9 research outputs found
Core-Free, Rank Two Coset Geometries from Edge-Transitive Bipartite Graphs
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
Locally arc-transitive graphs of valence with trivial edge kernel
In this paper we consider connected locally -arc-transitive graphs with
vertices of valence 3 and 4, such that the kernel of the action
of an edge-stabiliser on the neighourhood is
trivial. We find nineteen finitely presented groups with the property that any
such group is a quotient of one of these groups. As an application, we
enumerate all connected locally arc-transitive graphs of valence on at
most 350 vertices whose automorphism group contains a locally arc-transitive
subgroup with
Recognizing algebraically constructed graphs which are wreath products.
It is known that a Cayley digraph of an abelian group A is isomorphic to a nontrivial wreath product if and only if there is a proper nontrivial subgroup B of A such that the connection set without B is a union of cosets of B in A. We generalize this result to Cayley digraphs of nonabelian groups G by showing that such a digraph is isomorphic to a nontrivial wreath product if and only if there is a proper nontrivial subgroup H of G such that S without H is a union of double cosets of H in G. This result is proven in the more general situation of a double coset digraph (also known as a Sabidussi coset digraph.) We then give applications of this result which include obtaining a graph theoretic definition of double coset digraphs, and determining the relationship between a double coset digraph and its corresponding Cayley digraph. We further expand the result obtained for double coset digraphs to a collection of bipartite graphs called bi-coset graphs and the bipartite equivalent to Cayley graphs called Haar graphs. Instead of considering when this collection of graphs is a wreath product, we consider the more general graph product known as an X-join by showing that a connected bi-coset graph of a group G with respect to some subgroups L and R of G is isomorphic to an X-join of a collection of empty graphs if and only if the connection set is a union of double cosets of some subgroups N containing L and M containing R in G. The automorphism group of such -joins is also found. We also prove that disconnected bi-coset graphs are always isomorphic to a wreath product of an empty graph with a bi-coset graph