Combinatorial Matrix Classes, Volume 13
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 developed including the classes of matrices of 0's and 1's with a specified number of 1's in each row and column (equivalently, bipartite graphs with a specified degree sequence), symmetric matrices in such classes (equivalently, graphs with a specified degree sequence), tournament matrices with a specified number of 1's in each row (equivalently, tournaments with a specified score sequence), nonnegative matrices with specified row and column sums, and doubly stochastic matrices. Most of this material is presented for the first time in book format and the chapter on doubly stochastic matrices provides the most complete development of the topic to date.
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Preface page ix
Basic Existence Theorems for Matrices with
3TheClassARS of 01Matrices
More on the Class ARS of 01Matrices
5TheClassT R of Tournament Matrices
Classes of Symmetric Integral Matrices
Convex Polytopes of Matrices
Doubly Stochastic Matrices
2-neighborly adjacency matrix algorithm assume bipartite graph Bruhat order class A(R column sum vector combinatorial components construction contradiction convex combination Corollary corresponding cycle of length deﬁned denote diﬀerent digraph directed cycle Discrete Math doubly stochastic matrix edges equivalent exists a matrix extreme points ﬁrst following theorem fully indecomposable Hence holds implies induction inequalities integer interchange graph invariant positions irreducible Lemma Linear Algebra Appl main diagonal matrix in A(R matrix of order maximal minimal number nondecreasing nonnegative integral matrix nonnegative integral vectors nonnegative matrix nonzero elements number of 1’s partition permutation matrices positive integer Proof R.A. Brualdi replacing row and column row sum vector satisﬁes satisfying score vector sequence structure matrix submatrix Suppose symmetric matrix T(Rn term rank Theorem total support tournament matrix transitive tournament transportation polytope vertex set vertices Young tableau zero matrix