Front Cover

A Survey of Combinatorial Theory

Copyright Page

Table of Contents

Dedication

Preface

List of Publications by R. C. Bose

CHAPTER 1. Some Classical and Modern Topics in Finite Geometrical Structures

1. Introduction

2. Finite projective and affine planes

3. Other geometric structures

4. Galois geometries

References

CHAPTER 2. Balanced Hypergraphs and Some Applications to Graph Theory

1. Introduction

2. General definitions

3. Balanced hypergraphs

4. The main result

References

CHAPTER 3. A Strongly Regular Graph Derived from the Perfect Ternary Golay Code

1. Introduction

2. Strongly regular graphs with

3. The perfect ternary Golay code

4. Constitution of the 243-graph

References

CHAPTER 4. Characterization Problems of Combinatorial Graph Theory

1. Introduction

2. Association schemes and partially balanced designs

3. Two class association schemes and strongly regular graphs

4. The triangular association scheme and the line graph of a complete graph

5. The Lr(k) association scheme, and net graphs

6. Embedding theorems

7. Partial geometries

8. Characterization of some classes of edge regular, and other related graphs

9. Characterization of classes of graphs by the eigenvalues of their adjacency matrices

References

CHAPTER 5. Line-minimal Graphs with Cyclic Group

Abstract

1. Introduction

2. Line-minimal

3. A general lemma for n a prime power

4. Line-minimal p-cyclic graphs

5. Line-minimal n-cyclic graphs when n is a prime power

6. Line-minimal n-cyclic graphs for n not a prime power

7. The numbers

References

CHAPTER 6. Circle Geometry in Higher Dimensions

1. Introduction

2. The axioms

3. A class of circle-geometries

References

CHAPTER 7. Bose as Teacher—The Early Years

CHAPTER 8. Construction of Symmetric Hadamard Matrices

1. Introduction

References

CHAPTER 9. Cayley Diagrams and Regular Complex Polygons

Abstract

1. Cayley diagrams

2. Regular complex polygons

3. Real representations

References

CHAPTER 10. Combinatorial Problems in Finite Abelian Groups

References

CHAPTER 11. A q-Analog of the Partition Lattice

1. Introduction

2. Preliminaries

3. The lattice of partitions

4. The q-partition lattice Qn

5. Representation of Qn

6. The characteristic polynomial and Whitney numbers of Qn

7. An application to design

References

CHAPTER 12. Problems and Results on Combinatorial Number Theory

References

CHAPTER 13. Dissection Graphs of Planar Point Sets

1. Introduction

2. Some structural properties of k-graphs

3. On the number of vertices of Gk

4. On the number of edges of Gk

5. Generalizations and problems

CHAPTER 14. Some Combinatorial Problems and Results in Fractional Replication

1. Introduction

2. Some combinatorial problems of singular saturated main effect plans of the 2n factorial

3. Frequency distribution of ones in (0, 1)-matrices having distinct rows

Note added in proof

References

CHAPTER 15. The Bose-Nelson Sorting Problem

References

CHAPTER 16. Nombres d'Euler et Permutations Alternantes

1. Introduction

2. Les polynomes d'Andre

3. Les permutations d'Andre

4. Remarques

References

CHAPTER 17. Some Combinatorial Aspects of Coding Theory

1. Introduction

2. Plotkin's maximal binary codes

3. Majority decoding of linear block codes

Note added in proof

Acknowledgements

References

CHAPTER 18. Some Augmentations of Bose-Chaudhuri Error Correcting Codes

Summary

1. Introduction

2. Augmented Bose-Chaudhuri codes that correct single and multiple bursts of errors

3. Bose-Chaudhuri codes augmented by means of Galois cycles

References

CHAPTER 19. On Order Statistics and Some Applications of Combinatorial Methods in Statistics

