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PureMathematicsnØêÆ,2021,11(12),2069-2075
PublishedOnlineDecember2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1112230
¡N+Ú2Âoê+þk•éó
Vp4ã
¯õ
H㲌ÆÚO†êÆÆ§H&²
ÂvFϵ2021c1110F¶¹^Fϵ2021c1210F¶uÙFϵ2021c1217F
Á‡
©Ì‡‰Ñk•éóVp4ã˜‡d^‡§¡N+ Ú2Âoê+þéóV
p4ã5ŸÚ("
'…c
¡N+§2Âoê+§k•éóVp4㧌é¡ã
FiniteDualBi-CayleyGraphs
onDihedralGroupsand
GeneralizedQuaternion
Groups
HonglinNing
SchoolofStatisticsandMathematics,YunnanUniversityofFinanceandEconomics,Kunming
Yunnan
Received:Nov.10
th
,2021;accepted:Dec.10
th
,2021;published:Dec.17
th
,2021
©ÙÚ^:¯õ.¡N+Ú2Âoê+þk•éóVp4ã[J].nØêÆ,2021,11(12):2069-2075.
DOI:10.12677/pm.2021.1112230
¯õ
Abstract
Inthispaper,wegiveanequivalentconditionoffinitedualbi-Cayleygraphsand
obtainpropertiesandstructuresofdualbi-Cayleygraphsondihedralgroupsand
generalizedquaterniongroups.
Keywords
DihedralGroup,GeneralizedQuaternionGroup,FiniteDualBi-CayleyGraph,
SemisymmetricGraph
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
©ïÄãÑ´k•ëÏÕüã,©9+Ñ´k•+.Γ•˜‡ã,·‚^
VΓ, EΓ, AΓÚAut(Γ)©OL«§º:8, >8, l8ÚgÓ+, ¡Γº:‡ê|VΓ|•
Γê. †º:αƒº:|¤8Ü¡•α•,P•Γ(α), ¡|Γ(α)|•:αÝê,
P•val(α).XJz‡:ÝêуÓ,K¡Γ•Kã,džãΓÝêval(Γ)u:αÝê.
X≤Aut(Γ)•ãΓ˜‡gÓ+,XJX3VΓ, EΓ, ½AΓþD4, K©O¡Γ´X-:
D4ã, X->D4ã, ½X-lD4ã. XJX©O/3EΓþD 4,3VΓþØD4, K¡Γ´
X-Œé¡ã.
Vp4ã´“êãØ9€ïÄ••ƒ˜.‰½˜‡+H9Ùf8R, L,S,Ù¥R=R
−1
,
L= L
−1
,R∪L⊆H\{1}.½Â+HþVp4ãBiCay(H,R,L,S)Xe:º:8VΓ= H
0
∪H
1
,
Ù¥H
0
:={h
0
|h∈H},H
1
:={h
1
|h∈H},>8EΓ={{h
0
,g
0
}|gh
−1
∈R}∪{{h
1
,g
1
}|
gh
−1
∈L}∪{{h
0
,g
1
}|gh
−1
∈S}.
Vp4ã´“êãØ¥•-‡ãaƒ˜,•´“êã Ø9€ïÄ‘Kƒ˜.'uVp4ã
ïÄ, Cc 5ïÄÆö‚‰NõóŠ,32012 c, KovScs ÚKuzman <3©z[1]¥©a
Ì‚+þoÝ>D4Vp4 ã.32014 c,Zhou ÚFeng 3©z[2]¥©a †+þnÝ:D
4Vp4ã.32020 c, Conder ÚZhou <3©z[3]¥‰Ñ A5>D4Vp4ã˜‡
•x,Óž‰ÑŒ•Ø‡L6>D4ã©a.Š˜J´32016 c,ZhouÚFeng 3©
DOI:10.12677/pm.2021.11122302070nØêÆ
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z[4]¥éu+HþVp4ã, (½H3ãgÓ+¥5zf, ù˜(J?˜ÚíÄ
YVp4ãïÄ.'uVp4ã•#ïČ넩z[5–9].
3©z[10]¥éuk•+þéóp4ã?1ïÄ,˜Ð(J:‰Ñk •éó
p4ãƒ'5ŸÚ(,éÙé¡5•?1˜½ïÄ.aq/,·‚Œòk•éóp4ãï
Äí2k•éóVp4ã.©òé¡N+Ú2Âoê+þéóVp4ã?1?Ø.
