Algebraic Graph TheoryAlgebraic graph theory is a fascinating subject concerned with the interplay between algebra and graph theory. Algebraic tools can be used to give surprising and elegant proofs of graph theoretic facts, and there are many interesting algebraic objects associated with graphs. The authors take an inclusive view of the subject, and present a wide range of topics. These range from standard classics, such as the characterization of line graphs by eigenvalues, to more unusual areas such as geometric embeddings of graphs and the study of graph homomorphisms. The authors' goal has been to present each topic in a self-contained fashion, presenting the main tools and ideas, with an emphasis on their use in understanding concrete examples. A substantial proportion of the book covers topics that have not appeared in book form before, and as such it provides an accessible introduction to the research literature and to important open questions in modern algebraic graph theory. This book is primarily aimed at graduate students and researchers in graph theory, combinatorics, or discrete mathematics in general. However, all the necessary graph theory is developed from scratch, so the only pre-requisite for reading it is a first course in linear algebra and a small amount of elementary group theory. It should be accessible to motivated upper-level undergraduates. Chris Godsil is a full professor in the Department of Combinatorics and Optimization at the University of Waterloo. His main research interests lie in the interactions between algebra and combinatorics, in particular the application of algebraic techniques to graphs, designs and codes. He has published more than 70 papers in these areas, is a founding editor of "The Journal of Algebraic Combinatorics" and is the author of the book "Algebraic Combinatorics". Gordon Royle teaches in the Department of Computer Science & Software Engineering at the University of Western Australia. His main research interests lie in the application of computers to combinatorial problems, in particular the cataloguing, enumeration and investigation of graphs, designs and finite geometries. He has published more than 30 papers in graph theory, design theory and finite geometry. |
Contents
Preface | 1 |
2 | 20 |
Groups | 29 |
Transitive Graphs | 35 |
25 | 56 |
ArcTransitive Graphs | 61 |
5 | 76 |
Homomorphisms | 129 |
6 | 227 |
TwoGraphs | 249 |
Line Graphs and Eigenvalues | 265 |
The Laplacian of a Graph | 279 |
| 306 | |
Cuts and Flows | 307 |
The Rank Polynomial | 341 |
Notes | 371 |
Kneser Graphs | 135 |
Matrix Theory | 163 |
Interlacing | 193 |
Strongly Regular Graphs | 217 |
Knots | 373 |
Knots and Eulerian Cycles | 395 |
Glossary of Symbols | 427 |
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Common terms and phrases
1-factors adjacency matrix algebraic And(k Aut(X automorphism group bipartite graph Cayley graph characteristic vector chromatic number clique coloop columns components connected graph contains core Corollary cubic graph define deleting denote determined disjoint distance edge eigenvalues eigenvector elements entries equal equiangular lines exactly Exercise Figure flow space follows fractional colouring girth graph theory hence heptads homomorphism implies incidence matrix independent set induced subgraph integer isomorphic Kneser graphs knot Laplacian lattice least Lemma line graph linear link diagram loop matroid Moore graph multiplicity n-colourable neighbours nonnegative nonzero number of vertices orbit orientation orthogonal pair parameters partition path permutation Petersen graph planar points Proof prove q-critical quadrangle rank function rank polynomial result s-arc Section set of lines spanning trees strongly regular graph subconstituent subset subspace Suppose symmetric symmetric matrix Theorem transitive triples two-graph unique valency vertex set vertex-transitive graph zero