Combinatorics and graph theory Books
Cambridge University Press Numbers Groups and Codes
Book SynopsisA thoroughly revised and updated version of the popular textbook on abstract algebra. The material is introduced with clarity and reference to problems and concepts that students will easily understand. With many examples and exercises, it will serve as the ideal introduction to this important and ubiquitous subject.Trade Review'This book is a lucid introduction to the subject which would make a suitable text for a one-semester course in the undergraduate program. It is very useful to everybody who want to understand the concepts of algebra and group theory and their relation to applications, particularly in computer science.' Zentralblatt MATHTable of Contents1. Number theory; 2. Sets, functions and relations; 3. Logic and mathematical argument; 4. Examples of groups; 5. Group theory and error-correcting codes; 6. Polynomials.
£49.39
Cambridge University Press LMSST 35 Young Tableaux With Applications to Representation Theory and Geometry London Mathematical Society Student Texts Series Number 35
Book SynopsisThe aim of this book is to develop the combinatorics of Young tableaux and to show them in action in the algebra of symmetric functions, representations of the symmetric and general linear groups, and the geometry of flag varieties. The first part of the book is a self-contained presentation of the basic combinatorics of Young tableaux, including the remarkable constructions of 'bumping' and 'sliding', and several interesting correspondences. In Part II these results are used to study representations with geometry on Grassmannians and flag manifolds, including their Schubert subvarieties, and the related Schubert polynomials. Much of this material has never appeared in book form.There are numerous exercises throughout, with hints or answers provided. Researchers in representation theory and algebraic geometry as well as in combinatorics will find Young Tableaux interesting and useful; students will find the intuitive presentation easy to follow.Table of ContentsPart I. Calculus Of Tableux: 1. Bumping and sliding; 2. Words: the plactic monoid; 3. Increasing sequences: proofs of the claims; 4. The Robinson-Schensted-Knuth Correspondence; 5. The Littlewood-Richardson rule; 6. Symmetric polynomials; Part II. Representation Theory: 7. Representations of the symmetric group; 8. Representations of the general linear group; Part III. Geometry: 9. Flag varieties; 10. Schubert varieties and polynomials; Appendix A; Appendix B.
£42.74
Cambridge University Press Eigenspaces of Graphs 66 Encyclopedia of Mathematics and its Applications Series Number 66
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£105.45
Cambridge University Press Combinatorics Geometry and Probability A Tribute to Paul Erds
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£151.05
Cambridge University Press Bipartite Graphs and their Applications 131 Cambridge Tracts in Mathematics Series Number 131
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£105.45
Cambridge University Press LMS 245 Geom Combinatorial Design London Mathematical Society Lecture Note Series Series Number 245
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£61.71
Cambridge University Press LMS 241 Surveys Combinatorics 97 London Mathematical Society Lecture Note Series Series Number 241
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£69.99
Cambridge University Press Combinatorics Geometry and Probability
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£49.39
Cambridge University Press Surveys in Combinatorics 2005 327 London Mathematical Society Lecture Note Series Series Number 327
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£61.71
Cambridge University Press Dense Sphere Packings A Blueprint for Formal Proofs 400 London Mathematical Society Lecture Note Series Series Number 400
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£50.12
Cambridge University Press Surveys in Combinatorics 1999 267 London Mathematical Society Lecture Note Series Series Number 267
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£65.86
Cambridge University Press Games of No Chance 3 56 Mathematical Sciences Research Institute Publications
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£46.54
Cambridge University Press Combinatorics The Rota Way Cambridge Mathematical Library
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£55.09
Cambridge University Press Basic Phylogenetic Combinatorics
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£39.99
Cambridge University Press New Perspectives in Algebraic Combinatorics
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£122.55
Cambridge University Press Design Theory Volume 2 78 Encyclopedia of Mathematics and its Applications Series Number 78
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£133.95
Cambridge University Press Graph Theory
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£46.54
Cambridge University Press Topics in Algebraic Graph Theory Encyclopedia of Mathematics and its Applications Series Number 102
Book SynopsisThe rapidly expanding area of algebraic graph theory uses two different branches of algebra to explore various aspects of graph theory: linear algebra (for spectral theory) and group theory (for studying graph symmetry). These areas have links with other areas of mathematics, such as logic and harmonic analysis, and are increasingly being used in such areas as computer networks where symmetry is an important feature. Other books cover portions of this material, but this book is unusual in covering both of these aspects and there are no other books with such a wide scope. Peter J. Cameron, internationally recognized for his substantial contributions to the area, served as academic consultant for this volume, and the result is ten expository chapters written by acknowledged international experts in the field. Their well-written contributions have been carefully edited to enhance readability and to standardize the chapter structure, terminology and notation throughout the book. To help thTrade Review"...highly suitable for an advanced course or seminar series, but should also serve as a useful resource for mathematicians who need to find out about one or more of the topics presented, and it complements other recent texts on a subject of increasing interests and significance." -Mathematical Reviews, Marston ConderTable of ContentsForeword Peter J. Cameron; Introduction; 1. Eigenvalues of graphs Michael Doob; 2. Graphs and matrices Richard A. Brualdi and Bryan L. Shader; 3. Spectral graph theory Dragos Cvetkovic and Peter Rowlinson; 4. Graph Laplacians Bojan Mohar; 5. Automorphism groups Peter J. Cameron; 6. Cayley graphs Brian Alspach; 7. Finite symmetric graphs Cheryle E. Praeger; 8. Strongly regular graphs Peter J. Cameron; 9. Distance-transitive graphs Arjeh M. Cohen; 10. Computing with graphs and groups Leonard H. Soicher.
