Algebraic Graph Theory

Transcription

1 Chris Godsil Gordon Royle Algebraic Graph Theory With 120 Illustrations Springer

2 Preface 1 Graphs Graphs Subgraphs Automorphisms Homomorphisms Circulant Graphs Johnson Graphs Line Graphs Planar Graphs 12 Exercises 16 Notes 17 References 18 2 Groups Permutation Groups Counting Asymmetric Graphs Orbits on Pairs Primitivity... : Primitivity and Connectivity 29 Exercises 30 Notes 32 References 32 vii

3 xiv 3 Transitive Graphs Vertex-Transitive Graphs Edge-Transitive Graphs Edge Connectivity Vertex Connectivity Matchings Hamilton Paths and Cycles Cayley Graphs Directed Cayley Graphs with No Hamilton Cycles Retracts Transpositions 52 Exercises 54 Notes 56 References 57 4 Arc-Transitive Graphs Arc-Transitive Graphs Arc Graphs Cubic Arc-Transitive Graphs The Petersen Graph Distance-Transitive Graphs The Coxeter Graph Tutte's 8-Cage 71 Exercises 74 Notes 76 References 76 5 Generalized Polygons and Moore Graphs Incidence Graphs Projective Planes A Family of Projective Planes Generalized Quadrangles A Family of Generalized Quadrangles Generalized Polygons Two Generalized Hexagons Moore Graphs The Hoffman-Singleton Graph Designs 94 Exercises 97 Notes 100 References Homomorphisms The Basics Cores Products : 106

4 xv 6.4 The Map Graph Counting Homomorphisms Products and Colourings Uniquely Colourable Graphs Foldings and Covers Cores with No Triangles The Andrasfai Graphs Colouring Andrasfai Graphs A Characterization Cores of Vertex-Transitive Graphs Cores of Cubic Vertex-Transitive Graphs 125 Exercises 128 Notes 132 References 133 Kneser Graphs Fractional Colourings and Cliques Fractional Cliques Fractional Chromatic Number Homomorphisms and Fractional Colourings Duality Imperfect Graphs Cyclic Interval Graphs Erdos-Ko-Rado Homomorphisms of Kneser Graphs Induced Homomorphisms The Chromatic Number of the Kneser Graph Gale's Theorem Welzl's Theorem The Cartesian Product Strong Products and Colourings 155 Exercises 156 Notes 159 References 160 Matrix Theory The Adjacency Matrix The Incidence Matrix The Incidence Matrix of an Oriented Graph Symmetric Matrices Eigenvectors Positive Semidefinite Matrices Subharmonic Functions The Perron-Frobenius Theorem The Rank of a Symmetric Matrix The Binary Rank of the Adjacency Matrix 181

5 xvi 8.11 The Symplectic Graphs Spectral Decomposition Rational Functions 187 Exercises 188 Notes 192 References Interlacing Interlacing Inside and Outside the Petersen Graph Equitable Partitions Eigenvalues of Kneser Graphs More Interlacing More Applications Bipartite Subgraphs Fullerenes Stability of Fullerenes 210 Exercises 213 Notes 215 References Strongly Regular Graphs Parameters Eigenvalues Some Characterizations Latin Square Graphs Small Strongly Regular Graphs Local Eigenvalues The Krein Bounds Generalized Quadrangles Lines of Size Three Quasi-Symmetric Designs The Witt Design on 23 Points The Symplectic Graphs 242 Exercises 244 Notes 246 References Two-Graphs Equiangular Lines The Absolute Bound Tightness The Relative Bound Switching Regular Two-Graphs Switching and Strongly Regular Graphs 258

6 xvii 11.8 The Two-Graph on 276 Vertices 260 Exercises 262 Notes 263 References Line Graphs and Eigenvalues Generalized Line Graphs Star-Closed Sets of Lines Reflections Indecomposable Star-Closed Sets A Generating Set The Classification Root Systems Consequences A Strongly Regular Graph 276 Exercises 277 Notes 278 References The Laplacian of a Graph The Laplacian Matrix Trees Representations Energy and Eigenvalues Connectivity Interlacing Conductance and Cutsets How to Draw a Graph The Generalized Laplacian Multiplicities Embeddings 300 Exercises 302 Notes 305 References Cuts and Flows The Cut Space The Flow Space Planar Graphs Bases and Ear Decompositions Lattices Duality Integer Cuts and Flows Projections and Duals Chip Firing Two Bounds 323

7 xviii Recurrent States Critical States The Critical Group Voronoi Polyhedra Bicycles The Principal Tripartition 334 Exercises 336 Notes 338 References The Rank Polynomial Rank Functions Matroids Duality Restriction and Contraction Codes The Deletion-Contraction Algorithm Bicycles in Binary Codes Two Graph Polynomials Rank Polynomial Evaluations of the Rank Polynomial The Weight Enumerator of a Code Colourings and Codes Signed Matroids Rotors Submodular Functions 366 Exercises 369 Notes 371 References Knots Knots and Their Projections Reidemeister Moves Signed Plane Graphs Reidemeister moves on graphs Reidemeister Invariants The Kauffman Bracket The Jones Polynomial Connectivity 388 Exercises 391 Notes References Knots and Euleriari Cycles Eulerian Partitions and Tours The Medial Graph. 398

8 xix 17.3 Link Components and Bicycles Gauss Codes Chords and Circles Flipping Words Characterizing Gauss Codes Bent Tours and Spanning Trees Bent Partitions and the Rank Polynomial Maps Orientable Maps Seifert Circles Seifert Circles and Rank 420 Exercises 423 Notes 424 References 425 Glossary of Symbols 427 Index 433