Introduction

For many mathematicians the most noble activity lies in proving theorems. It must have come as a blow for them when Gödel [7] showed that there are unprovable theorems. At the beginning they still could find some consolation in hoping that such culprits might only occur in Peano arithmetics through esoteric diagonalization arguments. Nowadays there is a wealth of the most natural valid theorems that can be stated in the language of finite combinatorics but are not provable within that system.

Mathematicians understand to a certain extent how to find unprovable theorems and how to prove their unprovability within a formal system. In that sense we are relying on the classical work by Gentzen [5], Kreisel [15] and Wainer [31]. Moreover, we shall apply their beautiful ideas to something that seems to be well understood, viz to well-quasi-orderings. This is an old concept found in Gordan [6], and Kruskal [16] correctly pointed out that it was ‘a frequently discovered concept’. That is why we are not reinventing it and are well aware that any sequence (si) of specialists starting with the author must contain an arbitrary long subsequence of experts knowing more than s0, a fact, which gives a nice theme for this paper.

Introduction

How well can global properties of a graph be inferred from observations that are purely local? This general question gives rise to numerous interesting problems that we want to discuss here. Such a local-global approach is often taken in geometry, where it has a long and successful history, but a systematic study of graphs from this perspective has not begun until recently. Nevertheless, a number of older results in graph theory do fit very nicely into this framework, as we later point out. Most of the specific problems fall in two categories. In the first, local structural information on the graph is collected and then used to derive certain consequences for the graph as a whole. The other class of problems concerns consistency of local data. Namely, one asks to characterize those sets of local data that may come from some graphs.

As the reader will soon see, the local-global paradigm leads to many questions in which graphs are viewed as geometric objects, a point of view that we believe can greatly benefit graph theory. Besides the geometric connection, ties also exist with the theory of combinatorial algorithms.

Introduction

We shall be concerned with a graph G regarded as an electrical network in which each edge has resistance 1. A well-known result due to R.M. Foster [6] (see also [3, p.41] and [9]) asserts that if G has n vertices and m edges, the effective resistance between adjacent vertices is r1 = (n − 1)/m, provided that all edges are equivalent under the action of the automorphism group. In this paper I shall obtain formulae for ri, the effective resistance between vertices at distance i, for i ≥ 2, provided G is distance-transitive (DT). With hindsight, it will be clear that the same formulae hold if we assume only that G is distance-regular (DR). The case i = 2 was also studied by Foster [7].

Another well-known fact is that the electrical problem can be regarded as a case of solving Laplace's equation on the graph.

Introduction: on graph drawing

An old problem of geometry consists of representing a simple plane graph G by means of a collection of disks in one-to-one correspondence with the vertices of G. These disks may only intersect pairwise in at most one point, the corresponding contacts representing the edges of G. The case of disks with no prescribed shape is solved by merely drawing for each vertex v a closed curve around v and cutting the edges half way. The difficulty arises when the disks have to be of a specified shape. The famous case of circular disks, solved by the Andreev–Thurston circle packing theorem [1], involves questions of numerical analysis: the coordinates of the centers and radii are not rational, and are computed by means of convergent series. This problem is still up to date, and considered in many research works. In the present paper we will consider triangular disks.

During the last two decades a vast number of articles have been published in which probabilistic methods have been used successfully to solve combinatorial problems, and especially to obtain various asymptotic results concerning some or other characteristic of combinatorial objects. This book is aimed at readers interested in problems of this kind both from the theoretical point of view and from the point of view of possible applications.

The book may be used by students and postgraduates in combinatorics and other fields where asymptotic methods of probability theory are applied. In particular, the material contained in this book could be taught in courses on combinatorial structures such as graphs, trees and mappings with an emphasis on the asymptotic properties of their characterictics. We believe that the asymptotic results presented here provide specialists in probability theory with new examples of applications of general limit theorems.

This text assumes a standard graduate-level knowledge of probability theory and an aquaintance with typical facts drawn from a general introduction to functions of complex variables. For the reader's convenience, the relevant results of probability theory are briefly reviewed in Chapter 1. The preliminaries from combinatorial analysis are given in the introductions to each of the subsequent chapters. Readers who are interested in obtaining more detailed knowledge of the corresponding aspects of combinatorics are advised to study my book ‘Combinatorial Methods in Discrete Mathematics’ or any other basic course devoted to this subject.

The English translation differs slightly from the Russian edition of the book in the following. I have rewritten the Introduction, Subsections 2.2, 2.3 and 6.2.2. A number of minor changes have been made throughout the text to eliminate misprints and awkward proofs. Also, the list of references has been extended by the inclusion of articles and monographs devoted to relevant problems of probabilistic combinatorics which have appeared subsequent to publication of the Russian version of the book.

I am greatly indebted to Professor B. Bollobás and Professor V. Vatutin for their help and valuable advice during the preparation of the manuscript.

Introduction

This chapter is devoted to random graphs. A number of ways of introducing randomness for various classes of graph exists, one of which is to specify on some classes of graph (trees, forests, graphs of one-to-one mappings and so on) certain, as a rule uniform, probability distributions. The second way of constructing random graphs is defined by a stochastic process which gives a rule for joining a number of initially isolated vertices by edges. The third way, which is closely related to the second, is described by a random procedure of deletion of edges from a complete graph. Other methods for constructing random graphs exist but they are of little use.

Before proceeding to describe results in the field, we list a number of statements concerning the combinatorial properties of graphs that will be required in the sequel. In this chapter we deal mainly with labeled graphs and for this reason the results cited below are related, as a rule, to such combinatorial structures.

Trees

Labeled trees are, in a sense, the simplest labeled graphs. A tree is a connected graph with no cycles. A rooted tree is a tree which has a distinguished vertex called the root.