£105.45
Cambridge University Press Topics in Topological Graph Theory 129 Encyclopedia of Mathematics and its Applications Series Number 128
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£122.55
Cambridge University Press More Games of No Chance
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£133.95
Cambridge University Press Random Graphs 73 Cambridge Studies in Advanced Mathematics Series Number 73
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£186.20
Cambridge University Press Abstract Regular Polytopes 92 Encyclopedia of Mathematics and its Applications Series Number 92
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£151.05
Cambridge University Press Mathematical Constants
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£151.05
Cambridge University Press Global Methods for Combinatorial Isoperimetric Problems
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£85.49
Cambridge University Press Additive Combinatorics 105 Cambridge Studies in Advanced Mathematics Series Number 105
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£116.85
Cambridge University Press Lectures on the Combinatorics of Free Probability 335 London Mathematical Society Lecture Note Series Series Number 335
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£96.89
Cambridge University Press Games of No Chance 3
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£114.00
Cambridge University Press Geometry of Chemical Graphs
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£105.45
Cambridge University Press Algorithmic Aspects of Graph Connectivity 123 Encyclopedia of Mathematics and its Applications Series Number 123
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£83.60
Cambridge University Press Combinatorics The Rota Way Cambridge Mathematical Library
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£105.45
Cambridge University Press Groups Graphs and Trees An Introduction to the Geometry of Infinite Groups London Mathematical Society Student Texts Series Number 73
Book SynopsisThis book presents a modern, geometric approach to group theory. An accessible and engaging approach to the subject, with many exercises and figures to develop geometric intuition. Ideal for advanced undergraduates, it will also interest graduate students and researchers as a gentle introduction to geometric group theory.Trade Review'Meier has the rare ability to make complex concepts accessible to novices while simultaneously challenging the experienced. … With well-chosen illustrations and examples, Meier succeeds brilliantly in this unique approach.' SciTech Book News'… totally accessible to undergraduate students and would be a good textbook for an advanced undergraduate course in group theory, or a graduate course in geometric group theory.' Mathematical Review'… an excellent introduction to geometric group theory. … Carefully chosen examples are an essential part of the exposition and they really help to understand general constructions.' EMS NewsletterTable of ContentsPreface; 1. Cayley's theorems; 2. Groups generated by reflections; 3. Groups acting on trees; 4. Baumslag-Solitar groups; 5. Words and Dehn's word problem; 6. A finitely-generated, infinite, Torsion group; 7. Regular languages and normal forms; 8. The Lamplighter group; 9. The geometry of infinite groups; 10. Thompson's group; 11. The large-scale geometry of groups; Bibliography; Index.
£89.30
Cambridge University Press Compact Matrix Quantum Groups and Their Combinatorics
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£37.99
Cambridge University Press Compact Matrix Quantum Groups and Their Combinatorics
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£94.99
Cambridge University Press Finite Geometry and Combinatorial Applications 82 London Mathematical Society Student Texts Series Number 82
Book SynopsisThe projective and polar geometries that arise from a vector space over a finite field are particularly useful in the construction of combinatorial objects, such as latin squares, designs, codes and graphs. This book provides an introduction to these geometries and their many applications to other areas of combinatorics. Coverage includes a detailed treatment of the forbidden subgraph problem from a geometrical point of view, and a chapter on maximum distance separable codes, which includes a proof that such codes over prime fields are short. The author also provides more than 100 exercises (complete with detailed solutions), which show the diversity of applications of finite fields and their geometries. Finite Geometry and Combinatorial Applications is ideal for anyone, from a third-year undergraduate to a researcher, who wishes to familiarise themselves with and gain an appreciation of finite geometry.Table of Contents1. Fields; 2. Vector spaces; 3. Forms; 4. Geometries; 5. Combinatorial applications; 6. The forbidden subgraph problem; 7. MDS codes; Appendix A. Solutions to the exercises; Appendix B. Additional proofs; Appendix C. Notes and references; References; Index.
£133.95
Cambridge University Press Analytic Combinatorics in Several Variables
Book SynopsisDiscrete structures, like DNA sequences and the internet, are complex objects created from indivisible parts. Now more accessible to graduate students, this book introduces multivariate generating functions, which are used to create computational tools to detect and understand patterns in such structures.Trade Review'A definitive treatment of a challenging but very useful subject. There is a wide variety of situations calling for the estimation of the coefficients of a multivariate generating function. The authors have done a superb job of classifying and elucidating the myriad of available techniques for achieving this aim.' Richard P. Stanley, University of Miami'This book is an invaluable resource that is certain to have dramatic impact on research and teaching in this rapidly developing area of mathematics. The first edition broke new ground; this edition prepares the field for others to harvest new knowledge with important applications in many scientific disciplines.'Table of ContentsPart I. Combinatorial Enumeration: 1. Introduction; 2. Generating functions; 3. Univariate asymptotics; Part II. Mathematical Background: 4. Fourier–Laplace integrals in one variable; 5. Multivariate Fourier–Laplace integrals; 6. Laurent series, amoebas, and convex geometry; Part III. Multivariate Enumeration: 7. Overview of analytic methods for multivariate generating functions; 8. Effective computations and ACSV; 9. Smooth point asymptotics; 10. Multiple point asymptotics; 11. Cone point asymptotics; 12. Combinatorial applications; 13. Challenges and extensions; Appendices: A. Integration on manifolds; B. Algebraic topology; C. Residue forms and classical Morse theory; D. Stratification and stratified Morse theory; References; Author index; Subject index.
