Applications of Schur rings in algebraic combinatorics: graphs, partial difference sets and cyclotomic schemes (Q2753136)

The concept of Schur rings was introduced in 1933 by I. Schur. For several decades applications of Schur rings were restricted to the investigation of permutation groups. Starting in the fifties, similar concepts like association schemes, cellular algebras and coherent configurations were introduced independently by different authors. They were used for various questions in algebraic combinatorics and statistics. In this thesis three different tasks which are related to these concepts are considered: (1) characterization of commuting graphs, (2) consideration of strongly regular graphs and partial difference sets, and (3) investigation of cyclotomic schemes. The first part deals with graphs with commuting adjacency matrices. Here, we give results for commuting regular graphs and discuss the case of non-regular graphs. The second part deals with the construction of partial difference sets by using strongly regular Cayley graphs. Theoretical and computational approaches are discussed and all regular partial difference sets in groups up to order 49 are determined. Moreover, regular partial difference sets for strongly regular graphs up to 255 vertices which have primitive automorphism group, are constructed. In the third part an algorithm for the determination of cellular subrings of cellular rings is adopted for cyclotomic schemes. This algorithm uses the information given by cyclotomic numbers for the complete theoretical determination of all subschemes. The determination of subschemes for cyclotomic schemes with three, four and six classes is described in detail.