Properties of (0, 1)-matrices without certain configurations

We generalize results of Ryser on (0, 1)-matrices without triangles, 3 × 3 submatrices with row and column sums 2. The extremal case of matrices without triangles was previously studied by the author. Let the row intersection of row i and row j (i ≠ j) of some matrix, when regarded as a vector, have a 1 in a given column if both row i and row j do not 0 otherwise. For matrices satisfying some conditions on forbidden configurations and column sums ? 2, we find that the number of linearly independent row intersections is equal to the number of distinct columns. The extremal matrices with m rows and $\text{(}\text{m}\text{2}\text{)}$ distinct columns have a unique SDR of pairs of rows with 1's. A triangle bordered with a column of 0's and its (0, 1)-complement are also considered as forbidden configurations. Similar results are obtained and the extremal matrices are closely related to the extremal matrices without triangles.     

(0, 1)-matrices with minimal permanents

It is shown that the permanent of a totally indecomposable (0,1)-matrix is equal to its largest row sum if and only if all its other row sums are 2. This research was supported by the U.S. Air Force Office of Scientific Research under Grant AFOSR-72-2164.     

A characterization of convertible (0, 1)-matrices

We characterize those (0, 1)-matrices M whose elements can be given plus or minus signs so as to yield a matrix M′ for which det M′ = perm M.     

Monotone subsequences in (0, 1)-matrices

A (0, 1)-matrix contains anS ₀(k) if it has 0-cells (i, j ₁), (i + 1,j ₂),..., (i + k – 1,j _k) for somei andj ₁ < ... < j_k, and it contains anS ₁(k) if it has 1-cells (i ₁,j), (i ₂,j + 1),...,(i _k ,j + k – 1) for somej andi ₁ < ... <i _k . We prove that ifM is anm × n rectangular (0, 1)-matrix with 1 m n whose largestk for anS ₀(k) isk ₀ m, thenM must have anS ₁(k) withk m/(k ₀ + 1). Similarly, ifM is anm × m lower-triangular matrix whose largestk for anS ₀(k) (in the cells on or below the main diagonal) isk ₀ m, thenM has anS ₁(k) withk m/(k ₀ + 1). Moreover, these results are best-possible.     

Representing (0, 1)-matrices by boolean circuits

A boolean circuit represents an n by n(0,1)-matrix A if it correctly computes the linear transformation over GF(2) on all n unit vectors. If we only allow linear boolean functions as gates, then some matrices cannot be represented using fewer than Ω(n²/lnn) wires. We first show that using non-linear gates one can save a lot of wires: any matrix can be represented by a depth-2 circuit with O(nlnn) wires using multilinear polynomials over GF(2) of relatively small degree as gates. We then show that this cannot be substantially improved: If any two columns of an n by n(0,1)-matrix differ in at least d rows, then the matrix requires Ω(dlnn/lnlnn) wires in any depth-2 circuit, even if arbitrary boolean functions are allowed as gates.     

On (0, 1)-matrices with prescribed row and column sum vectors

Given partitions R and S with the same weight, the Robinson-Schensted-Knuth correspondence establishes a bijection between the class A(R,S) of (0, 1)-matrices with row sum R and column sum S and pairs of Young tableaux of conjugate shapes λ and λ^∗, with S?λ?R^∗. An algorithm for constructing a matrix in A(R,S) whose insertion tableau has a prescribed shape λ, with S?λ?R^∗, is provided. We generalize some recent constructions due to R. Brualdi for the extremal cases λ=S and λ=R^∗.     

A minimal completion of (0, 1)-matrices without total support

In this paper, we provide a method to complete a (0, 1)-matrix without total support via the minimal doubly stochastic completion of doubly substochastic matrices and show that the size of the completion is determined by the maximum diagonal sum or the term rank of the given (0, 1)-matrix.     

关于逆N0-矩阵的若干性质

1989年Meyer为计算马尔可夫链的平稳分布向量构造了一个算法,首次提出非负不可约矩阵的Perron补的概念.在非负不可约矩阵的广义Perron补若干性质的基础上,给出逆N0-矩阵的几个性质.     

An upper bound for theα-height of (0, 1)-matrices

We obtain an upper bound for theα-height of an arbitrary matrix of zeros and ones. We apply the result to a number of known combinatorial problems.     

On permanents of (1, − 1)-matrices

A preliminary study on permanents of (1, ? 1)-matrices is given. Some inequalities are derived and a few unsolved problems, believed to be new, are mentioned.     