£66.49
American Mathematical Society Coloring Mixed Hypergraphs
Book SynopsisThe theory of graph coloring has existed for more than 150 years. This book states that in the case of hypergraphs, there exist problems on both the minimum and the maximum number of colors. This feature pervades the theory, methods, algorithms, and applications of mixed hypergraph coloring.Table of ContentsIntroduction The lower chromatic number of a hypergraph Mixed hypergraphs and the upper chromatic number Uncolorable mixed hypergraphs Uniquely colorable mixed hypergraphs $\mathcal{C}$-perfect mixed hypergraphs Gaps in the chromatic spectrum Interval mixed hypergraphs Pseudo-chordal mixed hypergraphs Circular mixed hypergraphs Planar mixed hypergraphs Coloring block designs as mixed hypergraphs Modelling with mixed hypergraphs Bibliography List of figures Index.
£57.95
Precalculus
Book SynopsisYear after year, PRECALCULUS: FUNCTIONS AND GRAPHS leads the way in helping students like you succeed in their Precalculus courses. Its clear explanations and examples and exercises featuring a variety of real-life applications make the content understandable and relatable. This 13th edition of Swokowski and Cole's bestselling text is consistently praised for being at just the right level for Precalculus students. Perhaps most important, this book effectively prepares readers for further courses in mathematics.Table of Contents1. TOPICS FROM ALGEBRA. Real Numbers. Exponents and Radicals. Algebraic Expressions. Equations. Complex Numbers. Inequalities. 2. FUNCTIONS AND GRAPHS. Rectangular Coordinate Systems. Graphs of Equations. Lines. Definition of Function. Graphs of Functions. Quadratic Functions. Operations on Functions. 3. POLYNOMIAL AND RATIONAL FUNCTIONS. Polynomial Functions of Degree Greater Than 2. Properties of Division. Zeros of Polynomials. Complex and Rational Zeros of Polynomials. Rational Functions. Variation. 4. INVERSE, EXPONENTIAL, AND LOGARITHMIC FUNCTIONS. Inverse Functions. Exponential Functions. The Natural Exponential Function. Logarithmic Functions. Properties of Logarithms. Exponential and Logarithmic Equations. 5. TRIGONOMETRIC FUNCTIONS. Angles. Trigonometric Functions of Angles. Trigonometric Functions of Real Numbers. Values of the Trigonometric Functions. Trigonometric Graphs. Additional Trigonometric Graphs. Applied Problems. 6. ANALYTIC TRIGONOMETRY. Verifying Trigonometric Identities. Trigonometric Equations. The Additions and Subtraction of Formulas. Multiple-Angle Formulas. Product-To-Sum and Sum-To-Product Formulas. The Inverse Trigonometric Functions. 7. APPLICATIONS OF TRIGONOMETRY. The Law of Sines. The Law of Cosines. Vectors. The Dot Product. Trigonometric Form for Complex Numbers. De Moivre���s Theorem and nth Roots of Complex Numbers. 8. SYSTEMS OF EQUATIONS AND INEQUALITIES. Systems of Equations. Systems of Linear Equations in Two Variables. Systems of Inequalities. Linear Programming. Systems of Linear Equations in More Than Two Variables. The Algebra of Matrices. The Inverse of a Matrix. Determinants. Properties of Determinants. Partial Fractions. 9. SEQUENCES, SERIES, AND PROBABILITY. Infinite Sequences and Summation Notation. Arithmetic Sequences. Geometric Sequences. Mathematical Induction. The Binomial Theorem. Permutations. Distinguishable Permutations and Combinations. Probability. 10. TOPICS FROM ANALYTICAL GEOMETRY. Parabolas. Ellipses. Hyperbolas. Plane Curves and Parametric Equations. Polar Coordinates. Polar Equations of Conics. 11. LIMITS OF FUNCTIONS. Introductions to Limits. Definition of a Limit. Techniques for Finding Limits. Limits Involving Infinity. Appendix I: Common Graphs and Their Equations. Appendix II: A Summary of Graph Transformations. Appendix III: Graphs of the Trigonometric Functions and Their Inverses. Appendix IV: Values of the Trigonometric Functions of Special Angles on a Unit Circle. Appendix V: Theorems on Limits.
