Finite edge-transitive bi-circulants

Highlights

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Abstract

This work studies edge-transitive bi-circulants of order $2n$ with ${\mathbb{Z}}_n$ being a Hall subgroup of the full automorphism groups. The automorphism groups of such graphs are completely determined. As an application, a classification is given of bipartite edge-transitive bi-circulants of order $2n$ with valency less than the smallest prime divisors of $n$. This also leads to a new construction of an infinite family of arc-regular bi-circulants.

Introduction

For a undirected and simple graph $\Gamma ,$ we express by $\mathsf{Aut}\Gamma$ its full automorphism group. If there is an automorphism group $X$ acting transitively on $V\Gamma$ (vertex set), $E\Gamma$ (edge set) or $A\Gamma$ (arc set) of $\Gamma ,$ then $\Gamma$ is called $X$-vertex-transitive, $X$-edge-transitive or $X$-arc-transitive, respectively. If $X$ is transitive on $E\Gamma$ but intransitive on $V\Gamma ,$ then $\Gamma$ is called $X$-semisymmetric.

A graph $\Gamma$ admitting a semiregular automorphism group $G$ with exactly two orbits on $V\Gamma$ is called a bi-Cayley graph on $G$. In particular, $\Gamma$ is called normal if $G$ is normal in $\mathsf{Aut}\Gamma ,$ and $\Gamma$ is called a bi-circulant if $G$ 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 $\Gamma$ of order $2n$ with a Hall subgroup of $\mathsf{Aut}\Gamma$ isomorphic to ${\mathbb{Z}}_n$. (Recall a subgroup $H$ is called a Hall subgroup of a group $G$ if $|G:H|$ is coprime to $\left|H\right|$). The first assertion determines the automorphism groups of such graphs.

Theorem 1.1

Let $\Gamma$ be a connected bipartite edge-transitive bi-circulant on a group $G\cong {\mathbb{Z}}_n,$ with $G$ a Hall subgroup of $\mathsf{Aut}\Gamma$. Denote by ${\left(\mathsf{Aut}\Gamma \right)}^{+}$ the stabilizer of $\mathsf{Aut}\Gamma$ on the bipartitions of $\Gamma$. Then $\mathsf{Aut}\Gamma ={\left(\mathsf{Aut}\Gamma \right)}^{+}.{\mathbb{Z}}_2$ is arc-transitive on $\Gamma ,$ and the following statements hold.

• If $\mathsf{Aut}\Gamma$ is soluble, then ${\left(\mathsf{Aut}\Gamma \right)}^{+}={\mathbb{Z}}_n:H$ and $\Gamma$ is normal arc-regular, where $H\le {\mathbb{Z}}_n^{*}$.

• If $\mathsf{Aut}\Gamma$ is insoluble, then

  $${\left(\mathsf{Aut}\Gamma \right)}^{+}=(({\mathbb{Z}}_m:L)\times T_1\times T_2\times \cdots \times T_l).P,$$

  where $m\mid n,$ $l\ge 1,$ $L\le {\mathbb{Z}}_m^{*},$ $(T_i,{\left(T_i\right)}_{\alpha })=({\mathsf{M}}_{11},{\mathrm{A}}_6.{\mathbb{Z}}_2),({\mathsf{M}}_{23},{\mathsf{M}}_{22}),(\text{PSL}(2,11),{\mathrm{A}}_5),$ $({\mathrm{A}}_s,{\mathrm{A}}_{s-1})$ or $(\text{PSL}(d,q),P_1)$ with $P_1$ a maximal parabolic subgroup of $\text{PSL}(d,q),$ and $P\le \mathsf{Out}\left(T_1\right)\times \mathsf{Out}\left(T_2\right)\times \cdots \times \mathsf{Out}\left(T_l\right),$ such that $(n,|T_i\left|\right)>1$ but $(m,|T_i\left|\right)=(n,|T_i|,|T_j\left|\right)=1$ for $i\ne j$.

For convenience, we express by

$$spd\left(n\right)thesmallestprimedivisorofapositiveintegern.$$

As an application of Theorem 1.1, edge-transitive bipartite bi-circulants of order $2n$ with valency less than $spd\left(n\right)$ are determined. Graphs appearing in the theorem are described in Section 2.1.

Theorem 1.2

Let $\Gamma$ be a connected bipartite edge-transitive bi-circulant of order $2n,$ and suppose $3\le \mathsf{val}\left(\Gamma \right)<spd\left(n\right)$. Then $\Gamma$ is arc-transitive, and one of the following holds.

• $\Gamma =\mathsf{Bi}\mathsf{Cay}(G,\varnothing ,\varnothing ,S)$ is normal arc-regular, with $S\subseteq G$.

• $\Gamma ={\mathsf{K}}_{p,p}-p{\mathsf{K}}_2$ with $p\ge 5$ a prime, and $\mathsf{Aut}\Gamma ={\mathrm{S}}_p\times {\mathbb{Z}}_2$.

• $\Gamma =\mathcal{H}\mathcal{D}(22,5,2)$ or $\mathcal{H}\mathcal{D}(22,6,2),$ and $\mathsf{Aut}\Gamma =\text{PGL}(2,11)$.

• $\Gamma =\mathcal{P}\mathcal{H}(d,q)$ or ${\mathcal{P}\mathcal{H}}^{\prime }(d,q),$ and $\mathsf{Aut}\Gamma =\mathrm{P}\Gamma \mathrm{L}(d,q).{\mathbb{Z}}_2,$ where $d\ge 3$ and $\frac{q^d-1}{q-1}$ is a prime.

• $\Gamma ={\mathsf{C}}_{2m}{\times }_{bi}\mathcal{H}\mathcal{D}(22,5,2),$ with $spd\left(m\right)\ge 13$.

• $\Gamma ={\Sigma }_1{\times }_{bi}\mathcal{P}\mathcal{H}(d,q)$ or ${\Sigma }_1{\times }_{bi}{\mathcal{P}\mathcal{H}}^{\prime }(d,q),$ where ${\Sigma }_1$ is a normal arc-regular bi-circulant of order $2m,$ $d\ge 3,$ and $\frac{q^d-1}{q-1}$ is a prime such that $\mathsf{val}\left({\Sigma }_1\right)·\frac{q^{d-1}-1}{q-1}<spd\left(m\frac{q^d-1}{q-1}\right)$.

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 $G=〈a〉\cong {\mathbb{Z}}_n$. Suppose $n$ is not a prime and has no prime divisors 11 and $\frac{q^d-1}{q-1}$ with $d\ge 3$ and $q$ a prime power, and suppose $k<spd\left(n\right)$ and $l$ is a solution of the equation

$$x^{k-1}+x^{k-2}+\cdots +x+1\equiv 0\left(\mathsf{mod}n\right).$$

Then the bi-circulant

$$\mathsf{Bi}\mathsf{Cay}(〈a〉,\varnothing ,\varnothing ,\{1,a,a^{1+l},\cdots ,a^{1+l+\cdots +l^{k-2}}\})$$

is arc-regular.

We remark that the conditions ‘$n$ is not a prime, and has no prime divisors 11 and $\frac{q^d-1}{q-1}$’ are necessary. For otherwise, the bi-circulants in Theorem 1.3 includes the graphs ${\mathsf{K}}_{n,n}-n{\mathsf{K}}_2,$ $\mathcal{H}\mathcal{D}(22,5,2)$ and the Heawood graph of order 14, but all of them are not arc-regular.