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

On hypergroups of order at most 6

This thesis surveys recent results on hypergroups as defined by Frédéric Marty in [3] and [4] and their relation to association schemes as presented in [5]. We show that every association scheme is a hypergroup. Then, we compile a few general results on hypergroups needed for our investigation of hypergroups with three, four and six elements. From [1] and [7], we give examples of hypergroups that do not come from finite schemes and from no scheme at all. Our main result occurs when considering hypergroups S with six elements that have a non-normal closed subset T of order 2 with three cosets. Since such class of hypergroups is too large to be completely described, we investigate a subclass S determined in [7]. We found that at least four hypergroups in this class come from finite schemes. For such purposes, we use the Hanaki-Miyamoto Classification of Small Association Schemes; cf. [8]

On strong homotopy for quasi-schemoids

A quasi-schemoid is a small category with a particular partition of the set of morphisms. We define a homotopy relation on the category of quasi-schemoids and study its fundamental properties. As a homotopy invariant, the homotopy set of self-homotopy equivalences on a quasi-schemoid is introduced. The main theorem enables us to deduce that the homotopy invariant for the quasi-schemoid induced by a finite group is isomorphic to the automorphism group of the given group.Comment: 12 page

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

Data curation standards and social science occupational information resources

Occupational information resources - data about the characteristics of different occupational positions - are widely used in the social sciences, across a range of disciplines and international contexts. They are available in many formats, most often constituting small electronic files that are made freely downloadable from academic web-pages. However there are several challenges associated with how occupational information resources are distributed to, and exploited by, social researchers. In this paper we describe features of occupational information resources, and indicate the role digital curation can play in exploiting them. We report upon the strategies used in the GEODE research project (Grid Enabled Occupational Data Environment, http://www.geode.stir.ac.uk). This project attempts to develop long-term standards for the distribution of occupational information resources, by providing a standardized framework-based electronic depository for occupational information resources, and by providing a data indexing service, based on e-Science middleware, which collates occupational information resources and makes them readily accessible to non-specialist social scientists