In this paper we apply Pólya’s Theorem to the problem of enumerating Cayley graphs on permutation groups up to isomorphisms induced by conjugacy in the symmetric group. We report the results of a search of all three-regular Cayley graphs on permutation groups of degree at most nine for small diameter graphs. We explore several methods of constructing covering graphs of these Cayley graphs. Exam...

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...

Nathanson was the pioneer in introducing the concepts of Number Theory, particularly, the “Theory of Congruences” in Graph Theory, thus paved the way for the emergence of a new class of graphs, namely “Arithmetic Graphs”. Cayley graphs are another class of graphs associated with the elements of a group. If this group is associated with some arithmetic function then the Cayley graph becomes an A...

In this paper, we introduce super Cayley graphs, a class of communication-efficient networks for parallel processing. We show that super Cayley graphs can embed trees, meshes, hypercubes, as well as star, bubble-sort graphs, and transposition networks with constant dilation. We also show that algorithms developed for star graphs can be emulated on suitably constructed super Cayley graphs with a...

In this paper, a characterisation is given of finite s-arc transitive Cayley graphs with s ≥ 2. In particular, it is shown that, for any given integer k with k ≥ 3 and k 6= 7, there exists a finite set (maybe empty) of s-transitive Cayley graphs with s ∈ {3, 4, 5, 7} such that all s-transitive Cayley graphs of valency k are their normal covers. This indicates that s-arc transitive Cayley graphs...

These lecture notes provide an introduction to automorphism groups of graphs. Some special families of graphs are then discussed, especially the families of Cayley graphs generated by transposition sets. Keywords—Automorphism groups of graphs; Cayley graphs; transposition sets.

We construct an infinite family of half-transitive graphs, which contains infinitely many Cayley graphs, and infinitely many non-Cayley graphs.

In the search for regular, undirected dense graphs for interconnection networks, Chudnovsky et al. found certain Cayley graphs that are the densest degree-four graphs known for an interesting range of diameters [l]. However, the group theoretic representation of Cayley graphs makes the development of effective routing algorithms difficult. This paper shows that all finite Cayley graphs can be r...