£269.20
Barcharts, Inc Math Common Core For Kindergarten
Book SynopsisParents and teachers can use this laminated 6-page guide with their kindergartener to hit specific learning requirements of The Common Core State Standards for Mathematics. These expectations and skills are the stepping stones needed to succeed in school, college and the real world. Each guide in the series focuses on critical areas of the curriculum and features real-world problems, examples and illustrations, with parent and teacher guidance built in. Suggested uses: o Teacher/Parent â use as a quick way to review all of the standards for kindergarten, with or without guiding the student through the material o Student â you can spill anything you want on this and it will be ok, it's laminated so you can eat your lunch off it or use playdough on it
£999.99
MP-AMM American Mathematical An Introduction to Proof via InquiryBased
Book SynopsisA textbook for the transition to proof course for mathematics majors. Designed to promote active learning through inquiry, the book features a highly structured set of leading questions and explorations. The content ranges over topics traditionally included in transitions courses.Table of Contents Introduction Mathematics and logic Set theory Induction The real numbers Three famous theorems Relations and partitions Functions Cardinality Elements for syle for proofs Fancy mathematical terms Paradoxes Defintions in mathematics
£26.95
Charlesbridge Publishing,U.S. Show and Tell! Great Graphs and Smart Charts: An
Book SynopsisA visual-learning expert races up the charts and graphs math success with kid-friendly content sure to help with homework.Want to find the most popular meal in the cafeteria? Compare town sports enrollments? Or maybe you just want to know who burps the most in your family! Learn what line graphs, bar graphs, pie charts, and pictographs are and how and when to use them to represent data. Each project shows how to build a chart or graph and ties it all together with a creative infographic that really puts the A in STEAM (Science, Technology, Engineering, ARTS, and Mathematics). Whether used as an introductory aid or to underscore previous knowledge, the book prepares today''s visually savvy children to succeed in school and life by analyzing the world around them.
£15.29
ISTE Ltd and John Wiley & Sons Inc Graphs and Networks: Multilevel Modeling
Book SynopsisA transport network is typically a network of roads, streets, pipes, aqueducts, power lines, or nearly any structure that permits either vehicular movement or the flow of some commodity. Transport network analysis, a field of transport engineering that typically employs the use of graph theory, is used to determine the flow of vehicles, commodities, or people through these networks. It may combine different modes of transport - for example, walking and driving - to model multi-modal journeys. This edition is completely updated and contains two new chapters covering spatial analysis and urban management through graph theory simulation. Highly practical, the simulation approach allows readers to solve classic problems, such as placement of high-speed roads, the capacity of a network, pollution emission control, and more.
£170.95
ISTE Ltd and John Wiley & Sons Inc Mathematics for Informatics and Computer Science
Book SynopsisHow many ways do exist to mix different ingredients, how many chances to win a gambling game, how many possible paths going from one place to another in a network ? To this kind of questions Mathematics applied to computer gives a stimulating and exhaustive answer. This text, presented in three parts (Combinatorics, Probability, Graphs) addresses all those who wish to acquire basic or advanced knowledge in combinatorial theories. It is actually also used as a textbook. Basic and advanced theoretical elements are presented through simple applications like the Sudoku game, search engine algorithm and other easy to grasp applications. Through the progression from simple to complex, the teacher acquires knowledge of the state of the art of combinatorial theory. The non conventional simultaneous presentation of algorithms, programs and theory permits a powerful mixture of theory and practice. All in all, the originality of this approach gives a refreshing view on combinatorial theory.Trade Review"On the other hand if you are looking for an approach to combinatorics that is routed in applications and with lots of exercises then this is the book for you. Yes, dare I say it, it's fun." (I