We show that the directed labelled Cayley graphs coincide with the rooted deterministic vertextransitive simple graphs. The Cayley graphs are also the strongly connected deterministic simple graphs of which all vertices have the same cycle language, or just the same elementary cycle language. Under the assumption of the axiom of choice, we characterize the Cayley graphs for all group subsets as...

An old conjecture of Marušič, Jordan and Klin asserts that any finite vertextransitive graph has a non-trivial semiregular automorphism. Marušič and Scapellato proved this for cubic graphs. For these graphs, we make a stronger conjecture, to the effect that there is a semiregular automorphism of order tending to infinity with n. We prove that there is one of order greater than 2.

In this paper, we introduce a class of vertextransitive graphs induced by Quasigroups whose vertices are cosets. Also, many graph properties are expressed in terms of algebraic properties. This did not attract much attention in the literature.

Many large graphs can be constructed from existing smaller graphs by using graph operations, for example, the Cartesian product and the lexicographic product. Many properties of such large graphs are closely related to those of the corresponding smaller ones. In this short note, we give some properties of the lexicographic products of vertextransitive and of edge-transitive graphs. In particula...

The generalized Petersen graphs (GPGs) which have been invented by Watkins, may serve for perhaps the simplest nontrivial examples of “galactic” graphs, i.e. those with a nice property of having a semiregular automorphism. Some of them are also vertextransitive or even more highly symmetric, and some are Cayley graphs. In this paper, we study a further extension of the notion of GPGs with the e...

Mirror graphs were introduced by Brešar et al. in 2004 as an intriguing class of graphs: vertextransitive, isometrically embeddable into hypercubes, having a strong connection with regular maps and polytope structure. In this article we settle the structure of mirror graphs by characterizing them as precisely the Cayley graphs of the finite Coxeter groups or equivalently the tope graphs of refl...

In this paper, we prove that every vertex-transitive graph can be expressed as the edge-disjoint union of symmetric graphs. We define a multicycle graph and conjecture that every vertex-transitive graph cam be expressed as the edge-disjoint union of multicycles. We verify this conjecture for several subclasses of vertextransitive graphs, including Cayley graphs, multidimensional circulants, and...

In this paper, we extend the notion of a circulant to a broader class of vertex.transitive graphs, which we call multidimensional circulants. This new class of graphs is shown to consist precisely of those vertextransitive graphs with an automorphism group containing a regular abelian subgroup. The result is proved using a theorem of Sabidussi which shows how to recover any vertex-transitive gr...

We characterise connected cubic graphs admitting a vertextransitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph, settling [2, Problem ...

A graph is vertex-transitive if its automorphism group acts transitively on its vertices. A vertex-transitive graph is a Cayley graph if its automorphism group contains a subgroup acting regularly on its vertices. In this paper, the cubic vertextransitive non-Cayley graphs of order 8p are classified for each prime p. It follows from this classification that there are two sporadic and two infini...