Finite edge-transitive Cayley graphs and rotary Cayley maps

Abstract

Abstract. This paper aims to develop a theory for studying Cayley graphs, especially for those with a high degree of symmetry. The theory consists of analysing several types of basic Cayley graphs (normal, bi-normal, and corefree), and analysing several operations of Cayley graphs (core quotient, normal quotient, and imprimitive quotient). It provides methods for constructing and characterising various combinatorial objects, such as half-transitive graphs, (orientable and non-orientable) regular Cayley maps, vertex-transitive nonCayley graphs, and permutation groups containing certain regular subgroups.In particular, a characterisation is given of locally primitive holomorph Cayley graphs, and a classification is given of rotary Cayley maps of simple groups. Also a complete classification is given of primitive permutation groups that contain a regular dihedral subgroup.

Locally Primitive Graphs and Bidirect Products of Graphs

“…Let T be a nonabelian simple group with a subgroup H of index p e with p prime. Then T = A p e , or PSL(m, q) with (q m − 1)/(q − 1) = p e , or PSL (2,11) with p e = 11, or M 11 with p e = 11, or M 23 with p e = 23, or PSU(4, 2) with p e = 27. In particular, either T is 2-transitive on [T : H] or T = PSU(4, 2).…”

“…The graph of valency 5, denoted by D 1 2 (11,5), is the incidence graph of the wellknown 2-(11, 5, 1)-design; that of valency 6 is the complement of D 1 2 (11,5) in K 11,11 , denoted by D 1 2 (11,5). Both D 1 2 (11,5) and D 1 2 (11,5) are 2-arc-transitive. E 3.6.…”

“…L 3.7. Let X be almost simple, and assume that X u and X w are not conjugate in X for all u ∈ U and all w ∈ W. Then Γ = D 1 2 (11,5), D In the former case, Γ = D 1 2 (11,5) or D 1 2 (11,5), as in Example 3.5. These two graphs are locally (T, 2)-arc-transitive.…”

“…Here we characterise the family of graphs that are regular, bipartite, locally primitive, and of order 2p e , where p is prime. Typical examples include: (i) the complete bipartite graphs K p e ,p e ; (ii) the graphs K p e ,p e − p e K 2 obtained by deleting a 1-factor from K p e ,p e ; (iii) the incidence graph D 1 2 (11,5) and the nonincidence graph D 1 2 (11,5) of the 2-(11, 5, 1)-design; (iv) the incidence graph PH(d, q) and the nonincidence graph PH(d, q) of the projective geometry PG(d − 1, q), where d ≥ 3; and (v) the standard double cover of the Schläfli graph.…”

“…A bipartite graph Γ = (V, E) is called a Cayley graph of a group R if Aut Γ has a subgroup which is isomorphic to R and regular on V. If further Aut Γ has a subgroup R which is regular on V and R ∩ (Aut Γ) + is normal in Aut Γ, then Γ is called a binormal Cayley graph. Binormal Cayley graphs have many interesting properties; see [11].…”