Programmer, 21 January 2011)Table of ContentsGeneral Introduction xxiii Chapter 1. Some Historical Elements 1 PART 1. COMBINATORICS 17 Part 1. Introduction 19 Chapter 2. Arrangements and Combinations 21 Chapter 3. Enumerations in Alphabetical Order 43 Chapter 4. Enumeration by Tree Structures 63 Chapter 5. Languages, Generating Functions and Recurrences 85 Chapter 6. Routes in a Square Grid 105 Chapter 7. Arrangements and Combinations with Repetitions 119 Chapter 8. Sieve Formula 137 Chapter 9. Mountain Ranges or Parenthesis Words: Catalan Numbers 165 Chapter 10. Other Mountain Ranges 197 Chapter 11. Some Applications of Catalan Numbers and Parenthesis Words 215 Chapter 12. Burnside’s Formula 227 Chapter 13. Matrices and Circulation on a Graph 253 Chapter 14. Parts and Partitions of a Set 275 Chapter 15. Partitions of a Number 289 Chapter 16. Flags 305 Chapter 17. Walls and Stacks 315 Chapter 18. Tiling of Rectangular Surfaces using Simple Shapes 331 Chapter 19. Permutations 345 PART 2. PROBABILITY 387 Part 2. Introduction 389 Chapter 20. Reminders about Discrete Probabilities 395 Chapter 21. Chance and the Computer 427 Chapter 22. Discrete and Continuous 447 Chapter 23. Generating Function Associated with a Discrete Random Variable in a Game 469 Chapter 24. Graphs and Matrices for Dealing with Probability Problems 497 Chapter 25. Repeated Games of Heads or Tails 509 Chapter 26. Random Routes on a Graph 535 Chapter 27. Repetitive Draws until the Outcome of a Certain Pattern 565 Chapter 28. Probability Exercises 597 PART 3. GRAPHS 637 Part 3. Introduction 639 Chapter 29. Graphs and Routes 643 Chapter 30. Explorations in Graphs 661 Chapter 31. Trees with Numbered Nodes, Cayley’s Theorem and Prüfer Code 705 Chapter 32. Binary Trees 723 Chapter 33. Weighted Graphs: Shortest Paths and Minimum Spanning Tree 737 Chapter 34. Eulerian Paths and Cycles, Spanning Trees of a Graph 759 Chapter 35. Enumeration of Spanning Trees of an Undirected Graph 779 Chapter 36. Enumeration of Eulerian Paths in Undirected Graphs 799 Chapter 37. Hamiltonian Paths and Circuits 835 APPENDICES 867 Appendix 1. Matrices 869 Appendix 2. Determinants and Route Combinatorics 885 Bibliography 907 Index 911
£132.00
ISTE Ltd and John Wiley & Sons Inc Graph Partitioning
Book SynopsisGraph partitioning is a theoretical subject with applications in many areas, principally: numerical analysis, programs mapping onto parallel architectures, image segmentation, VLSI design. During the last 40 years, the literature has strongly increased and big improvements have been made. This book brings together the knowledge accumulated during many years to extract both theoretical foundations of graph partitioning and its main applications.Table of ContentsIntroduction xiii Charles-Edmond Bichot, Patrick Siarry Chapter 1. General Introduction to Graph Partitioning 1 Charles-Edmond Bichot 1.1. Partitioning 1 1.2. Mathematical notions 2 1.3. Graphs 4 1.4. Formal description of the graph partitioning problem 8 1.5. Objective functions for graph partitioning 11 1.6. Constrained graph partitioning 13 1.7. Unconstrained graph partitioning 14 1.8. Differences between constrained and unconstrained partitioning 16 1.9. From bisection to k-partitioning: the recursive bisection method 17 1.10. NP-hardness of graph partitioning optimization problems 19 1.11. Conclusion 22 1.12. Bibliography 22 Part 1: Graph Partitioning for Numerical Analysis 27 Chapter 2. A Partitioning Requiring Rapidity and Quality: The Multilevel Method and Partitions Refinement Algorithms 29 Charles-Edmond Bichot 2.1. Introduction 29 2.2. Principles of the multilevel method 30 2.3. Graph coarsening 33 2.4. Partitioning of the coarsened graph 37 2.5. Uncoarsening and partitions refinement 40 2.6. The spectral method 52 2.7. Conclusion 59 2.8. Bibliography 60 Chapter 3. Hypergraph Partitioning 65 Cédric Chevalier 3.1. Definitions and metrics 65 3.2. Connections between graphs, hypergraphs, and matrices 67 3.3. Algorithms for hypergraph partitioning 68 3.4. Purpose 72 3.5. Conclusion 77 3.6. Software references 78 3.7. Bibliography 78 Chapter 4. Parallelization of Graph Partitioning 81 François Pellegrini 4.1. Introduction 81 4.2. Distributed data structures 84 4.3. Parallelization of the coarsening phase 87 4.4. Folding 93 4.5. Centralization 95 4.6. Parallelization of the refinement phase 96 4.7. Experimental results 107 4.8. Conclusion 111 4.9. Bibliography 111 Chapter 5. Static Mapping of Process Graphs 115 François Pellegrini 5.1. Introduction 115 5.2. Static mapping models 