Book:Chris Godsil/Algebraic Graph Theory

Subject Matter

 * Graph Theory

Contents

 * Preface


 * 1 Graphs
 * 1.1 Graphs
 * 1.2 Subgraphs
 * 1.3 Automorphisms
 * 1.4 Homomorphisms
 * 1.5 Circulant Graphs
 * 1.6 Johnson Graphs
 * 1.7 Line Graphs
 * 1.8 Planar Graphs
 * Exercises
 * Notes
 * References


 * 2 Groups
 * 2.1 Permutation Groups
 * 2.2 Counting
 * 2.3 Asymmetric Graphs
 * 2.4 Orbits on Pairs
 * 2.5 Primitivity
 * 2.6 Primitivity and Connectivity
 * Exercises
 * Notes
 * References


 * 3 Transitive Graphs
 * 3.1 Vertex-Transitive Graphs
 * 3.2 Edge-Transitive Graphs
 * 3.3 Edge Connectivity
 * 3.4 Vertex Connectivity
 * 3.5 Matchings
 * 3.6 Hamilton Paths and Cycles
 * 3.7 Cayley Graphs
 * 3.8 Directed Cayley Graphs with No Hamilton Cycles
 * 3.9 Retracts
 * 3.10 Transpositions
 * Exercises
 * Notes
 * References


 * 4 Arc-Transitive Graphs
 * 4.1 Arc-Transitive Graphs
 * 4.2 Arc Graphs
 * 4.3 Cubic Arc-Transitive Graphs
 * 4.4 The Petersen Graph
 * 4.5 Distance-Transitive Graphs
 * 4.6 The Coxeter Graph
 * 4.7 Tutte's 8-Cage
 * Exercises
 * Notes
 * References


 * 5 Generalized Polygons and Moore Graphs
 * 5.1 Incidence Graphs
 * 5.2 Projective Planes
 * 5.3 A Family of Projective Planes
 * 5.4 Generalized Quadrangles
 * 5.5 A Family of Generalized Quadrangles
 * 5.6 Generalized Polygons
 * 5.7 Two Generalized Hexagons
 * 5.8 Moore Graphs
 * 5.9 The Hoffman-Singleton Graph
 * 5.10 Designs
 * Exercises
 * Notes
 * References


 * 6 Homomorphisms
 * 6.1 The Basics
 * 6.2 Cores
 * 6.3 Products
 * 6.4 The Map Graph
 * 6.5 Counting Homomorphisms
 * 6.6 Products and Colourings
 * 6.7 Uniquely Colourable Graphs
 * 6.8 Foldings and Covers
 * 6.9 Cores with No Triangles
 * 6.10 The Andrásfai Graphs
 * 6.11 Colouring Andrásfai Graphs
 * 6.12 A Characterization
 * 6.13 Cores of Vertex-Transitive Graphs
 * 6.14 Cores of Cubic Vertex-Transitive Graphs
 * Exercises
 * Notes
 * References


 * 7 Kneser Graphs
 * 7.1 Fractional Colourings and Cliques
 * 7.2 Fractional Cliques
 * 7.3 Fractional Chromatic Number
 * 7.4 Homomorphisms and Fractional Colourings
 * 7.5 Duality
 * 7.6 Imperfect Graphs
 * 7.7 Cyclic Interval Graphs
 * 7.8 Erdös-Ko-Rado
 * 7.9 Homomorphisms of Kneser Graphs
 * 7.10 Induced Homomorphisms
 * 7.11 The Chromatic Number of the Kneser Graph
 * 7.12 Gale's Theorem
 * 7.13 Welzl's Theorem
 * 7.14 Strong Products and Colourings
 * Exercises
 * Notes
 * References


 * 8 Matrix Theory
 * 8.1 The Adjacency Matrix
 * 8.2 The Incidence Matrix
 * 8.3 The Incidence Matrix of an Oriented Graph
 * 8.4 Symmetric Matrices
 * 8.5 Eigenvectors
 * 8.6 Positive Semidefinite Matrices
 * 8.7 Subharmonic Functions
 * 8.8 The Perron-Frobenius Theorem
 * 8.9 The Rank of a Symmetric Matrix
 * 8.10 The Binary Rank of the Adjacency Matrix
 * 8.11 The Symplectic Graphs
 * 8.12 Spectral Decomposition
 * 8.13 Rational Functions
 * Exercises
 * Notes
 * References


 * 9 Interlacing
 * 9.1 Interlacing
 * 9.2 Inside and Outside the Petersen Graph
 * 9.3 Equitable Partitions
 * 9.4 Eigenvalues of Kneser Graphs
 * 9.5 More Interlacing
 * 9.6 More Applications
 * 9.7 Bipartite Subgraphs
 * 9.8 Fullerenes
 * 9.9 Stability of Fullerenes
 * Exercises
 * Notes
 * References


