Automorphism Groups of Connected Cubic Cayley Graphs of Order 4p

Abstract

For a prime p, let D4p be the dihedral group 〈a,b | a2p = b2 = 1, b-1ab = a-1〉 of order 4p, and Cay (G,S) a connected cubic Cayley graph of order 4p. In this paper, it is shown that the automorphism group Aut ( Cay (G,S)) of Cay (G,S) is the semiproduct R(G) ⋊ Aut (G,S), where R(G) is the right regular representation of G and Aut (G,S) = {α ∈ Aut (G) | Sα = S}, except either G = D4p (p ≥ 3), Sβ = {b,ab,apb} for some β ∈ Aut (D4p) and [Formula: see text], or Cay (G,S) is isomorphic to the three-dimensional hypercube Q3[Formula: see text] and G = ℤ4 × ℤ2 or D8.

Automorphisms of a Family of Cubic Graphs

“…Let p and q be two primes. In [26,27,28], all connected cubic non-normal Cayley graphs of order 2pq are determined. For more results on the normality of Cayley graphs, we refer the reader to [11,19,23].…”

“…This family of graphs has been investigated in several recent research papers (see, for example, [3,28,29]). In this paper, it is shown that Γ n is a Cayley graph, and its full automorphism group is isomorphic to Z 3 2 S 3 for n = 2, and to Z n 2 D 2n for n > 2.…”

Cubic arc-transitive Cayley graphs on Frobenius groups

“…Since R = 1, we have N = T 1 × · · · × T ∼ = T , where T i ∼ = T is nonabelian simple. By [10], T is one of the following: (Cay(G, S)) is soluble, this case is also excluded, see [18,Lemma 2.2].…”

On cubic non‐Cayley vertex‐transitive graphs

“…The following six results about vertex‐transitive graphs will be needed in the remainder of the paper. The second one is obtained by combining together results from 8, 41, 42. The third one is obtained by combining together [9, Theorem 3.2 and Corollary 3.6] and [43, Theorem 2.1].…”