Richard A. Brualdi

University of Wisconsin–Madison | UW · Department of Mathematics

Keywords Matrix Patterns and Various Graphs Eigenvalues and Digraphs Sign-Nonsingular Matrices Doubly Stochastic Matrices See also References

A natural sequel to the author's previous book Combinatorial Matrix Theory written with H. J. Ryser, this is the first book devoted exclusively to existence questions, constructive algorithms, enumeration questions, and other properties concerning classes of matrices of combinatorial significance. Several classes of matrices are thoroughly develope...

We prove a theorem about the elementary divisors ot matrices associated with symmetric ranked posets. We then apply this theorem to matrix compounds for which the poset is the Young's lattice L(r s).

We define the r-th combinatorial compound C r * (A) of a matrix A, which can be viewed as the characteristic function of the subset of the r×r submatrices of A which are combinatorially nonsingular. We prove that for 1≤r<n, A is fully indecomposable if and only if C r * (A) is. We determine the minimum number of 2×2 and 3×3 combinatorially nonsingu...

This book is a continuation of Theory of Matroids (also edited by Neil White), and again consists of a series of related surveys that have been contributed by authorities in the area. The volume begins with three chapters on coordinatisations, followed by one on matching theory. The next two deal with transversal and simplicial matroids. These are...

To my way of thinking, it's a marvelously simple proof. It's over 50 years old, and when one uses the modem language of graph theory, it can be made very visual. The most difficult part of the proof is in achieving triangular form. After that the details are extremely easy to follow: elementary row operations followed by the corresponding elementar...

Let E=[eij] be a matrix with integral elements, and let x be an indeterminate defined over the rational field Q. We investigate matrices of the form X=[xeij] (i = 1,…, m; j = 1,…, n; m ⩽ n). We may multiply the lines (rows or columns) of the matrix X by suitable integral powers of x in various ways and thereby transform X into a matrix Y=[xfij] suc...

We determine the maximum spectral radius for (0,1)-matrices with k2 andk2+1 1's, respectively, and for symmetric (0,1)-matrices with zero trace and 1's (graphs with e edges). In all cases, equality is characterized.

The purpose of this note is to tie together some results concerning doubly stochastic matrices and their representations as convex combinations of permutation matrices.

A simple proof is given of a simplification of Haber's formula for the minimum term rank of matrices of 0's and 1's with a specified row and column sum vector. © 1981, Academic Press Inc. (London) Limited. All rights reserved.

This chapter describes diagonal hypergraph of a matrix. It associates, G = G(X, Y)⊆ Kn, n be a bipartite graph with bipartition X, Y where X={xl,. . . , xn} and Y ={yl,. . . , yn} with a hypergraph Hf(G) whose vertices are the edges of G and whose edges are the 1-factors of G. It is assumed that every edge of G belongs to a 1-factor and for simplic...

Let A ( R , S ) \mathfrak {A}(R,S) denote the class of all m × n m \times n matrices of 0’s and 1’s with row sum vector R and column sum vector S . A set I × J ( I ⊆ { 1 , … , m } , J ⊆ { 1 , … , n } ) I \times J(I \subseteq \{ 1, \ldots ,m\} ,J \subseteq \{ 1, \ldots ,n\} ) is said to be invariant if each matrix in A ( R , S ) \mathfrak {A}(R,S) c...

An n × n nonnegative matrix is called nearly reducible provided it is irreducible and the replacement of any positive entry by zero yields a reducible matrix. The purpose of this article is to investigate the exponent γ(A) of an n × n primitive, nearly reducible matrix A (aperiodic, minimally strong directed graph). We prove that γ(A) ≥ 6 and that...

Sufficient conditions for a sequence of numbers to be the degree sequence of a graph are derived from the Erdos-Gallai theorem on degree sequences of graphs.

A simple proof is given for the maximum term rank of matrices of 0's and 1's with a specified row and column sum vector.

This chapter discusses the chromatic index of the graph of the assignment polytope. Let n be a positive integer, and let Sn denote the set of permutations of {1, . . . , n}. A graph G is defined as follows. The set of vertices of Gn is Sn. Two verticesσ, τ ∈ Sn are joined by an edge in Gn if and only if the permutation σ-1τ has exactly one nontrivi...

We define various classes of hypergraphs associated with m × n matrices of 0's and 1's and determine for which classes the maximum cardinality of a set of pairwise disjoint edges equals the minimum cardinality of a set of nodes that cover all edges independently of the matrix.

Given an m x m symmetric nonnegative matrix A and a positive vector R - (r1,&, rm), necessary and sufficient conditions are obtained in order that there exist a diagonal matrix D with positive main diagonal such that DAD has row sum vector R.

Unified proofs of two theorems on fundamental transversal matroids are presented. A necessary condition for a matroid to be a fundamental transversal matroid with respect to a given basis is given.

We investigate join-semilattices in which each element is assigned a nonnegative weight in a strictly increasing way. A join-subsemi­lattice of a Boolean lattice is weighted by cardinality, and we give a char­acterization of these in terms of the notion of a spread. The collection of flats with no coloops (isthmuses) of a matroid or pregeometry, pa...

We investigate join-semilattices in which each element is assigned a nonnegative weight in a strictly increasing way. A join-subsemilattice of a Boolean lattice is weighted by cardinality, and we give a characterization of these in terms of the notion of a spread. The collection of flats with no coloops (isthmuses) of a matroid or pregeometry, part...

There are several known results concerning how matroids can be induced from given matroids by a bipartite graph and the properties that are inherited in this way. The purpose of this note is to extend some of these results to the situation where the bipartite graph is replaced by an arbitrary directed graph. We show how a directed graph and a matro...

In (8) M. V. Menon investigates the diagonal equivalence of a non-negative matrix A to one with prescribed row and column sums and shows that this equivalence holds provided there exists at least one non-negative matrix with these row and column sums and with zeros in exactly the same positions A has zeros. However, he leaves open the question of w...