Classification of small class association schemes coming from certain combinatorial objects

We explore two- or three-class association schemes. We study aspects of the structure of the relation graphs in association schemes which are not easily revealed by their parameters and spectra. The purpose is to develop some combinatorial methods to characterize the graphs and classify the association schemes, and also to delve deeply into several specific classification problems. We work with several combinatorial objects, including strongly regular graphs, distance-regular graphs, the desarguesian complete set of mutually orthogonal Latin squares, orthogonal arrays, and symmetric Bush-type Hadamard matrices, all of which give rise to many small-class association schemes. We work within the framework of the theory of association schemes.;Our focus is placed on the search for all isomorphism classes of association schemes and characterization of small-class association schemes of specific order. In particular, we examine two-class association schemes (strongly regular graphs) of order 64 and their three-class fission schemes. After we collect \u27feasible\u27 parameter sets for the putative association schemes, we make an attempt to check the realization (existence) of the parameter sets and describe the structure of the schemes chiefly by investigating the structure of their relation graphs. In the course of this thesis, we find a new way to construct orthogonal arrays and investigate their implications for strongly regular graphs, symmetric Bush-type Hadamard matrices, and three-class association schemes. We obtain several results regarding the characterization and classification of two- or three-class association schemes of order 64

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Permutation group approach to association schemes

AbstractWe survey the modern theory of schemes (coherent configurations). The main attention is paid to the schurity problem and the separability problem. Several applications of schemes to constructing polynomial-time algorithms, in particular, graph isomorphism tests, are discussed