Algebraic Graph Theory

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Cambridge University Press, 1993 - Mathematics - 205 pages
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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.
 

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Contents

Introduction
1
Linear algebra in graph theory
5
The spectrum of a graph
7
Regular graphs and line graphs
14
Cycles and cuts
23
Spanning trees and associated structures
31
The treenumber
38
Determinant expansions
44
Chromatic polynomials and spanning trees
106
Symmetry and regularity
113
Automorphisms of graphs
115
Vertextransitive graphs
122
Symmetric graphs
130
Symmetric graphs of degree three
138
The covering graph construction
149
Distancetransitive graphs
155

Vertexpartitions and the spectrum
52
Colouring problems
61
The chromatic polynomial
63
Subgraph expansions
73
The multiplicative expansion
81
The induced subgraph expansion
89
The Tutte polynomial
97
Feasibility of intersection arrays
164
Imprimitivity
173
Minimal regular graphs with given girth
180
REFERENCES
191
Index
202
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Page 198 - N. ROBERTSON, The smallest graph of girth 5 and valency 4, Bull. Amer. Math. Soc. 70 (1964), 824-825. 6. M. SPILL, "Ober minimale Graphen gegebener Taillenweite und Valenz,
Page 193 - PJ Cameron, CE Praeger, J. Saxl and GM Seitz, On the Sims' conjecture and distance-transitive graphs, Bull. London Math. Soc. 15 (1983), 499-506. A.

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