Locally s -distance transitive graphs and pairwise transitive designs

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 general construction for this case.

Pairwise transitive 2-designs

“…In this case, the group G is affine or almost simple. Recall from [18] the following definition (which is valid for any t-design not just 2-designs). (1) G is 2-transitive of affine type on P, and D is N-nicely affine for N the translation subgroup of G;…”

“…Assume (p, d) is one of (2, 4), (2, 10), (2,12), (2,18), so r is 5, 11, 13, 19 respectively. Considering for i all the prime powers dividing 2 d −1 d+1 , which are 3,9,27,5,7,31,73, and taking for k the values 2, 6, 18, 4, 3, 5, 18 respectively (regardless of the value of j), we see that condition (B) fails, so by our observation above i = 1 and G 0 = ΓL(1, 2 d ).…”

“…Our aim is to classify all 2-designs D = (P, B, I) which have a strong form of symmetry on pairs from P ∪ B, defined in the following paragraph. These so-called pairwise transitive designs were introduced in [18], where they arose in the study of locally 4-distance transitive graphs. The definition for 2-designs is a bit simpler than the one given in [18] since for a 2-design all point-pairs are incident with λ common blocks.…”

“…In [18], we showed that a graph is locally (G, 4)-distance transitive and has a normal star quotient K 1,r for r ≥ 3 if and only if its adjacency design is G-pairwise transitive and N-nicely affine with r parallel classes of blocks (see Definition 4.7). Tables 1 and 2 enables us to classify these graphs in the case where the adjacency design is a 2-design.…”

On 2-distance-transitive circulants

“…Both distance-transitive graphs and 2arc-transitive graphs have been extensively studied, see [3,15]. The investigation of 2-distance-transitive graphs was initiated recently, see [4][5][6].…”