On non-Cayley vertex-transitive graphs and the meta-Cayley graphs

Abstract

The pursuit to identify vertex-transitive non-Cayley graphs has been deliberate for some time now. In that vein, Alspach and Parsons [1] introduced metacirculant graphs. They are defined on two cyclic groups with adjacency resembling twisting that is typically used in defining semi-direct products of groups. In this sequel we generalise the construction to general groups and introduce a class of graphs we call meta-Cayley graphs. (2010): 05C25, 20B25. Mathematics Subject Classification

Some Meta-Cayley Graphs on Dihedral Groups

“…For groups A and A and mapping f : A → Aut A , we denote a semi-direct product of A and A as A × f A, where the elements of A × f A are of the form (a, f a (a )) with a ∈ A and a ∈ A . As alluded to in the introduction, a semi-direct product on two groups whose twisting map satisfies a weak condition to obtain a loop and not a group were introduced in [15] in the following way.…”

“…Proposition 1 [15] Let A and A be groups. Let f be a mapping f : A → Aut A such that f (e) is the identity map on A , where e is the identity element of A. Denote f (a) as f a and define Q by Q = A × f A .…”

“…Again, as was also alluded to earlier, determining quasi-associative Cayley subsets is at the crux of the matter. It has been observed in [15] that sets satisfying the following are quasi-associative Cayley subsets.…”

“…The concept of quasi-associative subsets was introduced by Gauyacq [4] who termed them right associative. As elsewhere [12][13][14][15], we will continue to call them quasiassociative. Cayley graphs on loops with quasi-associative Cayley subsets are termed quasi-Cayley.…”

“…In addition, the crux of the matter is to determine subsets of the product that are quasi-associative and Cayley to construct vertex-transitive graphs. Such graphs have been called meta-Cayley graphs [15]. In order to prove that graphs of this class are non-Cayley on groups, we first completely determine their automorphism groups.…”