1. Introduction

2. Basic distribution theory

3. Moments of order statistics and bounds

4. Asymptotic theory

5. Combinatorial methods in order statistics

6. Some combinatorial methods in fluctuation theory and the distribution of the maxima

7. Some estimation and hypothesis testing problems based on order statistics

8. Multiple decision (selection and ranking) problems

Acknowledgement

References

CHAPTER 20. Construction of Block Designst

1. Introduction

2. Block designs with r < 15

3. Some recent constructions of block designs

4. Various approaches to construction of designs

Note added in proof

References

CHAPTER 21. A Survey of Graphical Enumeration Problems

Abstract

1. Setting the stage

2. Digraphs

3. Graphs with given structural properties

4. Graphs with given parameter

5. Subgraphs of a given graph

6. Supergraphs of a given graph

7. Enumeration equivalent to the four color conjecture

Conclusion

8. Summary of problem areas

References

CHAPTER 22. On Spectrally Bounded Graphs

1. Introduction

2. Statement of characterizations

3. Proof of Theorem

4. Proof of the Theorem (continued)

References

CHAPTER 23. Combinatorial Search Problems

1. Introduction

2. Connections with noiseless encoding

3. Results

4. Random search

5. Open problems

References

CHAPTER 24. Some Problems in Combinatorial Geometry

Bibliography

CHAPTER 25. Recent Developments in India in the Construction of Confounded Asymmetrical Factorial Designs

1. Introduction

2. Definitions

3. The methods of finite fields and finite geometries

4. The use of incomplete block designs

5. The method of fractional replication of symmetrical factorial designs

6. Some further problems

References

CHAPTER 26. Combinatorial Problems in Communication Networks

Abstract

1. Introduction

2. Description of networks

3. Combinatorial problems in real time communications

4. Core storage and preprocessing

5. Related problems and technological advances

References

CHAPTER 27. On Hadamard Difference Sets

References

CHAPTER 28. On the Randomization of Block Designs

References

CHAPTER 29. Some Combinatorial Problems of Arrays and Applications to Design of Experimentst

1. Introduction

2. Orthogonal array of strength d

3. Semi orthogonal array

4. Balanced array

5. Semi balanced array

6. Partially balanced array

7. Orthogonal arrays with a variable number of symbols in rows

References

CHAPTER 30. The Existence of Resolvable Block Designs

1. Introduction

2. Necessary conditions and some historical remarks

3. Completed designs and some preliminary results

4. Special constructions; proof of the main theorem

5. An application: maximum subdesigns

6. Generalizations and unsolved problems

References

CHAPTER 31. Variants of (v, k, λ)-Designs

Summary

1. Introduction

2. Extensions of a theorem of De Bruijn and Erdös

3. The matrix equation

4. Multiplicative designs

5. The matrix equation

References

CHAPTER 32. Coding Need: A System's Concept

1. Introduction

2. A coder/multiplexer concept

3. The general problem

4. Summary

References

CHAPTER 33. On the Problem of Construction and Uniqueness of Saturated Designs

1. Introduction and summary

2. The values

References

CHAPTER 34. Strongly Regular Graphs and Symmetric 3-Designs

Main results

References

CHAPTER 35. Balanced Arrays and Orthogonal Arrays

Summary

1. Introduction

2. Preliminaries

3. Analysis of the SDE

4. Arrays with μ2 = 1

5. The intermediate diophantine equations (IDE)

6. Arrays with μι = 2, 3

7. Arrays with μ2 = 4,5,6

8. Orthogonal arrays

References

CHAPTER 36. On Software Validation

References

CHAPTER 37. The Enumerative Theory of Planar Maps

1. Planar maps

2. Rooted maps

3. Rooted trees

4. Rooted 2-connected inner-triangulations

5. The quadratic method

6. Changes of connectivity

7. Planar maps with a partition

8. Unrooted planar maps

References

CHAPTER 38. A Construction for Room Squares

The Room square existence problem

References

Author Index