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Random graphs

Author: Béla Bollobás
Publisher: Cambridge ; New York : Cambridge University Press, ©2001.
Series: Cambridge studies in advanced mathematics, 73.
Edition/Format:   Print book : English : 2nd edView all editions and formats
Summary:
"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 date and comprehensive account of random graph theory. The theory (founded by Erdös and Re;nyi in the late fifties) aims to estimate the number of graphs of a  Read more...
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: Béla Bollobás
ISBN: 0521809207 9780521809207 0521797225 9780521797221
OCLC Number: 45583496
Description: xviii, 498 pages : illustrations ; 24 cm
Contents: 1 Probability Theoretic Preliminaries 1 --
1.1 Notation and Basic Facts 1 --
1.2 Some Basic Distributions 5 --
1.3 Normal Approximation 9 --
1.4 Inequalities 15 --
1.5 Convergence in Distribution 25 --
2 Models of Random Graphs 34 --
2.1 Basic Models 34 --
2.2 Properties of Almost All Graphs 43 --
2.3 Large Subsets of Vertices 46 --
2.4 Random Regular Graphs 50 --
3 Degree Sequence 60 --
3.1 Distribution of an Element of the Degree Sequence 60 --
3.2 Almost Determined Degrees 65 --
3.3 Shape of the Degree Sequence 69 --
3.4 Jumps and Repeated Values 72 --
3.5 Fast Algorithms for the Graph Isomoprhism Problem 74 --
4 Small Subgraphs 78 --
4.1 Strictly Balanced Graphs 79 --
4.2 Arbitrary Subgraphs 85 --
4.3 Poisson Approximation 91 --
5 Evolution of Random Graphs--Sparse Components 96 --
5.1 Trees of Given Sizes As Components 96 --
5.2 Number of Vertices on Tree Components 102 --
5.3 Largest Tree Components 110 --
5.4 Components Containing Cycles 117 --
6 Evolution of Random Graphs--the Giant Component 130 --
6.1 A Gap in the Sequence of Components 130 --
6.2 Emergence of the Giant Component 138 --
6.3 Small Components after Time n/2 143 --
6.4 Further Results 148 --
6.5 Two Applications 153 --
7 Connectivity and Matchings 160 --
7.1 Connectedness of Random Graphs 161 --
7.2 K-Connectedness of Random Graphs 166 --
7.3 Matchings in Bipartite Graphs 171 --
7.4 Matchings in Random Graphs 178 --
7.5 Reliable Networks 189 --
7.6 Random Regular Graphs 195 --
8 Long Paths and Cycles 201 --
8.1 Long Paths in G[subscript c/n]--First Approach 202 --
8.2 Hamilton Cycles--First Approach 206 --
8.3 Hamilton Cycles--Second Approach 212 --
8.4 Long Paths in G[subscript c/n]--Second Approach 219 --
8.5 Hamilton Cycles in Regular Graphs--First Approach 221 --
8.6 Hamilton Cycles in Regular Graphs--Second Approach 224 --
9 Automorphism Group 229 --
9.1 Number of Unlabelled Graphs 229 --
9.2 Asymptotic Number of Unlabelled Regular Graphs 241 --
9.3 Distinguishing Vertices by Their Distance Sequences 243 --
9.4 Asymmetric Graphs 245 --
9.5 Graphs with a Given Automorphism Group 248 --
10 Diameter 251 --
10.1 Large Graphs of Small Diameter 251 --
10.2 Diameter of G[subscript p] 254 --
10.3 Diameter of Random Regular Graphs 264 --
10.4 Graph Processes 267 --
10.5 Related Results 271 --
10.6 Small Worlds 276 --
11 Cliques, Independent Sets and Colouring 282 --
11.1 Cliques in G[subscript p] 282 --
11.2 Poisson Approximation 290 --
11.3 Greedy Colouring of Random Graphs 294 --
11.4 Chromatic Number of Random Graphs 298 --
11.5 Sparse Graphs 303 --
12 Ramsey Theory 319 --
12.1 Bounds on R(s) 320 --
12.2 Off-Diagonal Ramsey Numbers 324 --
12.3 Triangle-Free Graphs 332 --
12.4 Dense Subgraphs 339 --
12.5 Size-Ramsey Number of a Path 341 --
13 Explicit Constructions 348 --
13.1 Character Sums 348 --
13.2 Paley Graph P[subscript q] 357 --
13.3 Dense Graphs 365 --
13.4 Sparse Graphs 373 --
13.5 Pseudorandom Graphs 376 --
14 Sequences, Matrices and Permutations 383 --
14.1 Random Subgraphs of the Cube 384 --
14.2 Random Matrices 394 --
14.3 Balancing Families of Sets 399 --
14.4 Random Elements of Finite Groups 408 --
14.5 Random Mappings 412 --
15 Sorting Algorithms 425 --
15.1 Finding Most Comparisons in One Round 426 --
15.2 Sorting in Two Rounds 431 --
15.3 Sorting with Width n/2 435 --
15.4 Bin Packing 442 --
16 Random Graphs of Small Order 447 --
16.1 Connectivity 447 --
16.2 Independent Sets 448 --
16.3 Colouring 451 --
16.4 Regular Graphs 455.
Series Title: Cambridge studies in advanced mathematics, 73.
Responsibility: Béla Bollobás.
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Abstract:

In this second edition of a now classic text, the addition of two new sections, numerous new results and over 150 references mean that this represents a comprehensive account of random graph theory.  Read more...

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'... contains an enormous amount of material, assembled by one who has played a leading role in the development of the area.' Zentralblatt MATH 'This book, written by one of the leaders in the field, Read more...

 
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