 * 10 Strongly Regular Graphs
 * 10.1 Parameters
 * 10.2 Eigenvalues
 * 10.3 Some Characterizations
 * 10.4 Latin Square Graphs
 * 10.5 Small Strongly Regular Graphs
 * 10.6 Local Eigenvalues
 * 10.7 The Krein Bounds
 * 10.8 Generalized Quadrangles
 * 10.9 Lines of Size Three
 * 10.10 Quasi-Symmetric Designs
 * 10.11 The Witt Design on 23 Points
 * 10.12 The Symplectic Graphs
 * Exercises
 * Notes
 * References


 * 11 Two-Graphs
 * 11.1 Equiangular Lines
 * 11.2 The Absolute Bound
 * 11.3 Tightness
 * 11.4 The Relative Bound
 * 11.5 Switching
 * 11.6 Regular Two-Graphs
 * 11.7 Switching and Strongly Regular Graphs
 * 11.8 The Two-Graph on 276 Vertices
 * Exercises
 * Notes
 * References


 * 12 Line Graphs and Eigenvalues
 * 12.1 Generalized Line Graphs
 * 12.2 Star-Closed Sets of Lines
 * 12.3 Reflections
 * 12.4 Indecomposable Star-Closed Sets
 * 12.5 A Generating Set
 * 12.6 The Classification
 * 12.7 Root Systems
 * 12.8 Consequences
 * 12.9 A Strongly Regular Graph
 * Exercises
 * Notes
 * References


 * 13 The Laplacian of a Graph
 * 13.1 The Laplacian Matrix
 * 13.2 Trees
 * 13.3 Representations
 * 13.4 Energy and Eigenvalues
 * 13.5 Connectivity
 * 13.6 Interlacing
 * 13.7 Conductance and Objects
 * 13.8 How to Draw a Graph
 * 13.9 The Generalized Laplacian
 * 13.10 Multiplicities
 * 13.11 Embeddings
 * Exercises
 * Notes
 * References


 * 14 Cuts and Flows
 * 14.1 The Cut Space
 * 14.2 The Flow Space
 * 14.3 Planar Graphs
 * 14.4 Bases and Ear Decompositions
 * 14.5 Lattices
 * 14.6 Duality
 * 14.7 Integer Cuts and Flows
 * 14.8 Projections and Duals
 * 14.9 Chip Firing
 * 14.10 Two Bounds
 * 14.11 Recurrent States
 * 14.12 Critical States
 * 14.13 The Critical Group
 * 14.14 Voronoi Polyhedra
 * 14.15 Bicycles
 * 14.16 The Principal Tripartition
 * Exercises
 * Notes
 * References


 * 15 The Rank Polynomial
 * 15.1 Rank Functions
 * 15.2 Matroids
 * 15.3 Duality
 * 15.4 Restriction and Contraction
 * 15.5 Codes
 * 15.6 The Deletion-Contraction Algorithm
 * 15.7 Bicycles in Binary Codes
 * 15.8 Two Graph Polynomials
 * 15.9 Rank Polynomial
 * 15.10 Evaluations of the Rank Polynomial
 * 15.11 The Weight Enumerator of a Code
 * 15.12 Colourings and Codes
 * 15.13 Signed Matroids
 * 15.14 Rotors
 * 15.15 Submodular Functions
 * Exercises
 * Notes
 * References


 * 16 Knots
 * 16.1 Knots and Their Projections
 * 16.2 Reidemeister Moves
 * 16.3 Signed Plane Graphs
 * 16.4 Reidemeister Moves on Graphs
 * 16.5 Reidemeister Invariants
 * 16.6 The Kauffman Bracket
 * 16.7 The Jones Polynomial
 * 16.8 Connectivity
 * Exercises
 * Notes
 * References


 * 17 Knots and Eulerian Cycles
 * 17.1 Eulerian Partitions and Tours
 * 17.2 The Medial Graph
 * 17.3 Link Components and Bicycles
 * 17.4 Gauss Codes
 * 17.5 Chords and Circles
 * 17.6 Flipping Words
 * 17.7 Characterizing Gauss Codes
 * 17.8 Bent Tours and Spanning Trees
 * 17.9 Bent Partitions and the Rank Polynomial
 * 17.10 Maps
 * 17.11 Orientable Maps
 * 17.12 Seifert Circles
 * 17.13 Seifert Circles and Rank
 * Exercises
 * Notes
 * References


 * Glossary of Symbols
 * Index