  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.

Algebraic Geometry;Combinatorics;Group Theory