116 5.3. Exact algorithms 121 5.4. Approximation algorithms 123 5.5. Conclusion 133 5.6. Bibliography 134 Part 2: Optimization Methods for Graph Partitioning 137 Chapter 6. Local Metaheuristics and Graph Partitioning 139 Charles-Edmond Bichot 6.1. General introduction to metaheuristics 140 6.2. Simulated annealing 141 6.3. Iterated local search 149 6.4. Other local search metaheuristics 158 6.5. Conclusion 159 6.6. Bibliography 159 Chapter 7. Population-based Metaheuristics, Fusion-Fission and Graph Partitioning Optimization 163 Charles-Edmond Bichot 7.1. Ant colony algorithms 163 7.2. Evolutionary algorithms 165 7.3. The fusion-fission method 182 7.4. Conclusion 195 7.5. Acknowledgments 196 7.6. Bibliography 196 Chapter 8. Partitioning Mobile Networks into Tariff Zones 201 Mustapha Oughdi, Sid Lamrous, Alexandre Caminada 8.1. Introduction 201 8.2. Spatial division of the network 208 8.3. Experimental results 220 8.4. Conclusion 222 8.5. Bibliography 223 Chapter 9. Air Traffic Control Graph Partitioning Application 225 Charles-Edmond Bichot, Nicolas Durand 9.1. Introduction 225 9.2. The problem of dividing up the airspace 227 9.3. Modeling the problem 231 9.4. Airspace partitioning: towards a new optimization metaheuristic 237 9.5. Division of the central European airspace 240 9.6. Conclusion 246 9.7. Acknowledgments 247 9.8. Bibliography 247 Part 3: Other Approaches to Graph Partitioning 249 Chapter 10. Application of Graph Partitioning to Image Segmentation 251 Amir Nakib, Laurent Najman, Hugues Talbot, Patrick Siarry 10.1. Introduction 251 10.2. The image viewed in graph form 251 10.3. Principle of image segmentation using graphs 254 10.4. Image segmentation via maximum flows 257 10.5. Unification of segmentation methods via graph theory 265 10.6. Conclusions and perspectives 269 10.7. Bibliography 271 Chapter 11. Distances in Graph Partitioning 275 Alain Guénoche 11.1. Introduction 275 11.2. The Dice distance 276 11.3. Pons-Latapy distance 281 11.4. A partitioning method for distance arrays 283 11.5. A simulation protocol 286 11.6. Conclusions 292 11.7. Acknowledgments 293 11.8. Bibliography 293 Chapter 12. Detection of Disjoint or Overlapping Communities in Networks 297 Jean-Baptiste Angelelli, Alain Guénoche, Laurence Reboul 12.1. Introduction 297 12.2. Modularity of partitions and coverings 299 12.3. Partitioning method 301 12.4. Overlapping partitioning methods 307 12.5. Conclusion 311 12.6. Acknowledgments 312 12.7. Bibliography 312 Chapter 13. Multilevel Local Optimization of Modularity 315 Thomas Aynaud, Vincent D. Blondel, Jean-Loup Guillaume and Renaud Lambiotte 13.1. Introduction 315 13.2. Basics of modularity 317 13.3. Modularity optimization 319 13.4. Validation on empirical and artificial graphs 327 13.5. Discussion 333 13.6. Conclusion 341 13.7. Acknowledgments 342 13.8. Bibliography 342 Appendix. The Main Tools and Test Benches for Graph Partitioning 347 Charles-Edmond Bichot A.1. Tools for constrained graph partitioning optimization 348 A.2. Tools for unconstrained graph partitioning optimization 350 A.3. Graph partitioning test benches 351 A.4. Bibliography 354 Glossary 357 List of Authors 361 Index 365
£125.06
ISTE Ltd and John Wiley & Sons Inc Probabilistic Combinatorial Optimization on
Book SynopsisThis title provides a comprehensive survey over the subject of probabilistic combinatorial optimization, discussing probabilistic versions of some of the most paradigmatic combinatorial problems on graphs, such as the maximum independent set, the minimum vertex covering, the longest path and the minimum coloring. Those who possess a sound knowledge of the subject mater will find the title of great interest, but those who have only some mathematical familiarity and knowledge about complexity and approximation theory will also find it an accessible and informative read.Table of ContentsPreface 11 Chapter 1. A Short Insight into Probabilistic Combinatorial Optimization 15 1.1. Motivations and applications 15 1.2. A formalism for probabilistic combinatorial optimization 19 1.3. The main methodological issues dealing with probabilistic combinatorial optimization 24 1.3.1. Complexity issues 24 1.3.1.1. Membership in NPO is not always obvious 24 1.3.1.2. Complexity of deterministic vs. complexity of probabilistic optimization problems 24 1.3.2. Solution issues 26 1.3.2.1. Characterization of optimal a priori solutions 26 1.3.2.2. Polynomial subcases 28 1.3.2.3. Exact solutions and polynomial approximation issues 29 1.4. Miscellaneous and bibliographic notes 31 