Model 1. Consider the complete graph Kn, with vertex set [n] = {1, 2,…, n}, in which each edge e is assigned a length Xe. Colour the k shortest edges incident with each vertex green and the remaining edges blue. The graph made up of the green edges only, will be referred to as the k-th nearest neighbour graph. This graph has been studied in a variety of contexts both computational and statistical.

A new graph theoretic proof of the convergence of Markov chains with variable transition probabilities and a new algorithm for computing the limiting distributions are presented.

For every integer N, we explicitly construct a subset of residues mod N of size(log N)o(1) which is nearly uniformly distributed in every arithmetic progression modulo N.

We consider transition operators P on a countable set, which are reversible, irreducible and invariant under a group G of permutations of X with compact point stabilizers. We relate the computation of the spectral radius (norm) of P with the spectral radii of certain matrices defined over the factor set G\X. In various cases, this allows easy computation of the norm of P.

Suppose each vertex of a bipartite multigraph (with partition (X, Y)) is assigned a set of colours; we say this colour scheme is feasible if the edges of the graph can be properly coloured so that each receives a colour in both of its endpoints' sets. We prove various results showing that certain types of colour scheme are always feasible. For instance, we prove that if the colour scheme obtained by assigning the set {1,…, d(x)} of colours to each vertex x of X and the set T = {1,…, t} (t < Δ(X)) to each vertex of Y is feasible, then so is every colour scheme where each vertex x of X gets d(x) colours from T and each vertex of Y gets the set T.

We present a method for obtaining upper bounds for the connective constant of self-avoiding walks. The method works for a large class of lattices, including all that have been studied in connection with self-avoiding walks. The bound is obtained as the largest eigenvalue of a certain matrix. Numerical application of the method has given improved bounds for all lattices studied, e.g. μ < 2.696 for the square lattice, μ < 4.278 for the triangular lattice and μ < 4.756 for the simple cubic lattice.

Let f(t) be the largest integer such that every graph with average degree t has a topological clique with f(i) vertices. It is widely believed that . Here we prove the weaker estimate .

We describe two computational methods for the construction of cubic graphs with given girth. These were used to produce two independent proofs that the (3,9)-cages, defined as the smallest cubic graphs of girth 9, have 58 vertices. There are exactly 18 such graphs. We also show that cubic graphs of girth 11 must have at least 106 vertices and cubic graphs of girth 13 must have at least 196 vertices.

Paul Erdős has conjectured that Menger's theorem extends to infinite graphs in the following way: whenever A, B are two sets of vertices in an infinite graph, there exist a set of disjoint A−B paths and an A−B separator in this graph such that the separator consists of a choice of precisely one vertex from each of the paths. We prove this conjecture for graphs that contain a set of disjoint paths to B from all but countably many vertices of A. In particular, the conjecture is true when A is countable.

We show that, for any finite set P of points in the plane and for any integer k ≥ 2, there is a finite set R = R(P, k) with the following property: for any k-colouring of R there is a monochromatic set , ⊆ R, such that is combinatorially equivalent to the set P, and the convex hull of P contains no point of R \ . We also consider related questions for colourings of p-element subsets of R (p > 1), and show that these analogues have negative solutions.

In this paper we prove that given a finite collection of finite graphs, and the subsets of vertices of a random graph G that induce those graphs, it is almost always possible to uniquely reconstruct a class of graphs equivalent to G.

Analogues of the Erdős-Ko-Rado theorem are proved for the Boolean algebra of all subsets of {1,…n} and in this algebra truncated by the removal of the empty set and the whole set.