Core-Free, Rank Two Coset Geometries from Edge-Transitive Bipartite Graphs

TL;DR

This paper explores the construction of core-free, rank two coset geometries from edge-transitive bipartite graphs, especially focusing on 3-valent and 4-valent cases, and provides detailed properties of these geometries.

Contribution

It introduces a method to derive all associated coset geometries from given edge-transitive graphs, reversing the known Levi graph relationship.

Findings

Constructed coset geometries from 3-valent and 4-valent graphs.

Provided comprehensive tables of properties for these geometries.

Focused on graphs with small vertex-stabilizers in the 4-valent case.

Abstract

It is known that the Levi graph of any rank two coset geometry is an edge-transitive graph, and thus coset geometries can be used to construct many edge transitive graphs. In this paper, we consider the reverse direction. Starting from edge- transitive graphs, we construct all associated core-free, rank two coset geometries. In particular, we focus on 3-valent and 4-valent graphs, and are able to construct coset geometries arising from these graphs. We summarize many properties of these coset geometries in a sequence of tables; in the 4-valent case we restrict to graphs that have relatively small vertex-stabilizers.