Cayley graphs of abelian groups which are not normal edge-transitive (original) (raw)

Arc-transitive and s-regular Cayley graphs of valency 5 on Abelian groups

Discussiones Mathematicae Graph Theory, 2006

Let G be a finite group, and let 1 G ∈ S ⊆ G. A Cayley di-graph Γ = Cay(G, S) of G relative to S is a di-graph with a vertex set G such that, for x, y ∈ G, the pair (x, y) is an arc if and only if yx −1 ∈ S. Further, if S = S −1 := {s −1 |s ∈ S}, then Γ is undirected. Γ is conected if and only if G = s. A Cayley (di)graph Γ = Cay(G, S) is called normal if the right regular representation of G is a normal subgroup of the automorphism group of Γ. A graph Γ is said to be arc-transitive, if Aut(Γ) is transitive on an arc set. Also, a graph Γ is s-regular if Aut(Γ) acts regularly on the set of s-arcs. In this paper, we first give a complete classification for arc-transitive Cayley graphs of valency five on finite Abelian groups. Moreover, we classify s-regular Cayley graph with valency five on an abelian group for each s ≥ 1.

Non-normal edge-transitive directed Cayley graphs of abelian groups

Mathematical Sciences, 2013

We consider signed graphs, i.e., graphs with positive or negative signs on their edges. The notion of signed strongly regular graph is recently defined by the author (Signed strongly regular graphs, Proceeding of 48th Annual Iranian Mathematical Conference, 2017). We construct some families of signed strongly regular graphs with only two distinct eigenvalues. The construction is based on the well-known method known as star complement technique.

Edge-transitive bi-Cayley graphs

Journal of Combinatorial Theory, Series B, 2020

A graph Γ admitting a group H of automorphisms acting semi-regularly on the vertices with exactly two orbits is called a bi-Cayley graph over H. Such a graph Γ is called normal if H is normal in the full automorphism group of Γ, and normal edge-transitive if the normaliser of H in the full automorphism group of Γ is transitive on the edges of Γ. In this paper, we give a characterisation of normal edgetransitive bi-Cayley graphs, and in particular, we give a detailed description of 2-arc-transitive normal bi-Cayley graphs. Using this, we investigate three classes of bi-Cayley graphs, namely those over abelian groups, dihedral groups and metacyclic p-groups. We find that under certain conditions, 'normal edgetransitive' is the same as 'normal' for graphs in these three classes. As a by-product, we obtain a complete classification of all connected trivalent edge-transitive graphs of girth at most 6, and answer some open questions from the literature about 2-arc-transitive, half-arc-transitive and semisymmetric graphs.

Classification of tetravalent 2-transitive non-normal Cayley graphs of finite simple groups

2021

A graph Γ is called (G, s)-arc-transitive if G ≤Aut(Γ) is transitive on the set of vertices of Γ and the set of s-arcs of Γ, where for an integer s ≥ 1 an s-arc of Γ is a sequence of s+1 vertices (v_0,v_1,…,v_s) of Γ such that v_i-1 and v_i are adjacent for 1 ≤ i ≤ s and v_i-1 v_i+1 for 1 ≤ i ≤ s-1. Γ is called 2-transitive if it is (Aut(Γ), 2)-arc-transitive but not (Aut(Γ), 3)-arc-transitive. A Cayley graph Γ of a group G is called normal if G is normal in Aut(Γ) and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if Γ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either Γ is normal or G is one of the groups PSL_2(11), M_11, M_23 and A_11. However, it was unknown whether Γ is normal when G is one of these four groups. In the present paper we answer this question by proving that among these four groups only M_11 produces connected tetravalent 2-transitive non-normal Cayley graphs. We prove further that there are exactly tw...

Normal edge-transitive Cayley graphs and Frattini-like subgroups

2021

For a finite group G and an inverse-closed generating set C of G, let Aut ( G ; C ) consist of those automorphisms of G which leave C invariant. We define an Aut ( G ; C ) -invariant normal subgroup Φ ( G ; C ) of G which has the property that, for any Aut ( G ; C ) -invariant normal set of generators for G, if we remove from it all the elements of Φ ( G ; C ) , then the remaining set is still an Aut ( G ; C ) -invariant normal generating set for G. The subgroup Φ ( G ; C ) contains the Frattini subgroup Φ ( G ) but the inclusion may be proper. The Cayley graph Cay ( G , C ) is normal edge-transitive if Aut ( G ; C ) acts transitively on the pairs { c , c − 1 } from C. We show that, for a normal edge-transitive Cayley graph Cay ( G , C ) , its quotient modulo Φ ( G ; C ) is the unique largest normal quotient which is isomorphic to a subdirect product of normal edge-transitive Cayley graphs of characteristically simple groups. In particular, we may therefore view normal edge-transiti...

On normal 2-geodesic transitive Cayley graphs

Journal of Algebraic Combinatorics, 2013

We investigate connected normal 2-geodesic transitive Cayley graphs Cay(T , S). We first prove that if Cay(T , S) is neither cyclic nor K 4[2] , then a \ {1} S for all a ∈ S. Next, as an application, we give a reduction theorem proving that each graph in this family which is neither a complete multipartite graph nor a bipartite 2arc transitive graph, has a normal quotient that is either a complete graph or a Cayley graph in the family for a characteristically simple group. Finally we classify complete multipartite graphs in the family.

Classification of Tetravalent -Transitive Nonnormal Cayley Graphs of Finite Simple Groups

Bulletin of The Australian Mathematical Society, 2021

A graph Γ is called (G, s)-arc-transitive if G ≤ Aut(Γ) is transitive on the set of vertices of Γ and the set of s-arcs of Γ, where for an integer s ≥ 1 an s-arc of Γ is a sequence of s + 1 vertices (v 0 , v 1 ,. .. , v s) of Γ such that v i−1 and v i are adjacent for 1 ≤ i ≤ s and v i−1 = v i+1 for 1 ≤ i ≤ s− 1. Γ is called 2-transitive if it is (Aut(Γ), 2)-arc-transitive but not (Aut(Γ), 3)arc-transitive. A Cayley graph Γ of a group G is called normal if G is normal in Aut(Γ) and non-normal otherwise. It was proved by X. G. Fang, C. H. Li and M. Y. Xu that if Γ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either Γ is normal or G is one of the groups PSL 2 (11), M 11 , M 23 and A 11. However, it was unknown whether Γ is normal when G is one of these four groups. In the present paper we answer this question by proving that among these four groups only M 11 produces connected tetravalent 2-transitive non-normal Cayley graphs. We prove further that there are exactly two such graphs which are non-isomorphic and both determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.

Nonnormal Edge-Transitive Cubic Cayley Graphs of Dihedral Groups

ISRN Algebra, 2011

A Cayley graph of a finite group is called normal edge transitive if its automorphism group has a subgroup which both normalizes and acts transitively on edges. In this paper we determine all cubic, connected, and undirected edge-transitive Cayley graphs of dihedral groups, which are not normal edge transitive. This is a partial answer to the question of Praeger (1999).