Search the repository

Abstract: We study n-dimensional matrices with {0, 1}-entries (n-cubes) such that all their 2-dimensional slices are incidence matrices of symmetric designs. A known construction of these objects obtained from difference sets is generalized so that the resulting n-cubes may have inequivalent slices. For suitable parameters, they can be transformed into n-dimensional Hadamard matrices with this property. In contrast, previously known constructions of n-dimensional designs all give examples with equivalent slices.
Keywords: symmetric design, difference set, Hadamard matrix
Published in RUP: 21.10.2025; Views: 778; Downloads: 8
Full text (121,51 KB)

Abstract: The main result of this article is a classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such graphs, which are called trivial. These are: complete graphs, complete bipartite graphs, complete bipartite graphs with the edges of a 1-factor removed, and cycles. It is proved that every non-trivial distance-regular Cayley graph on a dihedral group is bipartite, non-antipodal, has diameter 3 and arises either from a cyclic di#erence set, or possibly (if any such exists) from a dihedral difference set satisfying some additional conditions. Finally, all distance-transitive Cayley graphs on dihedral groups are determined. It transpires that a Cayley graph on a dihedral group is distance-transitive if and only if it is trivial, or isomorphic to the incidence or to the non-incidence graph of a projective space ▫$\mathrm{PG}_{d-1} (d,q)$▫, ▫$d \ge 2$▫, or the unique pair of complementary symmetric designs on 11 vertices.
Keywords: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set
Published in RUP: 10.07.2015; Views: 6139; Downloads: 121
Link to full text