Finite locally primitive abelian Cayley graphs

Abstract

Let Γ be a finite connected locally primitive Cayley graph of an abelian group. It is shown that one of the following holds: (1) Γ = Kn, Kn,n, Kn,n − nK 2 , Kn × · · · × Kn; (2) Γ is the standard double cover of Kn × · · · × Kn; (3) Γ is a normal or a bi-normal Cayley graph of an elementary abelian or a meta-abelian 2-group.

Pentavalent Symmetric Graphs of Order Twice a Prime Square

“…If p = 5, we have i = 6, 11, 16 or 21. Then a computation shows that S = a σ = {a, a 6 , a 11 , a 16 , a 21 } for all these four cases, that is, Σ satisfies (i) in Theorem 1.1(c).…”

“…Recall the socle of a group is the product of all its minimal normal subgroups. According to the structure and to the action of the socle, the O'Nan-Scott theorem divides quasiprimitive permutation groups into eight types (see [29] [29] or [21]). Proof.…”

Two-Distance-Primitive Graphs

“…For instance, the cyclic graph C n is 2-distanceprimitive whenever n 4; the icosahedron (the graph in Figure 1) is 2-distance-primitive of valency 5; the family of 2-geodesic-transitive but not 2-arc-transitive graphs of prime valency provides an infinite family of such examples, refer to [13]. This family of graphs is also related to the class of well-known 'locally primitive graphs', see [19,22,23,24,30].…”

Finite 2-Geodesic Transitive Abelian Cayley Graphs

“…Godsil [7,Lemma 2.1] observed that N Aut( ) (T ) = T : Aut(T, S) where Aut(T, S) = {σ ∈ Aut(T )|S σ = S}. If Aut( ) = N Aut( ) (T ), then the graph was called a normal Cayley graph by Xu [24] and such graphs have been studied under various additional conditions, see [6,11,14,17,19]. In this paper, we classify the family of 2-geodesic transitive abelian Cayley graphs.…”