Presented by Mr. Daniel KALMANOVICH
Type: Oral presentation
Track: Association Schemes
The starting point of this subject are the famous combinatorial objects known as two-graphs. We introduce a generalization of two-graphs due to D. G. Higman, called $t$-graphs, where our emphasis is on regular $t$-graphs with $t=3$. Higman's definition is initially formulated in cohomological terms. It turns out that regular 3-graphs are closely related to a special class of antipodal distance regular graphs (ADRG's) of diameter 3. To reveal this relation, we (following Higman) study certain rank 4 and rank 6 association schemes. We prove the equivalence of regular $3$-graphs, association schemes of rank 6 with certain structure constants, and cyclic $(n,3,c_2)$-covers (in the sense of C. D. Godsil and A. D. Hensel). Finally, new examples of regular $3$-graphs are presented, based on the construction of ADRG's suggested by M. Klin and C. Pech, and classifications of class-regular symmetric transversal designs, due to V. Tonchev et al. This is a joint project with M. Klin.