Here we are interested in those results which concern bounds on the condition number of the chain and on the error in the computation of the stationary distribution vector. We show that a lemma due to Paz can be used to improve, sometimes by a factor of 2, some of the constants in the bounds obtained in the aforementioned papers.

ON THE MINIMUM RANK OF NOT NECESSARILY SYMMETRIC MATRICES: A PRELIMINARY STUDY ∗

The minimum rank of a directed graph Γ is defined to be the smallest possible rank over all real matrices whose ijth entry is nonzero whenever (i,j) is an arc in Γ and is zero otherwise. The symmetric minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for ij) is nonzero whenever {i,j} is an edge in G and is zero otherwise. Maximum nullity is equal to the difference between the order of the graph and minimum rank in either case. Definitions of various graph parameters used to bound symmetric maximum nullity, including path cover number and zero forcing number, are extended to digraphs, and additional parameters related to minimum rank are introduced. It is shown that for directed trees, maximum nullity, path cover number, and zero forcing number are equal, providing a method to compute minimum rank for directed trees. It is shown that the minimum rank problem for any given digraph or zero-nonzero pattern may be converted into a symmetric minimum rank problem.

Exponents of nonnegative matrix pairs

The notions of primitivity and exponent of a square nonnegative matrix A are classical: A is primitive provided there is a nonnegative integer k such that A k is entrywise positive and in the case A is primitive the exponent of A is the smallest such k. Fornasini and Valcher have extended the notion of primitivity to pairs (A, B) of square nonnegative matrices of the same order. The pair (A, B) is primitive provided there exist nonnegative integers h and k such that the sum of all products formed by words consisting of hA ’s and kB ’s is entrywise positive. This paper defines the exponent of a nonnegative matrix pair to be the smallest value of h + k over all such h and k. It is then shown that the largest exponent of a primitive pair of n by n nonnegative matrices lies in the interval [(n 3 − 5n 2 )/2 ,( 3n 3 + 2n 2 − 2n)/2]. In addition, the exponent of a pair of nonnegative matrices is related to properties of an associated twodimensional dynamical system.

Biclique decompositions and Hermitian rank

The Hermitian rank, h(A), of a Hermitian matrix A is defined and shown to equal max{n+(A),n−(A)}, the maximum of the numbers of positive and negative eigenvalues of A. Properties of Hermitian rank are developed and used to obtain results on the minimum number, b(G), of complete bipartite subgraphs needed to partition the edge set of a graph G. Witsenhausens inequality b(G)⩾max{n+(G),n−(G)} is reproved and conditions necessary for equality to hold are given. The results are then used to estimate b(G) for several classes of graphs. For example, if G is the complement of a path then b(G)=⌊23(n−1)⌋, while if G is the complement of a cycle then b(G)=2⌊13(n−1)⌋ or ⌊13(2n−1)⌋.

Tournament matrices with extremal spectral properties

Abstract For a tournament matrix M of order n, we define its walk space WM to be Span {M j 1 : j = 0 ,…, n − 1 } where 1 is the all ones vector. We show that the dimension of WM equals the number of eigenvalues of M whose real parts are greater than - 1 2 . We then focus on tournament matrices whose walk space has particularly simple structure, and characterize them in terms of their spectra. Specifically, we characterize those tournament matrices such that Mj1 is an eigenvector of M for some j ⩾ 0. We also characterize the tournament matrices M such that Jn − 2M is a skew-Hadamard matrix. Throughout, we illustrate our results with examples.

Exponents of tuples of nonnegative matrices

Abstract The notions of irreducibility, primitivity, and exponent of a nonegative matrix are generalized to k -tuples of non-negative matrices of the same order. It is shown that for each positive integer k , the maximum exponent of a primitive k -tuple of n by n nonnegative matrices is Θ ( n k +1 ).

Generating potentially nilpotent full sign patterns

A sign pattern is a matrix with entries in {+,�,0}. A full sign pattern has no zero entries. The refined inertia of a matrix pattern is defined and techniques are developed for constructing potentially nilpotent full sign patterns. Such patterns are spectrally arbitrary. These techniques can also be used to construct potentially nilpotent sign patterns that are not full, as well as potentially stable sign patterns.