Neal Brand

Almost all Steinhaus graphs have diameter 2

A Steinhaus graph is a graph with n vertices whose adjacency matrix (ai, j) satisfies the condition that ai, j aa--1, j--1 + ai--1, j (mod 2) for each 1 < i < j ≤ n. It is clear that a Steinhaus graph is determined by its first row. In [3] Bringham and Dutton conjecture that almost all Steinhaus graphs have diameter 2. That is, as n approaches infinity, the ratio of the number of Steinhaus graphs with n vertices having diameter 2 to the total number of Steinhaus graphs approaches 1. Here we prove Bringham and Duttons conjecture.

On the Bays-Lambossy theorem

The Bays-Lambossy Theorem states that if p is a prime then any pair of cyclic isomorphic t −( p, k, λ) designs are isomorphic by a multiplier map. For each prime, p ≡ 1(6), and n ⩾ 2 this paper gives examples of cyclic 2 − ( p π , 3, 1) designs (or Steiner triple systems) which are isomorphic but not isomorphic by any multiplier. These examples show that any generalization of the Bays-Lambossy theorem for the stated parameters would have to involve more than just multiplier maps.

Polynomial isomorphisms of combinatorial objects

LetB andB′ be any isomorphic cyclic combinatorial objects such as graphs, digraphs, designs, or simplicial complexes. It is shown that if there are certain polynomials of degreen in the automorphism group ofB and some other technical conditions are satisfied thenB andB′ are isomorphic by a polynomial of degreen + 1. The technical conditions involve the order of the automorphism group ofB.

Isomorphisms of cyclic combinatorial objects

Abstract Let υ = qp r where p is a prime number and p does not divide q . Let B and B ′ be isomorphic combinatorial objects whose vertex sets is Z υ , the integers modulo υ, and further assume that translations in Z υ are automorphisms of the objects. Conditions are given which imply that B and B ′ must be isomorphic by the composition of a polynomial in Z υ and a map which is a translation when restricted to each of the cosets of Z υ modulo q .

Isomorphic designs that are not multiplier equivalent

Abstract In this paper interesting families of designs are constructed and studied. First, a family of cyclic 2-(4n, 3, 2) designs is constructed with the property that each design has two different cyclic structures. This appears to be the first example of any cyclic τ-(ν, κ, γ) designs which are isomorphic but not isomorphic by a multiplier. A related family of 1-rotational 2-(2·4n + 1, 3, 2) designs is also constructed with each having two different 1-rotational structures.

Probability of diameter two for Steinhaus graphs

Abstract A Steinhaus graph is a graph with n vertices whose adjacency matrix ( a i,j ) satisfies the condition that a i,j ≡ a i − 1, j − 1 + a i − 1, j (mod 2) for each 1 i j ≤ n . It is clear that a Steinhaus graph is determined by its first row. In “Almost all Steinhaus graphs have diameter two”, J. Graph Theory 16 (1992) 213–219 it is shown that almost all Steinhaus graphs have diameter two. Here we generalize to the case where the j th entry of the first row has probability p j of being 1. Under reasonable conditions it is shown that the probability measure of the set of Steinhaus graphs with diameter two approaches 1 as the number of vertices in the graph approaches infinity.

Constructions and topological invariants of 2-(n,3,l) designs with group actions

Connections between 2-(ν3, λ) designs and topology are exploited to produce topological invariants of these designs. When these designs have a natural group action with 0 or 1 fixed point, these invariants are easily computed. Methods of constructing 2-(ν, 3, λ) designs with such a group action are given. Examples of this construction are given and the invariants are used to distinguish the designs.

Invariants and constructions of mendelsohn designs

In this paper we examine Mendelsohn designs and some connections to topology which lead to an easily described algorithm for computing invariants of these designs. The results are applied to designs which have natural group actions. We also use the topology to describe when ordinary two-fold triple systems with a group action lead to Mendelsohn designs with the same group action. Procedures for constructing Mendelsohn designs are also given. In particular, we give necessary and sufficient conditions for constructing 2-(v, 4, 1) Mendelsohn designs.

Construction of universal branched coverings

Abstract A construction for the classifying spaces for branched coverings with branch set a codimension 2 submanifold is given by Brand (1978, 1980). Using this result as a first step we inductively construct universal branched coverings with branch set a stratified set. We also give some of the lower homotopy groups of the classifying spaces which correspond to branched coverings of spheres.

One-factors and the existence of affine designs

Abstract Kohler (1982) defines a graph G p for p a prime which he relates to affine designs. He shows that if the graph has a 1-factor then there exists an affine 3-( p ,4,λ) design with λ satisfying the usual necessary conditions. Here we generalize Kohlers results to include finite fields.