FIRST PART. PROBABILISTIC GRAPH-PROBLEMS 35 Chapter 2. The Probabilistic Maximum Independent Set 37 2.1. The modification strategies and a preliminary result 39 2.1.1. Strategy M1 39 2.1.2. Strategies M2 and M3 39 2.1.3. Strategy M4 41 2.1.4. Strategy M5 41 2.1.5. A general mathematical formulation for the five functionals 42 2.2. PROBABILISTIC MAX INDEPENDENT SET1 44 2.2.1. Computing optimal a priori solutions 44 2.2.2. Approximating optimal solutions 45 2.2.3. Dealing with bipartite graphs 46 2.3. PROBABILISTIC MAX INDEPENDENT SET2 and 3 47 2.3.1. Expressions for E(G, S, M2) and E(G, S, M3) 47 2.3.2. An upper bound for the complexity of E(G, S, M2) 48 2.3.3. Bounds for E(G, S, M2) 49 2.3.4. Approximating optimal solutions 51 2.3.4.1. Using argmax{_vi∈S pi} as an a priori solution 51 2.3.4.2. Using approximations of MAX INDEPENDENT SET 53 2.3.5. Dealing with bipartite graphs 53 2.4. PROBABILISTIC MAX INDEPENDENT SET4 55 2.4.1. An expression for E(G, S, M4) 55 2.4.2. Using S∗ or argmax{_vi∈S pi} as an a priori solution 56 2.4.3. Dealing with bipartite graphs 57 2.5. PROBABILISTIC MAX INDEPENDENT SET5 58 2.5.1. In general graphs 58 2.5.2. In bipartite graphs 60 2.6. Summary of the results 61 2.7. Methodological questions 63 2.7.1. Maximizing a criterion associated with gain 65 2.7.1.1. The minimum gain criterion 65 2.7.1.2. The maximum gain criterion 66 2.7.2. Minimizing a criterion associated with regret 68 2.7.2.1. The maximum regret criterion 68 2.7.3. Optimizing expectation 70 2.8. Proofs of the results 71 2.8.1. Proof of Proposition 2.1 71 2.8.2. Proof of Theorem 2.6 74 2.8.3. Proof of Proposition 2.3 77 2.8.4. Proof of Theorem 2.13 78 Chapter 3. The Probabilistic Minimum Vertex Cover 81 3.1. The strategies M1, M2 and M3 and a general preliminary result 82 3.1.1. Specification of M1, M2 and M3 82 3.1.1.1. Strategy M1 82 3.1.1.2. Strategy M2 83 3.1.1.3. Strategy M3 83 3.1.2. A first expression for the functionals 84 3.2. PROBABILISTIC MIN VERTEX COVER1 84 3.3. PROBABILISTIC MIN VERTEX COVER2 86 3.4. PROBABILISTIC MIN VERTEX COVER3 87 3.4.1. Building E(G, C, M3) 87 3.4.2. Bounds for E(G, C, M3) 88 3.5. Some methodological questions 89 3.6. Proofs of the results 91 3.6.1. Proof of Theorem 3.3 91 3.6.2. On the the bounds obtained in Theorem 3.3 93 Chapter 4. The Probabilistic Longest Path 99 4.1. Probabilistic longest path in terms of vertices 100 4.2. Probabilistic longest path in terms of arcs 102 4.2.1. An interesting algebraic expression 104 4.2.2. Metric PROBABILISTIC ARC WEIGHTED LONGEST PATH 105 4.3. Why the strategies used are pertinent 109 4.4. Proofs of the results 110 4.4.1. Proof of Theorem 4.1 110 4.4.2. Proof of Theorem 4.2 112 4.4.3. An algebraic proof for Theorem 4.3 114 4.4.4. Proof of Lemma 4.1 116 4.4.5. Proof of Lemma 4.2 117 4.4.6. Proof of Theorem 4.4 117 Chapter 5. Probabilistic Minimum Coloring 125 5.1. The functional E(G,C) 127 5.2. Basic properties of probabilistic coloring 131 5.2.1. Properties under non-identical vertex-probabilities 131 5.2.2. Properties under identical vertex-probabilities 131 5.3. PROBABILISTIC MIN COLORING in general graphs 132 5.3.1. The complexity of probabilistic coloring 132 5.3.2. Approximation 132 5.3.2.1. The main result 132 5.3.2.2. Further approximation results 137 5.4. PROBABILISTIC MIN COLORING in bipartite graphs 139 5.4.1. A basic property 139 5.4.2. General bipartite graphs 141 5.4.3. Bipartite complements of bipartite matchings 147 5.4.4. Trees 151 5.4.5. Cycles 154 5.5. Complements of bipartite graphs 155 5.6. Split graphs 156 5.6.1. The complexity of PROBABILISTIC MIN COLORING 156 5.6.2. Approximation results 159 5.7. Determining the best k-coloring in k-colorable graphs 164 5.7.1. Bipartite graphs 164 5.7.1.1. PROBABILISTIC MIN 3-COLORING 164 5.7.1.2. PROBABILISTIC MIN k-COLORING fork > 3 168 5.7.1.3. Bipartite complements of bipartite matchings 171 5.7.2. The complements of bipartite graphs 171 5.7.3. Approximation in particular classes of graphs 174 5.8. Comments and open problems 175 5.9. Proofs of the different results 178 5.9.1. Proof of [5.5] 178 5.9.2. Proof of [5.4] 179 5.9.3. Proof of Property 5.1 180 5.9.4. Proof of Proposition 5.2 181 5.9.5. Proof of Lemma 5.11 183 SECOND PART. STRUCTURAL RESULTS 185 Chapter 6. Classification of Probabilistic Graph-problems 187 6.1. When MS is feasible 187 6.1.1. The a priori solution is a subset of the initial vertex-set 188 6.1.2. The a priori solution is a collection of subsets of the initial vertex-set 191 6.1.3. The a priori solution is a subset of the initial