Infinite primitive and distance transitive directed graphs of finite out-valency

TL;DR

This paper characterizes the descendant sets of vertices in infinite, primitive, distance transitive directed graphs with finite out-valency, establishing their structural properties and confirming a conjecture about their countability.

Contribution

It provides a detailed structure theory for these digraphs and proves that only countably many such descendant sets exist, confirming a previous conjecture.

Findings

Countably many isomorphism types of descendant sets.

Structural properties characterizing descendant sets.

Partial converse conditions for realization as descendant sets.

Abstract

We give certain properties which are satisfied by the descendant set of a vertex in an infinite, primitive, distance transitive digraph of finite out-valency and provide a strong structure theory for digraphs satisfying these properties. In particular, we show that there are only countably many possibilities for the isomorphism type of such a descendant set, thereby confirming a conjecture of the first Author. As a partial converse, we show that certain related conditions on a countable digraph are sufficient for it to occur as the descendant set of a primitive, distance transitive digraph.