Profile minimization problem for matrices and graphs

This project is supported by the National Natural Science Foundation of China.     

A combinatorial ranking problem

An algorithm is presented for finding annth-best spanning tree of an edge-weighted graphG. In sharp contrast to related ranking algorithms, the number of steps is a linear function of the parametern. The results apply as well to ranking the bases of an abstract matroid.     

A combinatorial property and Cayley graphs of semigroups

We describe semigroups satisfying a combinatorial property defined in terms of Cayley graphs.     

A combinatorial property for semigroups of matrices

In this paper we study the semigroups of matrices over a commutative semiring. We prove that a semigroup of matrices over a tropical semiring satisfies a combinatorial property called weak permutation property. We consider an application of this result to the Burnside problem for groups.     

Type-II matrices and combinatorial structures

Type-II matrices are a class of matrices used by Jones in his work on spin models. In this paper we show that type-II matrices arise naturally in connection with some interesting combinatorial and geometric structures.     

A combinatorial identity involving relatively prime integers

In this note we establish an identity concerning the number of ways an element in the semigroup generated by two relatively prime integers can be written in terms of the other elements of the semigroup.     

The inverse eigenvalue problem for Hermitian matrices whose graphs are cycles

In 1979, Ferguson characterized the periodic Jacobi matrices with given eigenvalues and showed how to use the Lanzcos Algorithm to construct each such matrix. This article provides general characterizations and constructions for the complex analogue of periodic Jacobi matrices. As a consequence of the main procedure, we prove that the multiplicity of an eigenvalue of a periodic Jacobi matrix is at most 2.     

Triangular matrices and combinatorial inversion formulas

The paper is devoted to the construction of the matrix inverse of an infinite triangular matrix and to finding the connection coefficients between polynomial sequences and general combinatorial inversion formulas.     

Invariant factors of combinatorial matrices

The Smith normal forms of an Hadamard matrix of order 4m (m square-free), and of the incidence matrix of a (ν, k, λ) configuration (n=k−λ square-free (n, λ)=1), are determined.     

A combinatorial problem associated with nonograms

Associated with an m × n matrix with entries 0 or 1 are the m-vector of row sums and n-vector of column sums. In this article we study the set of all pairs of these row and column sums for fixed m and n. In particular, we give an algorithm for finding all such pairs for a given m and n.     

A problem concerning infinite graphs

We show that there are graphs G and H which satisfy: (I) for every integer n, H contains n disjoint graphs each isomorphic to G, and (II) H does not contain infinitely many disjoint graphs each isomorphic to G. This answers one of the questions raised by Halin in the Graph Theory Newsletter.     

A maximum problem for matrices

The following maximum problem is considered: To find among all contractions T on an n-dimensional Hilbert space whose spectral radius does not exceed a given number p< 1, the operator T for which |Tⁿ| is maximum. A matrix T of Toeplitz type is constructed for which this maximum is attained.     

Incidence matrices and inequalities for combinatorial designs

In this paper we use incidence matrices of block designs and row–column designs to obtain combinatorial inequalities. We introduce the concept of nearly orthogonal Latin squares by modifying the usual definition of orthogonal Latin squares. This concept opens up interesting combinatorial problems and is expected to be useful in planning experiments by statisticians. © 2002 John Wiley & Sons, Inc. J Combin Designs 10: 17–26, 2002