In this substantial revision of a much-quoted monograph first published in 1974, Dr. Biggs aims to express properties of graphs in algebraic terms, then to deduce theorems about them. In the first section, he tackles the applications of linear algebra and matrix theory to the study of graphs; algebraic constructions such as adjacency matrix and the incidence matrix and their applications are discussed in depth. There follows an extensive account of the theory of chromatic polynomials, a subject that has strong links with the "interaction models" studied in theoretical physics, and the theory of knots. The last part deals with symmetry and regularity properties. Here there are important connections with other branches of algebraic combinatorics and group theory. The structure of the volume is unchanged, but the text has been clarified and the notation brought into line with current practice. A large number of "Additional Results" are included at the end of each chapter, thereby covering most of the major advances in the past twenty years. This new and enlarged edition will be essential reading for a wide range of mathematicians, computer scientists and theoretical physicists.
Author(s): Norman Biggs
Series: Cambridge Tracts in Mathematics
Edition: 2
Publisher: Cambridge University Press
Year: 1974
Language: English
Pages: 178
Series: Cambridge Tracts in Mathematics (Volume 67)......Page 1
Title: Algebraic Graph Theory......Page 3
Copyright......Page 4
Contents......Page 5
Preface......Page 7
1. Introduction......Page 9
PART ONE. Linear algebra in graph theory......Page 15
2. The spectrum of a graph......Page 17
3. Regular graphs & line graphs......Page 22
4. The homology of graphs......Page 30
5. Spanning trees & associated structures......Page 37
6. Complexity......Page 42
7. Determinant expansions......Page 48
PART TWO. Colouring problems......Page 55
8. Vertex-colourings & the spectrum......Page 57
9. The chromatic polynomial......Page 65
10. Edge-subgraph expansions......Page 72
11. The logarithmic transformation......Page 80
12. The vertex-subgraph expansion......Page 86
13. The Tutte polynomial......Page 94
14. The chromatic polynomial & spanning trees......Page 102
PART THREE. Symmetry & regularity of graphs......Page 107
15. General properties of graph automorphisms......Page 109
16. Vertex-transitive graphs......Page 114
17. Symmetric graphs......Page 120
18. Trivalent symmetric graphs......Page 127
19. The covering-graph construction......Page 135
20. Distance-transitive graphs......Page 140
21. The feasibility of intersection arrays......Page 148
22. Primitivity & imprimitivity......Page 155
23. Minimal regular graphs with given girth......Page 162
Bibliography......Page 173
Index......Page 177
