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9084 results in Discrete Mathematics, Information Theory and Coding

Page 425 of 455

  • 11 - The multiplicative expansion

  • Norman Biggs, London School of Economics and Political Science
  • Book:
    Algebraic Graph Theory
    Published online:
    05 August 2012
    Print publication:
    16 May 1974, pp 81-88

  • Summary

    In this chapter and the next one we shall investigate expansions of the chromatic polynomial which involve relatively few subgraphs in comparison with the expansion of Chapter 10. The idea first appeared in the work of Whitney (1932b), and it was developed independently by Tutte (1967) and researchers in theoretical physics who described the method as a ‘linked-cluster expansion’ (Baker 1971). The simple version given here is based on a paper by the present author (Biggs 1973a). There are other approaches which use more algebraic machinery; see Biggs (1978) and lle.

    We begin with some definitions. Recall that if a connected graph Г is separable then it has a certain number of cut-vertices, and the removal of any cut-vertex disconnects the graph. A non-separable subgraph of Г which is non-empty and maximal (considered as a subset of the edges) is known as a block. Every edge is in just one block, and we may think of Г as a set of blocks ‘stuck together’ at the cut-vertices. In the case of a disconnected graph we define the blocks to be the blocks of the components. It is worth remarking that this means that isolated vertices are disregarded, since every block must have at least one edge.

    Let Y be a real-valued function defined for all graphs, and having the following two properties.

    P1: Y(Г) = 1 if Г has no edges;

    P2: Y(Г) is the product of the numbers Y(B) taken over all blocks B of Г.

  • Preface

  • Norman Biggs, London School of Economics and Political Science
  • Book:
    Algebraic Graph Theory
    Published online:
    05 August 2012
    Print publication:
    16 May 1974, pp vii-viii

  • Summary

    This book is a substantially enlarged version of the Cambridge Tract with the same title published in 1974. There are two major changes.

    • The main text has been thoroughly revised in order to clarify the exposition, and to bring the notation into line with current practice. In the course of revision it was a pleasant surprise to find that the original text remained a fairly good introduction to the subject, both in outline and in detail. For this reason I have resisted the temptation to reorganise the material in order to make the book rather more like a standard textbook.

    • Many Additional Results are now included at the end of each chapter. These replace the rather patchy selection in the old version, and they are intended to cover most of the major advances in the last twenty years. It is hoped that the combination of the revised text and the additional results will render the book of service to a wide range of readers.

    I am grateful to all those people who have helped by commenting upon the old version and the draft of the new one. Particular thanks are due to Peter Rowlinson, Tony Gardiner, Ian Anderson, Robin Wilson, and Graham Brightwell. On the practical side, I thank Alison Adcock, who prepared a TEX version of the old book, and David Tranah of Cambridge University Press, who has been constant in his support.

Page 425 of 455