edge-set 193 6.2. When application of MS itself does not lead to feasible solutions 198 6.2.1. The functional associated with MSC 198 6.2.2. Applications 199 6.2.2.1. The a priori solution is a cycle 200 6.2.2.2. The a priori solution is a tree 201 6.3. Some comments 205 6.4. Proof of Theorem 6.4 206 Chapter 7. A Compendium of Probabilistic NPO Problems on Graphs 211 7.1. Covering and partitioning 214 7.1.1. MIN VERTEX COVER 214 7.1.2. MIN COLORING 214 7.1.3. MAX ACHROMATIC NUMBER 215 7.1.4. MIN DOMINATING SET 215 7.1.5. MAX DOMATIC PARTITION 216 7.1.6. MIN EDGE-DOMINATING SET 216 7.1.7. MIN INDEPENDENT DOMINATING SET 217 7.1.8. MIN CHROMATIC SUM 217 7.1.9. MIN EDGE COLORING 218 7.1.10. MIN FEEDBACK VERTEX-SET 219 7.1.11. MIN FEEDBACK ARC-SET 220 7.1.12. MAX MATCHING 220 7.1.13. MIN MAXIMAL MATCHING 220 7.1.14. MAX TRIANGLE PACKING 220 7.1.15. MAX H-MATCHING 221 7.1.16. MIN PARTITION INTO CLIQUES 222 7.1.17. MIN CLIQUE COVER 222 7.1.18. MIN k-CAPACITED TREE PARTITION 222 7.1.19. MAX BALANCED CONNECTED PARTITION 223 7.1.20. MIN COMPLETE BIPARTITE SUBGRAPH COVER 223 7.1.21. MIN VERTEX-DISJOINT CYCLE COVER 223 7.1.22. MIN CUT COVER 224 7.2. Subgraphs and supergraphs 224 7.2.1. MAX INDEPENDENT SET 224 7.2.2. MAX CLIQUE 224 7.2.3. MAX INDEPENDENT SEQUENCE 225 7.2.4. MAX INDUCED SUBGRAPH WITH PROPERTY π 225 7.2.5. MIN VERTEX DELETION TO OBTAIN SUBGRAPH WITH PROPERTY π 225 7.2.6. MIN EDGE DELETION TO OBTAIN SUBGRAPH WITH PROPERTY π 226 7.2.7. MAX CONNECTED SUBGRAPH WITH PROPERTY π 226 7.2.8. MIN VERTEX DELETION TO OBTAIN CONNECTED SUBGRAPH WITH PROPERTY π 226 7.2.9. MAX DEGREE-BOUNDED CONNECTED SUBGRAPH 226 7.2.10. MAX PLANAR SUBGRAPH 227 7.2.11. MIN EDGE DELETION k-PARTITION 227 7.2.12. MAX k-COLORABLE SUBGRAPH 227 7.2.13. MAX SUBFOREST 228 7.2.14. MAX EDGE SUBGRAPH or DENSE k-SUBGRAPH 228 7.2.15. MIN EDGE K-SPANNER 228 7.2.16. MAX k-COLORABLE INDUCED SUBGRAPH 229 7.2.17. MIN EQUIVALENT DIGRAPH 229 7.2.18. MIN CHORDAL GRAPH COMPLETION 229 7.3. Iso- and other morphisms 229 7.3.1. MAX COMMON SUBGRAPH 229 7.3.2. MAX COMMON INDUCED SUBGRAPH 230 7.3.3. MAX COMMON EMBEDDED SUBTREE 230 7.3.4. MIN GRAPH TRANSFORMATION 230 7.4. Cuts and connectivity 231 7.4.1. MAX CUT 231 7.4.2. MAX DIRECTED CUT 231 7.4.3. MIN CROSSING NUMBER 231 7.4.4. MAX k-CUT 232 7.4.5. MIN k-CUT 233 7.4.6. MIN NETWORK INHIBITION ON PLANAR GRAPHS 233 7.4.7. MIN VERTEX k-CUT 234 7.4.8. MIN MULTI-WAY CUT 234 7.4.9. MIN MULTI-CUT 234 7.4.10. MIN RATIO-CUT 235 7.4.11. MIN b-BALANCED CUT 236 7.4.12. MIN b-VERTEX SEPARATOR 236 7.4.13. MIN QUOTIENT CUT 236 7.4.14. MIN k-VERTEX CONNECTED SUBGRAPH 236 7.4.15. MIN k-EDGE CONNECTED SUBGRAPH 237 7.4.16. MIN BICONNECTIVITY AUGMENTATION 237 7.4.17. MIN STRONG CONNECTIVITY AUGMENTATION 237 7.4.18. MIN BOUNDED DIAMETER AUGMENTATION 237 Appendix A. Mathematical Preliminaries 239 A.1. Sets, relations and functions 239 A.2. Basic concepts from graph-theory 242 A.3. Elements from discrete probabilities 246 Appendix B. Elements of the Complexity and the Approximation Theory 249 B.1. Problem, algorithm, complexity 249 B.2. Some notorious complexity classes 250 B.3. Reductions and NP-completeness 251 B.4. Approximation of NP-hard problems 252 Bibliography 255 Index 261
£125.06
Springer Nature Switzerland AG Random Graphs, Phase Transitions, and the Gaussian Free Field: PIMS-CRM Summer School in Probability, Vancouver, Canada, June 5–30, 2017
Book SynopsisThe 2017 PIMS-CRM Summer School in Probability was held at the Pacific Institute for the Mathematical Sciences (PIMS) at the University of British Columbia in Vancouver, Canada, during June 5-30, 2017. It had 125 participants from 20 different countries, and featured two main courses, three mini-courses, and twenty-nine lectures. The lecture notes contained in this volume provide introductory accounts of three of the most active and fascinating areas of research in modern probability theory, especially designed for graduate students entering research: Scaling limits of random trees and random graphs (Christina Goldschmidt) Lectures on the Ising and Potts models on the hypercubic lattice (Hugo Duminil-Copin) Extrema of the two-dimensional discrete Gaussian free field (Marek Biskup) Each of these contributions provides a thorough introduction that will be of value to beginners and experts alike.Table of ContentsScaling Limits of Random Trees and Random Graphs (C. Goldschmidt).- Lectures on the Ising and Potts Models on the Hypercubic Lattice (H. Duminil-Copin).- Extrema of the Two-Dimensional Discrete Gaussian Free Field (M. Biskup).
£151.99
