The Isomorphic Classes of Small Degree Cayley Graphs over Dicyclic Groups

2020-07-08 03:25:02ChengTaoFengLihuaLiuWeijunChenQuanguoTangLang

數(shù)學(xué)理論與應(yīng)用 2020年4期

Cheng Tao Feng Lihua Liu Weijun Chen Quanguo Tang Lang

(1.School of Mathematics and Statistics, Shandong Normal University, Jinan, Shandong 250014, China;2.School of Mathematics and Statistics, HNP-LAMA, Central South University New Campus, Changsha, Hunan 410083, China;3.School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China;4.Mathematics and Computational Science, Hunan First Normal University, Changsha, Hunan 410205, China)

Abstract In this paper, we completely classify the isomorphic classes of the small degree(quartic, quintic and sextic)Cayley graphs over the dicyclic group T4p=(for an odd prime p)by means of spectral method and the CI-group properties.Moreover, by making use of the law of quadratic reciprocity, we obtain the exact number of isomorphic classes of the small degree(quartic, quintic and sextic)Cayley graphs on T4p.

Key words Cayley graph Dicyclic group Spectrum Isomorphic class Quadratic reciprocity

1 Introduction

In this paper, we are interested in Cayley graphs-a special class of regular graphs.Given a finite groupGand a subsetS?G{1} withS=S-1, the Cayley graphX(G,S)has vertex setGand two verticesa,bare adjacent ifa-1b∈S.X(G,S)is connected ifSgeneratesG.

For a finite groupG, let Γ=Cay(G,S)for some subsetS?G{1}.Letσbe an automorphism ofG.Thenσnaturally acts on the vertex setV=G.LetT=Sσ.Then it is easily shown thatσinduces an isomorphism fromCay(G,S)to the Cayley graphCay(G,T), such an isomorphism is called a Cayley isomorphism.However, it is of course possible that two Cayley graphsCay(G,S)andCay(G,T)are isomorphic but there are no Cayley isomorphisms mappingStoT.This leads us to try to find the conditions under whichCay(G,S)?Cay(G,S)if and only ifSσ=Tfor someσ∈Aut(G).

A Cayley graphCay(G,S)is called a CI-graph ofGif, wheneverCay(G,S)?Cay(G,T), there is an elementσ∈Aut(G)such thatT=Sσ.(CI stands for Cayley Isomorphism).A finite groupGis called a CI-group if all Cayley graphs ofGare CI-graphs.

The problem of seeking CI-groups has received considerable amount of attention over the past fourty years, beginning with a conjecture ofdám[1]that all finite cyclic groups were CI-groups, one may refer to the surveys in[2,14,23,24].dám’s conjecture was disproved by Elspas and Turner[8].Since then, many mathematician are devoted to classifying cyclic CI-groups(see, for example, Djokovic[7], Babai[4], Alspach and Parsons[3], Palfy[22]and Godsil[10]), and finally, Muzychuk[20,21]obtained a complete classification of cyclic CI-groups.For other related results, one may refer to[19]and their references.

The investigation of CI-graphs is also closely related to the determination of the isomorphic classes of Cayley graphs.In view of the definition, ifX(G,S)is a CI-graph, then to judgeX(G,S)is isomorphic toX(G,T), we only need to decide whether or not there exists an automorphismσsuch thatσ(S)=T.Li[16]and[26]determined the isomorphic classes of some families of Cayley graphs which are edge-transitive but not arc-transitive.For more results about CI-graphs and the isomorphic classes of Cayley graphs, one may refer to[14]and the references therein.

Huang et al.[12]determined the isomorphic classes of cubic Cayley graphs on dihedral groups.Inspired by their results, we are interested in the dicyclic group of order 4p.For an odd primep, the dicyclic group of order 4pis presented asT4p=.This paper is organized as follows.After collecting some preliminary results in Section 2, the isomorphic classes of quartic Cayley graphs onT4p(Theorem 3.3)and the isomorphic classes of quintic Cayley graphs onT4p(Theorem 4.3)will be presented in Section 3 and Section 4, respectively.In Section 5, the isomorphic classes of sextic Cayley graphs onT4p(Theorem 5.3)will be given, and the number of isomorphic classes(Theorem 5.11)will be got by using Gauss’s celebrated law of quadratic reciprocity.

2 Preliminaries

In this section, we introduce some notations and present several results which will be used later.

For a finite groupG, we denote byIRR(G)andIrr(G)the complete set of non-equivalent irreducible representations ofGand the complete set of non-equivalent irreducible characters ofG, respectively.We need the characters of cyclic groups.

The following lemma is essential for our results, which involves spectral graph theory and representations and characters of finite groups.

Lemma 2.2([5]) LetGbe a finite group of ordern,S?G{1} be such thatS=S-1, andIrr(G)={χ1,…,χh} withχi(1)=di(i=1,…,h).Then the spectrum of the Cayley graphX(G,S)can be arranged as

Spec(X(G,S))={[λ11]d1,…,[λ1d1]d1,…,[λh1]dh,…,[λhdh]dh}.

Furthermore, for any natural numbert, we have

Our focus is on the dicyclic group.In general, the presentation for dicyclic groupT4nis given by

T4n=.

Fornodd, settingx2=aandt=b, then we may rewriteT4nas

T4n=.

This group has order 4n, and we have

T4n={ak,akb,akb2,akb3| 0≤k≤n-1}.

Lemma 2.3For the dicyclic groupT4n=, wherenis odd, we have

(1)akb=ba-k;akb2=b2ak;akb3=b3a-k;

(2)(akb)-1=akb3;(akb2)-1=a-kb2.

ProofBy relationsan=b4=1 andb-1ab=a-1, the results follow immediately.

Then+3 conjugacy classes ofT4nare listed as follows

{ajb| 0≤j≤n-1}, {ajb3| 0≤j≤n-1}.

Now we get the character table ofT4n.

Lemma 2.4([13]) Fornodd, the character table ofT4n= is given in Table 1,whereχgandψhare respectively irreducible characters of degree one and two(g=1,2,3,4,h=1,2,…,n-1).

Table 1 The character table of T4n for n odd

By Lemmas 2.2 and 2.4, we have

Theorem 2.5([6]) LetT4nbe the dicyclic group andS?T4n{1} satisfyingS=S-1.Then we have

Spec(X(T4n,S))={[λg]1;[μh1]2,[μh2]2g=1,2,3,4;h=1,2,…,n-1},

and

(1)

We also need some notations for our convenience.For any characterχand two subsetsA,Bof a groupG, we denoteχ(A)=∑a∈Aχ(a)andχ(AB)=∑a∈A,b∈Bχ(ab).In particular,χ(A2)=∑a1,a2∈Aχ(a1a2).

For oddn, in the dicyclic groupT4n, we setS1?∪b2andS2?b∪b3.By Theorem 2.5, we obtain the following explicit formula for the spectrum ofX(T4n,S).

Corollary 2.6For oddn, letT4n= be the dicyclic group, andS=S1∪S2?T4n{1} be such thatS=S-1.Then we have

Spec(X(T4n,S))={[λg]1;[μh1]2,[μh2]2|g=1,2,3,4;h=1,2,…,n-1},