Isomorphisms, Symmetry and Computations in Algebraic Graph Theory: Pilsen, Czech Republic, October 3–7, 2016 (Springer Proceedings in Mathematics & Statistics)

Preface
Acknowledgements
Contents
Orientably-Regular Maps on Twisted Linear Fractional Groups
	1 Introduction
	2 The Twisted Linear Groups M(q2)
	3 Twisted Subgroups of M(q2)
	4 Representatives of Twisted Elements
	5 Conjugacy of Representatives of Twisted Elements
	6 Conjugacy Classes of Twisted Elements
	7 Non-singular Pairs and Twisted Subgroups
	8 Orbits of Non-singular Pairs: The Diagonal Case
	9 Orbits of Non-singular Pairs: The Off-Diagonal Case
	10 Enumeration of Orientably-Regular Maps on M(q2)
	11 Enumeration of Reflexible Maps
		11.1 Conjugating Involutions
		11.2 Enumeration
		11.3 Counting Reflexible Maps
	12 Remarks
	References
From Schur Rings to Constructive  and Analytical Enumeration of Circulant Graphs with Prime-Cubed Number  of Vertices
	1 Introduction
		1.1 First Definitions
		1.2 The Structural Approach: An Introduction
		1.3 The Multiplier Approach: An Introduction
	2 Automorphism Groups of Prime-Cubed Circulants
		2.1 Wreath Products
		2.2 Affine Overgroups of (mathbbZn, mathbbZn)
		2.3 Main Results
		2.4 Brief Historical Summary
	3 The Structural Approach: The Case p3 for p=3
	4 The Multiplier Approach for n=p3
		4.1 The Main Isomorphism Theorem
		4.2 Representation and Computational Implementation  of the Main Isomorphism Theorem, p=3,5
		4.3 Results for the Cases p=3, 5, Undirected and Directed
		4.4 Discussion of Results: Unexpected Patterns  and Corollaries
	References
A Note on a Problem of L. Martínez on Almost-Uniform Partial Sum Families
	1 Introduction
	2 Preliminaries
	3 The Main Result
	References
The Paulus–Rozenfeld–Thompson Graph on 26 Vertices Revisited and Related Combinatorial Structures
	1 Introduction
	2 Basic Concepts
	3 Computer-Aided Tools and Catalogues with Data
	4 Brief Historical Overview
	5 Outline of the Main Goal
	6 Alternating Group A5 and Its Subgroups
	7 Transitive Actions of A5
		7.1 Action of Degree 5
		7.2 Action of Degree 6
		7.3 Action of Degree 10
		7.4 Action of Degree 15
		7.5 Action of Degree 20
		7.6 Action of Degree 30
		7.7 Action of Degree 60
	8 Master Association Scheme of Rank 8 on 20 Vertices
	9 Mergings of mathcalM and Their Automorphism Groups
	10 A5 as the Group of Rotations of the Dodecahedron
	11 Some Diagrams of the Dodecahedron and Related Graphs
	12 More About the Basic Graphs of the Master Association Scheme mathcalM
	13 Directed Strongly Regular Graphs on 20 Vertices
	14 Brouwer\'s (0,2)-Graph N6.3
	15 Master Scheme mathcalM as Infima of Rank 7 and Rank 6 Schemes
	16 Four Regular Two-Graphs on 26 Vertices and Beyond
	17 Regular Two-Graph D and Its Group
	18 From the Two-Graph D to the Steiner System S(3,6,26)
	19 Association Schemes on Circles of S(3,6,26)
	20 Some Association Schemes on 52 Points and Beyond
	21 Master Coherent Configuration W on 26 Vertices
	22 The Graph T as Merging of W
	23 Some Diagrams of the Graph T
	24 Interactions Between Some Diagrams of T, Its Models and Properties
	25 Hoffman Cocliques and Association Schemes on 260 Copies of Graph T
	26 A Model for the Pair of Classical Generalized Quadrangles of Order 5
	27 Miscellanea
	28 Concluding Discussion
	References
Paley and the Paley Graphs
	1 Introduction
	2 Definition and Properties of the Paley Graphs
	3 Jacobsthal and Sums of Squares
	4 Perron, Brauer, Hopf and Schur
	5 Hadamard Matrices and Designs
	6 Paley
	7 Paley\'s Hadamard Matrix Constructions
		7.1 Paley\'s First Construction
		7.2 Paley\'s Second Construction
		7.3 Paley\'s Third Construction
		7.4 Todd\'s Paper
		7.5 Coxeter\'s Paper
	8 The Origin of the Paley Graphs
		8.1 Sachs
		8.2 Erdős and Rényi
		8.3 Naming the Paley Graphs
		8.4 Pseudo-Paley Graphs
	9 The Automorphism Group of a Paley Graph
		9.1 Characterising the Automorphisms
		9.2 Carlitz
		9.3 Subsequent Proofs
		9.4 Generalisations of the Main Theorem
		9.5 Muzychuk\'s Proof of the Main Theorem
		9.6 Generalised Paley Graphs
		9.7 The Automorphism Group of the Paley Tournament
		9.8 Automorphism Groups of Hadamard Matrices
	10 Attribution and Terminology
	References
Automorphism Groups of Paley Graphs and Cyclotomic Schemes
	1 Introduction
	2 Cyclotomic Schemes and Their Automorphisms
		2.1 Field Functions with Small |A|
		2.2 Proof of the Theorem 1
	References
Recognizing and Testing Isomorphism  of Cayley Graphs over an Abelian Group  of Order 4p in Polynomial Time
	1 Introduction
	2 Coherent Configurations
		2.1 Notation
		2.2 Main Definitions
		2.3 Fibers
		2.4 Restrictions and Quotients
		2.5 Isomorphisms
		2.6 Direct Sum
		2.7 Tensor Product
		2.8 Generalized Wreath Product
		2.9 Algorithms
	3 Quasi-trivial Coherent Configurations
	4 Cayley Schemes
		4.1 General Facts
		4.2 Cayley Schemes over E4timesCp
	5 Extension of Coherent Configuration and WL-Algorithm
		5.1 Partial Order
		5.2 WL-Algorithm
		5.3 Examples of Extensions
		5.4 Extension of Cayley Schemes
	6 Structure of Cayley Schemes over E4timesCp
		6.1 Principal Equivalence Relation
		6.2 A Principal Equivalence Relation of Large Valency
		6.3 Summary
	7 Finding a Representative Set of Semiregular Cp-Subgroups
		7.1 The Main Algorithm
		7.2 Analysis of the Running Time
		7.3 The Correctness of the MS
	8 Proof of Theorem 2
	References
Tatra Schemes and Their Mergings
	1 Introduction
	2 Preliminaries
		2.1 Association Schemes
		2.2 Difference Sets
	3 The Main Construction
	4 Distance-Regular Basis Graphs of W
	5 A Rank 4 Merging Based on Difference Sets
	6 Mergings Based on S-Rings
	7 Non-commutative Schemes of Rank 6
	8 Discussion
	References