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Random Graphs Book

Random Graphs
Random Graphs, This is a new edition of the now classic text. The already extensive treatment given in the first edition has been heavily revised by the author. The addition of two new sections, numerous new results and 150 references means that this represents an up-to, Random Graphs has a rating of 5 stars
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Random Graphs, This is a new edition of the now classic text. The already extensive treatment given in the first edition has been heavily revised by the author. The addition of two new sections, numerous new results and 150 references means that this represents an up-to, Random Graphs
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  • Random Graphs
  • Written by author Bela Bollobas
  • Published by Cambridge University Press, August 2001
  • This is a new edition of the now classic text. The already extensive treatment given in the first edition has been heavily revised by the author. The addition of two new sections, numerous new results and 150 references means that this represents an up-to
  • This is a revised and updated version of the classic first edition. Booknews Probabilistic concepts have proven important in the study of graphs since Erdós and Rényi's founding of the theory of random graphs in the early 196
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Preface
Notation
1Probability Theoretic Preliminaries1
1.1Notation and Basic Facts1
1.2Some Basic Distributions5
1.3Normal Approximation9
1.4Inequalities15
1.5Convergence in Distribution25
2Models of Random Graphs34
2.1The Basic Models34
2.2Properties of Almost all Graphs43
2.3Large Subsets of Vertices46
2.4Random Regular Graphs50
3The Degree Sequence60
3.1The Distribution of an Element of the Degree Sequence60
3.2Almost Determined Degrees65
3.3The Shape of the Degree Sequence69
3.4Jumps and Repeated Values72
3.5Fast Algorithms for the Graph Isomorphism Problem74
4Small Subgraphs78
4.1Strictly Balanced Graphs79
4.2Arbitrary Subgraphs85
4.3Poisson Approximation91
5The Evolution of Random Graphs - Sparse Components96
5.1Trees of Given Sizes As Components96
5.2The Number of Vertices on Tree Components102
5.3The Largest Tree Components110
5.4Components Containing Cycles117
6The Evolution of Random Graphs - the Giant Component130
6.1A Gap in the Sequence of Components130
6.2The Emergence of the Giant Component138
6.3Small Components after Time n/2143
6.4Further Results148
6.5Two Applications153
7Connectivity and Matchings160
7.1The Connectedness of Random Graphs161
7.2The k-Connectedness of Random Graphs166
7.3Matchings in Bipartite Graphs171
7.4Matchings in Random Graphs178
7.5Reliable Networks189
7.6Random Regular Graphs195
8Long Paths and Cycles201
8.1Long Paths in G[subscript c/n] - First Approach202
8.2Hamilton Cycles- First Approach206
8.3Hamilton Cycles- Second Approach212
8.4Long Paths in G[subscript c/n]-Second Approach219
8.5Hamilton Cycles in Regular Graphs - First Approach221
8.6Hamilton Cycles in Regular Graphs - Second Approach224
9The Automorphism Group229
9.1The Number of Unlabelled Graphs229
9.2The Asymptotic Number of Unlabelled Regular Graphs241
9.3Distinguishing Vertices by Their Distance Sequences243
9.4Asymmetric Graphs245
9.5Graphs with a Given Automorphism Group248
10The Diameter251
10.1Large Graphs of Small Diameter251
10.2The Diameter of G[subscript p]254
10.3The Diameter of Random Regular Graphs264
10.4Graph Processes267
10.5Related Results271
10.6Small Worlds276
11Cliques, Independent Sets and Colouring282
11.1Cliques in G[subscript p]282
11.2Poisson Approximation290
11.3Greedy Colouring of Random Graphs294
11.4The Chromatic Number of Random Graphs298
11.5Sparse Graphs303
12Ramsey Theory319
12.1Bounds on R(s)320
12.2Off-Diagonal Ramsey Numbers324
12.3Triangle-Free Graphs332
12.4Dense Subgraphs339
12.5The Size-Ramsey Number of a Path341
13Explicit Constructions348
13.1Character Sums348
13.2The Paley Graph P[subscript q]357
13.3Dense Graphs365
13.4Sparse Graphs373
13.5Pseudorandom Graphs376
14Sequences, Matrices and Permutations383
14.1Random Subgraphs of the Cube384
14.2Random Matrices394
14.3Balancing Families of Sets399
14.4Random Elements of Finite Groups408
14.5Random Mappings412
15Sorting Algorithms425
15.1Finding Most Comparisons in One Round426
15.2Sorting in Two Rounds431
15.3Sorting with Width n/2435
15.4Bin Packing442
16Random Graphs of Small Order447
16.1Connectivity447
16.2Independent Sets448
16.3Colouring451
16.4Regular Graphs455
References457
Index496


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