Upper bounds for permanents of (1, − 1)-matrices

Let Ω _n denote the set of alln×n (1, ? 1)-matrices. In 1974 E. T. H. Wang posed the following problems: Is there a decent upper bound for |perA| whenAσΩ _n is nonsingular? We recently conjectured that the best possible bound is the permanent of the matrix with exactlyn?1 negative entries in the main diagonal, and affirmed that conjecture by the study of a large class of matrices in Ω _n . Here we prove that this conjecture also holds for another large class of (1, ?1)-matrices which are all nonsingular. We also give an upper bound for the permanents of a class of matrices in Ω _n which are not all regular.     

Balanced 0, ±1-matrices,bicoloring and total dual integrality

A 0, ±1-matrixA is balanced if, in every submatrix with two nonzero entries per row and column, the sum of the entries is a multiple of four. This definition was introduced by Truemper (1978) and generalizes the notion of a balanced 0, 1-matrix introduced by Berge (1970). In this paper, we extend a bicoloring theorem of Berge (1970) and total dual integrality results of Fulkerson, Hoffman and Oppenheim (1974) to balanced 0, ±1-matrices.This work was supported in part by NSF grants DDM-8800281, ECS-8901495 and DDM-9001705.     

Some properties of the permanent of (1, -1)-matrices

Some convergence properties for the Campbell-Baker-Hausdorff series for the logarithm of a product of exponentials are established.     

A note on the proportion of singular (0, 1)-matrices in a class associated with fractional 2n factorial designs

Let the set T = {(x₁, x₂,…, x_n): x_i = 0, 1}. Since the elements of T can be seen as binary representations of integers, we order them with their corresponding integer values. Let $\text{Γ}{}^1$ be the set of (n + 1) × (n + 1) matrices of the form [1 … D], where the n + 1 rows of D are distinct ordered elements of T. We show that the proportion of singular matrices in $\text{Γ}{}^1$ approaches 0 as n → ∞.     

On some questions concerning permanents of (1,−1)-matrices

Let Ω _n denote the set of alln×n-(1,?1)-matrices. E.T.H. Wang has posed the following problem: For eachn≧4, can one always find nonsingularA∈Ω _n such that |perA|=|detA| (*)? We present a solution forn≦6 and, more generally, we show that (*) does not hold ifn=2 ^k ?1,k≧2, even for singularA∈Ω _n . Moreover, we prove that perA≠0 ifA∈Ω _n ,n=2 ^k ?1, and we derive new results concerning the divisibility of the permanent in Ω _n by powers of 2.     

The class A(R,S) of (0, 1)-matrices

Let R and S be two vectors whose components are m and n non-negative integers, respectively. Let $\text{A}$(R, S) be the class consisting of all (0, 1)-matrices of size m by n with row sum vector R and column sum vector S. In this paper we derive a lower bound to the cardinality of the class $\text{A}$(R, S), which can be computed readily.     

Hamiltonian cycles in planar triangulations with no separating triangles

A classical theorem of Hassler Whitney asserts that any maximal planar graph with no separating triangles is Hamiltonian. In this paper, we examine the problem of generalizing Whitney's theorem by relaxing the requirement that the triangulation be a maximal planar graph (i.e., that its outer boundary be a triangle) while maintaining the hypothesis that the triangulation have no separating triangles. It is shown that the conclusion of Whitney's theorem still holds if the chords satisfy a certain sparse-ness condition and that a Hamiltonian cycle through a graph satisfying this condition can be found in linear time. Upper bounds on the shortness coefficient of triangulations without separating triangles are established. Several examples are given to show that the theorems presented here cannot be extended without strong additional hypotheses. In particular, a 1-tough, non-Hamiltonian triangulation with no separating triangles is presented.     

Even signings,signed switching classes,and (−1, 1)-matrices

In this paper, algebraic and combinatorial techniques are used to establish results concerning even signings of graphs, switching classes of signed graphs, and (?1, 1)-matrices. These results primarily deal with enumeration of isomorphism types, and determining whether there are fixed elements under the action of automorphisms. A formula is given for the number of isomorphism types of even signings of any fixed simple graph. This is shown to be equal to the number of isomorphism types of switching classes of signings of the graph. A necessary and sufficient criterion is found for all switching classes fixed by a given graph automorphism to contain signings fixed by that automorphism. It is determined whether this criterion is met for all automorphisms of various graphs, including complete graphs, which yields a known result of Mallows and Sloane. As an application, a formula is developed for the number of H-equivalence classes of (?1, 1)-matrices of fixed size. Independently, using Molien's theorem and following a suggestion of Cameron's, generating series for these numbers are given. As a final application, a necessary and sufficient condition that a square (?1, 1)-matrix be switching equivalent to a symmetric matrix is given.