Finite edge-transitive bi-circulants☆
Introduction
For a undirected and simple graph we express by its full automorphism group. If there is an automorphism group acting transitively on (vertex set), (edge set) or (arc set) of then is called -vertex-transitive, -edge-transitive or -arc-transitive, respectively. If is transitive on but intransitive on then is called -semisymmetric.
A graph admitting a semiregular automorphism group with exactly two orbits on is called a bi-Cayley graph on . In particular, is called normal if is normal in and is called a bi-circulant if is cyclic. The family of bi-circulants involves Cayley graphs on dihedral groups, the generalized Petersen graphs [8], Rose-Window graphs [25] and Tabačjn graphs [1]. Edge-transitive bi-circulants have obtained much attention in the literature, see [8], [23] for valency 3, [17] for valency 4, [1], [5] for valency 5 and [13], [28] for valency 6. For more results, refer to [4], [21], [22], [26], [27].
Bi-circulant is a natural generalization of circulant (Cayley graph on a cyclic group). However, in sharp contrast with edge-transitive circulants (which were classified by Kovaćs [16] and Li [19]), the problem for classifying edge-transitive bi-circulants is widely open. In fact, the known results are mainly for small valency. This work aims to study edge-transitive bipartite bi-circulants of order with a Hall subgroup of isomorphic to . (Recall a subgroup is called a Hall subgroup of a group if is coprime to ). The first assertion determines the automorphism groups of such graphs. Theorem 1.1 Let be a connected bipartite edge-transitive bi-circulant on a group with a Hall subgroup of . Denote by the stabilizer of on the bipartitions of . Then is arc-transitive on and the following statements hold. If is soluble, then and is normal arc-regular, where . If is insoluble, thenwhere or with a maximal parabolic subgroup of and such that but for .
For convenience, we express byAs an application of Theorem 1.1, edge-transitive bipartite bi-circulants of order with valency less than are determined. Graphs appearing in the theorem are described in Section 2.1. Theorem 1.2 Let be a connected bipartite edge-transitive bi-circulant of order and suppose . Then is arc-transitive, and one of the following holds. is normal arc-regular, with . with a prime, and . or and . or and where and is a prime. with . or where is a normal arc-regular bi-circulant of order and is a prime such that .
Using properties of normal edge-transitive bi-Cayley graphs (see [4, Proposition 3.4] and [27, Theorem 1.1]), the normal arc-regular bi-circulants in parts (1) and (6) can be constructed in a straightforward way, see Lemma 3.4 for a description of such graphs for prime valency.
Constructing arc-regular graphs is a hot topic in the literature. Kwak et al. [18] constructed an infinite family of arc-regular dihedrants for any prescribed valency, and Feng and Li [7] determined arc-regular dihedrants of any prime valency. Our Theorem 1.2 leads to a new construction of an infinite family of arc-regular bi-circulants. Theorem 1.3 Let . Suppose is not a prime and has no prime divisors 11 and with and a prime power, and suppose and is a solution of the equationThen the bi-circulantis arc-regular.
We remark that the conditions ‘ is not a prime, and has no prime divisors 11 and ’ are necessary. For otherwise, the bi-circulants in Theorem 1.3 includes the graphs and the Heawood graph of order 14, but all of them are not arc-regular.
Section snippets
Examples
There are some typical bi-circulants: the cycle of length ; the complete graphs of vertices; the complete bipartite graph of valency ; the graph deleted a 1-match from . It is easy to see the graphs and are bi-circulants on and and are bi-circulants on .
Two sporadic examples are arisen from bipartite designs: and the incidence and non-incidence graphs of Hadamard design on 22 points. It is known that
A few lemmas
We prove several lemmas in this section that will be used in the sequel. Express by the center and the commutator subgroup of a group . Lemma 3.1 Let where and are nonabelian simple groups. Then and either and ; or there is a prime such that Sylow -subgroups of are nonabelian.
Proof
Set . Then . Observe and we easily deduce namely . Similarly, because and
Proof of Theorem 1.1
In this section, we prove Theorem 1.1 by a series of lemmas.
Let be a connected bipartite -arc-transitive bi-circulant on with odd, and suppose is a Hall subgroup of .
Set the set of primes dividing and set the bipartitions of . Let the stabilizer of on and . Then is of index 2 in and for . Since and is of odd order, we deduce is regular on and . By Frattini argument, so is a -group.
Proofs of Theorems 1.2 and 1.3
Let be a connected bipartite edge-transitive bi-circulant on with and let and be the bipartitions of . Write and . Then is odd and for . Denote by the set of primes dividing . The edge-transitivity of implies and the connectivity of forces that and have the same prime divisors, we deduce is a -group because . Since is odd, and we get is regular on . By
Acknowledgments
The authors thank the referee for some helpful comments.
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