Richard A. Brualdi

Matrices of zeros and ones with fixed row and column sum vectors

Abstract Let m and n be positive integers, and let R =( r 1 ,…, r m ) and S =( s 1 ,…, s n ) be nonnegative integral vectors. We survey the combinational properties of the set of all m × n matrices of 0s and 1s having r i 1s in row i and s i 1s in column j . A number of new results are proved. The results can be also be formulated in terms of a set of bipartite graps with bipartition into m and n vertices having degree sequence R and S , respectively. They can also be formulated in terms of the set of hypergraphs with m vertices having degree sequence R and n edges whose cardinalities are given by S .

Nested species subsets, gaps, and discrepancy

Abstract The nested-subset hypothesis of Patterson and Atmar states that species composition on islands with less species richness is a proper subset of those on islands with greater species richness. The sum of species absences, referred to as gaps, was suggested as a metric for nestedness, and null models have been used to test whether or not island species exhibited nestedness. Simberloff and Martin stated that finding examples of non-nested faunas was difficult. We revisit previous analyses of nested faunas and introduce a new metric we call “discrepancy” which we recommend as a measure for nestedness. We also recommend that the sample spaces conserve both row sums (number of species per site) and column sums (number of sites per species) derived from the incidence matrix. We compare our results to previous analyses.

The diagonal equivalence of a nonnegative matrix to a stochastic matrix

In [l] R. Sinkhorn proved the following theorem: Let A be a positive square matrix. Then there exist two diagonal matrices D, , D, whose diagonal elements are positive such that D,AD, is doubly stochastic. Moreover, these matrices are uniquely determkd up to scalar factors. In addition, Sinkhorn gave some examples which show that the theorem fails for some nonnegative matrices A. Marcus and Newman [2] and Maxfield and Mint [3] also studied this problem. Recently M. V. Menon [4] gave a simplified proof of Sinkhorn’s theorem based on the Brouwer fixed-point theorem. Perfect and Mirsky [5] have shown that given a fully indecomposable matrix B, there exists a doubly stochastic nonnegative matrix with the same zero pattern. The operator T defined by Menon in his proof of Sinkhorn’s theorem is a homogeneous positive nonlinear operator. Morishima [6] and Thompson [7] have studied such operators in extending the theorems of Perron and Frobenius. We define the operator T in the case when A is an irreducible matrix with a positive main diagonal. Using the Wielandt approach to the PerronFrobenius theory, we show that T has some but not all of the properties of of an irreducible nonnegative matrix. Thus T has a unique eigenvector in the interior of the positive cone, but there may also exist eigenvectors on the boundary. We then deduce Sinkhorn’s theorem when A is a nonnegative fully indecomposable matrix. It is an easy matter to establish that this condition is essentially necessary (see [5] or 6.2). After completing our work we learned that Sinkhorn and Knopp [8] have also obtained a proof of the D,AD, theorem under the full indecomposability assumption. Their method of proof is quite different from ours, and we feel

Codes with a poset metric

Abstract Niederreiter generalized the following classical problem of coding theory: given a finite field F q and integers n > k ⩾ 1, find the largest minimum distance achievable by a linear code over F q of length n and dimension k . In this paper we place this problem in the more general setting of a partially ordered set and define what we call poset-codes. In this context, Niederreiters setting may be viewed as the disjoint union of chains. We extend some of Niederreiters bounds and also obtain bounds for posets which are the product of two chains.

Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function

Abstract The permanent function is used to determine geometrical properties of the set Ω n of all n × n nonnegative doubly stochastic matrices. If F is a face of Ω n , then F corresponds to an n × n (0, 1)-matrix A, where the permanent of A is the number of vertices of F . If A is fully indecomposable, then the dimension of F equals σ(A) − 2n + 1, where σ(A) is the number of 1s in A. The only two-dimensional faces of Ω n are triangles and rectangles. For n ⩾ 6, Ω n has four types of three-dimensional faces. The facets of the faces of Ω n are characterized. Faces of Ω n which are simplices are determined. If F is a face of Ω n which is two-neighborly but not a simplex, then F has dimension 4 and six vertices. All k-dimensional faces with k + 2 vertices are determined. The maximum number of vertices of a k-dimensional face is 2k. All k-dimensional faces with at least 2k−1 + 1 vertices are determined.

On the spectral radius of (0,1)-matrices

We determine the maximum spectral radius for (0,1)-matrices with k2 andk2+1 1s, respectively, and for symmetric (0,1)-matrices with zero trace and e=k21s (graphs with e edges). In all cases, equality is characterized.

Matrices eigenvalues, and directed graphs

We show how several classical results concerning inclusion regions and estimates for the eigenvalues of matrices can be unified and generalized by the use of directed graphs. Applications to nonnegative matricesM-matrices, and the spectra of graphs are given.

Comments on bases in dependence structures

Dependence structures (in the finite case, matroids) arise when one tries to abstract the properties of linear dependence of vectors in a vector space. With the help of a theorem due to P. Hall and M. Hall, Jr concerning systems of distinct representatives of families of finite sets, it is proved that if B 1 and B 2 are bases of a dependence structure, then there is an injection σ: B 1 → B 2 such that ( B 2 / {σ( e )}) ∩ { e } is a basis for all e in B 1 . A corollary is the theorem of R. Rado that all bases have the same cardinal number. In particular, it applies to bases of a vector space. Also proved is the fact that if B 1 and B 2 are bases of a dependence structure then given e in B 1 there is an f in B 2 such that both ( B 1 / { e }) ∩ { f } and ( B 2 / { f }) ∩ { e } are bases. This is a symmetrical kind of replacement theorem.

Incidence and strong edge colorings of graphs

Abstract We define the incidence coloring number of a graph and bound it in terms of the maximum degree. The incidence coloring number turns out to be the strong chromatic index of an associated bipartite graph. We improve a bound for the strong chromatic index of bipartite graphs all of whose cycle lengths are divisible by 4.

Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley

Abstract We give a common, concise derivation of some important determinantal identities attributed to the mathematicians in the title. We also give a formal treatment of determinantal identities of the minors of a matrix.