Guo-Hui Zhang

Single change covering designs

A single-change covering designSC (v, k, b) on av-setV is a sequence ofb k-sets (blocks) onV which together cover every pair of elements ofV at least once, such that two successive blocks havek−1 elements in common. It is desirable to minimizeb. Some constructions and lower bounds forb are given.

Combinatorially orthogonal matrices and related graphs

Abstract Let G be a graph and let c ( x , y ) denote the number of vertices in G adjacent to both of the vertices x and y . We call G quadrangular if c ( x , y ) ≠ 1 whenever x and y are distinct vertices in G . Reid and Thomassen proved that | E ( G )| ⩾ 2| V ( G )| −4 for each connected quadrangular graph G , and characterized those graphs for which the lower bound is attained. Their result implies lower bounds on the number of 1s in m × n combinatorially orthogonal (0,1)-matrices, where a (0,1)-matrix A is said to be combinatorially orthogonal if the inner product of each pair of rows and each pair of columns of A is never one. Thus the result of Reid and Thomassen is related to a paper of Beasley, Brualdi and Shader in which they show that a fully indecomposable, combinatorially orthogonal (0,1)-matrix of order n ⩾ 2 has at least 4 n − 4 ls and characterize those matrices for which equality holds. One of the results obtained here is equivalent to the result of Beasley, Brualdi and Shader. We also prove that | E ( G )| ⩾ 2| V ( G )| − 1 for each connected quadrangular nonbipartite graph G with at least 5 vertices, and characterize the graphs for which the lower bound is attained. In addition, we obtain optimal lower bounds on the number of 1s in m × n combinatorially row-orthogonal (0,1)-matrices.

Parity Dimension for Graphs - A Linear Algebraic Approach

For a graph G with closed neighborhood matrix N , the parity dimension of G , denoted PD( G ), is the dimension of the null space of N over the field

Smallest regular graphs with prescribed odd girth

{\cal Z}_2

Some New Bounds on Single-Change Covering Designs

. Equivalently, the number of vertex sets S in G with the property that S dominates each vertex an even number of times is 2 k for some value of k , and PD( G ) = k . Using primarily linear algebraic techniques, we investigate the parity dimension of graphs.

On the partition and coloring of a graph by cliques

The odd girth of a graph G gives the length of a shortest odd cycle in G. Let f(k,g) denote the smallest n such that there exists a k-regular graph of order n and odd girth g. The exact values of f(k,g) are determined if one of the following holds: (i) k > 2g −5 and k is a prime number, (ii) k > (2⌊(g + 1)/4⌋ −1)2, and (iiii) k is a perfect square.

Column orthogonal (0,1,-1)-matrices

A single-change covering design SCD(v, k, b) on a c-set V is a sequence of b k-sets (blocks) on V that together cover every pair of elements of V at least once, such that two successive blocks have k-1 elements in common. Let f(v,k) denote the smallest b for which there exists an SCD(v, k, b). Some new upper and lower bounds for f(v,k) are given.

Inseparable Orthogonal Matrices over Z2

Abstract We first introduce the concept of the k -chromatic index of a graph, and then discuss some of its properties. A characterization of the clique partition number of the graph G ⋁ K c m for any simple graph G is given, together with some of its applications. Graphs with maximum valency 3 are also considered.

The maximum valency of regular graphs with given order and odd girth

The weight of a matrix A, ω (A), is the total number of nonzero entries in A; A is called column orthogonal if AT A is a diagonal matrix; and A is called inseparable if A contains no zero row nor zero column, and there do not exist permutation matrices P and Q such that . Let f(m,n) denote the smallest t for which there exists an m × n inseparable, column orthogonal (0,1, −l)-matrix A with ω(A)= t. In this paper we determine the formula for f(m, n), along with a characterization of those matrices that attain the lower bound. In addition, we point out a relationship between trees and certain matrices.

Extremal Regular Graphs with Prescribed Odd Girth

A (0, 1)-matrixAis called orthogonal over Z2if bothAATandATAare diagonal matrices. A matrixAis called inseparable ifAcontains no zero row or zero column and there do not exist permutation matricesPandQsuch that[formula]A matrixAis said to be of type 0 ifAAT=OandATA=O. A square matrixAof ordernis said to be of type 1 ifAAT=In. It turns out that an inseparable orthogonal matrix over Z2is either of type 0 or of type 1. Letf0(m,=n) (respectively,F0(m,n)) denote the smallest (respectively, largest) number of 1s in anm×ninseparable orthogonal matrix of type 0 over Z2, andf1(n) (respectively,F1(n)) denote the smallest (respectively, largest) number of 1s in ann×ninseparable orthogonal matrix of type 1 over Z2. The formulas forf0(m,n),F0(m,n),f1(n), andF1(n) are completely determined in this paper.