Locally s-distance transitive graphs and pairwise transitive designs

TL;DR

This paper explores the relationship between locally s-distance transitive graphs with star quotients and highly symmetric pairwise transitive designs, revealing structural properties and group actions involved.

Contribution

It establishes an equivalence between certain graphs and designs, characterizes the acting groups, and provides a construction method for these symmetric structures.

Findings

Graphs with star quotients correspond to nicely affine pairwise transitive designs.

A group acting regularly on design points must be abelian.

A general construction method for such designs is provided.

Abstract

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.