2 research outputs found
Locally s-distance transitive graphs and pairwise transitive designs
The study of locally s-distance transitive graphs initiated by the authors in
previous work, identified that graphs with a star quotient are of particular
interest. This paper shows that the study of locally s-distance transitive
graphs with a star quotient is equivalent to the study of a particular family
of designs with strong symmetry properties that we call nicely affine and
pairwise transitive. We show that a group acting regularly on the points of
such a design must be abelian and give a general construction for this case.Comment: 22 pages, has been accepted for publication in J. Comb. Theory Series
Designs from Paley graphs and Peisert graphs
Fix positive integers and so that is prime, , and
(mod ). Fix a graph as follows: If is odd or
(mod ), let be the -vertex Paley graph; if is
even and (mod ), let be either the -vertex Paley graph or
the -vertex Peisert graph. We use the subgraph structure of to construct
four sequences of -designs, and we compute their parameters. Letting
denote the number of -vertex cliques in , we create additional
sequences of -designs from , and show how to express their parameters in
terms of only and . We find estimates and precise asymptotics for
in the case that is a Paley graph. We also explain how the presented
techniques can be used to find many additional -designs in . All
constructed designs contain no repeated blocks