Distance Sequences of Locally Infinite Primitive Graphs

TL;DR

This paper classifies the possible distance sequences in locally infinite primitive graphs, revealing that locally uncountable graphs have constant distance sequences until termination, and establishing constraints for locally finite infinite graphs.

Contribution

It provides the first classification of distance sequences in locally infinite primitive graphs, including new constraints and properties.

Findings

Locally uncountable primitive graphs have constant distance sequences until termination.

Constraints are established for the distance sequences of locally finite infinite graphs.

The classification advances understanding of the structure of primitive graphs.

Abstract

A graph is called primitive if its automorphism group acts primitively on the vertex set. In this paper, we prove a classification of the possible distance sequences of locally infinite primitive graphs. In particular we show that if a primitive graph is locally uncountable, the distance sequence is constant until it terminates. We also prove a constraint on the distance sequences of locally finite infinite graphs.