Advanced Graph Theory and Combinatorics by Rigo, Michel -- Read -- Imperial Library of Trantor

Index

Cover Table of Contents Dedication Title Copyright Foreword Introduction 1 A First Encounter with Graphs

1.1. A few definitions 1.2. Paths and connected components 1.3. Eulerian graphs 1.4. Defining Hamiltonian graphs 1.5. Distance and shortest path 1.6. A few applications 1.7. Comments 1.8. Exercises

2 A Glimpse at Complexity Theory

2.1. Some complexity classes 2.2. Polynomial reductions 2.3. More hard problems in graph theory

3 Hamiltonian Graphs

3.1. A necessary condition 3.2. A theorem of Dirac 3.3. A theorem of Ore and the closure of a graph 3.4. Chvátal’s condition on degrees 3.5. Partition of Kn into Hamiltonian circuits 3.6. De Bruijn graphs and magic tricks 3.7. Exercises

4 Topological Sort and Graph Traversals

4.1. Trees 4.2. Acyclic graphs 4.3. Exercises

5 Building New Graphs from Old Ones

5.1. Some natural transformations 5.2. Products 5.3. Quotients 5.4. Counting spanning trees 5.5. Unraveling 5.6. Exercises

6 Planar Graphs

6.1. Formal definitions 6.2. Euler’s formula 6.3. Steinitz’ theorem 6.4. About the four-color theorem 6.5. The five-color theorem 6.6. From Kuratowski’s theorem to minors 6.7. Exercises

7 Colorings

7.1. Homomorphisms of graphs 7.2. A digression: isomorphisms and labeled vertices 7.3. Link with colorings 7.4. Chromatic number and chromatic polynomial 7.5. Ramsey numbers 7.6. Exercises

8 Algebraic Graph Theory

8.1. Prerequisites 8.2. Adjacency matrix 8.3. Playing with linear recurrences 8.4. Interpretation of the coefficients 8.5. A theorem of Hoffman 8.6. Counting directed spanning trees 8.7. Comments 8.8. Exercises

9 Perron–Frobenius Theory

9.1. Primitive graphs and Perron’s theorem 9.2. Irreducible graphs 9.3. Applications 9.4. Asymptotic properties 9.5. The case of polynomial growth 9.6. Exercises

10 Google’s Page Rank

10.1. Defining the Google matrix 10.2. Harvesting the primitivity of the Google matrix 10.3. Computation 10.4. Probabilistic interpretation 10.5. Dependence on the parameter α 10.6. Comments

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