Finite graphs of valency 4 and girth 4 admitting 1 2-transitive group actions, that is vertex-and edge-but not arc-transitive group actions, are investigated. A graph is said to be 1 2-transitive if its automorphism group acts 1 2-transitively. There is a natural (balanced) orientation of the edge set of a 1 2-transitive graph induced and preserved by its auto-morphism group. Among others it is...

An in nite family of cubic edge transitive but not vertex transitive graph graphs with edge stabilizer isomorphic to Z is constructed

A general method is described which gives rise to highly symmetric tessellations of the Cantor sphere, i.e., the 2-sphere with the Cantor set removed and endowed with the hyperbolic geometry with constant negative curvature. These tessellations correspond to almost vertextransitive planar graphs with infinitely many ends. Their isometry groups have infinitely many ends and are free products wit...

Slovenija Abstract An innnite family of cubic edge-transitive but not vertex-transitive graph graphs with edge stabilizer isomorphic to Z Z 2 is constructed.

Abstract We consider a finite, connected and simple graph Γ that admits vertex-transitive group of automorphisms

The class of all connected vertex-transitive graphs with finite valency forms a metric space under a natural combinatorially defined metric. We prove some basic properties of this metric space and discuss the structure of graphs which are limit points of the subset consisting of all finite graphs that admit a vertex-primitive group of automorphisms. A description of these limit graphs would pro...

An interesting fact is that almost all the connected 2-arc-transitive nonnormal Cayley graphs on nonabelian simple groups with small valency or prime (provided solvable vertex stabilizers) are alternating An. This naturally motivates study of for arbitrary valency. In this paper, we characterize automorphism such graphs. particular, show a non-complete (G,2)-arc-transitive graph G simple, socle...

A digraph D is primitive if, for some positive integer r, there is a u ! v walk of length r for every pair u; v of vertices of D. The minimum such r is called the exponent of D, denoted exp(D). We prove that if a primitive digraph D of order n is an Abelian Cayley digraph with degree k 4, then exp(D) max n 5; n dk=2e 1 o : If we assume only that the primitive digraph D is vertex-transitive, the...

When one thinks of objects with a significant level of symmetry it is natural to expect there to be a simple classification. However, this leads to an interesting problem in that research has revealed the existence of highly symmetric objects which are very complex when considered within the framework of Borel complexity. The tension between these two seemingly contradictory notions leads to a ...

We show that the critical value for the contact process on a vertex-transitive graph G with finitely many edges added and/or removed is the same as the critical value for the contact process on G. This gives a partial answer to a conjecture of Pemantle and Stacey.