Separating rank 3 graphs

TL;DR

This paper classifies rank 3 graphs that do not meet certain bounds, resolving related group separation questions and providing new examples of specific group types.

Contribution

It offers a classification of rank 3 graphs failing key bounds and addresses open questions in group separation and synchronization.

Findings

Classified rank 3 graphs failing Delsarte or Hoffman bounds

Resolved separation questions for rank 3 primitive groups

Provided new examples of affine type groups that are synchronising but not $\

Abstract

We classify, up to some notoriously hard cases, the rank 3 graphs which fail to meet either the Delsarte or the Hoffman bound. As a consequence, we resolve the question of separation for the corresponding rank 3 primitive groups and give new examples of synchronising, but not $\mathbb{Q}\mathrm{I}$, groups of affine type.