©¤¦^âŠÚÎÒÑ´IO,Œ±ë•ÍŠ[11].éu+G, ·‚^Z(G)L«G¥
%.éuü‡+NÚH, ·‚^N: HL«NHŒ*Ü.éu˜‡ên,·‚^K
2n
L
«2nã, K
n,n
L«2nÜã, K
n,n
-nK
2
L«3K
n,n
¥K˜‡š¤
ã.
2.ý•£
¡N+Ú2Âoê+´+Øp›©-‡+.e¡·‚‰Ñ¡N+Ú2Âoê+½
Â.
2n¡N+D
2n
½ÂXe:
D
2n
= ha,b|a
n
= b
2
= 1,b
−1
ab= a
−1
i(n>2).
4n2Âoê+D
2n
½ÂXe:
Q
4n
= ha,b|a
2n
= 1,b
2
= a
n
,b
−1
ab= a
−1
i(n≥2).
´•¡N+Ú2Âoê+Ñ´š†+.
Γ= BiCay(H,R,L,S) ´k•+HþVp4ã,
ˆ
H= {
ˆ
h|
ˆ
h: x
0
7→(xh)
0
,x
1
7→(xh)
1
,x∈H,h∈H},
ˇ
H= {
ˇ
h|
ˇ
h: x
0
7→(h
−1
x)
0
,x
1
7→(h
−1
x)
1
,x∈H,h∈H}.
©O´HmKL«Ú†KL«.N´ymKL«
ˆ
Ho´Vp4ãΓgÓ+f+,
†KL«
ˇ
HKؘ½.XJ
ˇ
H≤Aut(Γ),K¡Γ´k•éóVp4ã.džN´y
ˆ
H
ˇ
H´
Aut(Γ)f+.
du©9+Ñ´k•+,3©Qã¥òk•éóVp4ã{¡•éóp4ã.e
H´†+,K
ˇ
H=
ˆ
H≤Aut(BiCay(H,R,L,S)),ŠâéóVp4ã½ÂŒ†+þVp
4ãÑ´éóVp4ã.Ïd·‚̇ïÄš†+þéóVp4ã.d©z[3]Vp 4ãäk±
eÄ5Ÿ:
Ún2.1[3] Γ=BiCay(H,R,L,S) ´+HþëÏVp4ã, Ù¥R,L,S´Hf8.
K±e·K¤á:
DOI:10.12677/pm.2021.11122302071nØêÆ
¯õ
(1)hR,L,Si= H;
(2)3ãgÓ¿Âe,SŒ•¹H¥ü ;
(3)BiCay(H,R,L,S)
∼
=
BiCay(H,R
α
,L
α
,S
α
),Ù¥α∈Aut(H);
(4)BiCay(H,R,L,S)
∼
=
BiCay(H,R,L,S
−1
).
3.̇(J
3Vp4ã¥,+HmKL«Ú†KL«k±e·K:
·K3.1
ˆ
H∩
ˇ
H
∼
=
Z(H).AO/,
ˆ
H6=
ˇ
H…=H´š†+.
y²
ˆ
h= ˇg,h,g∈H,K
(xh)
0
= (x
0
)
ˆ
h
= (x
0
)
ˇg
= (g
−1
x)
0
,
(xh)
1
= (x
1
)
ˆ
h
= (x
1
)
ˇg
= (g
−1
x)
1
,
?¿x∈H,-x=1, Kg
−1
= h. Ïd(xh)
0
= (hx)
0
, (xh)
1
= (hx)
1
,=kxh=hx, Kh∈Z(H).

ˆ
H∩
ˇ
H= {
ˆ
h|h∈Z(H)}
∼
=
Z(H).
©1˜‡Ì‡(Ø´‰ÑéóVp4ã˜‡d^‡.
½n3.2Γ=BiCay(H,R,L,S)´+HþVp4ã,Γ´éóVp4ã…=
R,L,S©O´+H,Ýa¿.
y²¿©5.dΓ´éóVp4ã,k
ˇ
H≤Aut(Γ), 
˜
h=
ˇ
h
ˆ
h∈
ˇ
H
ˆ
H≤Aut(Γ).
dΓ(1
0
) = R
0
∪S
1
,
˜
H= {
˜
h|h∈H}≤Aut(Γ),k(Γ(1
0
))
˜
H
= Γ(1
0
) = R
0
∪S
1
,?¿r∈R,
˜
h∈
˜
H,
d1
0
∼r
0
k(1
0
)
˜
h
∼(r
0
)
˜
h
= (r
h
)
0
,Ïdr
h
∈R,u´R
h
⊆R,qÏ•
˜
h∈Aut(Γ),KkR
h
= R,
dh?¿5ŒR´H˜Ýa¿.ÓnŒL, S•´H˜Ýa¿.
7‡5. eR,L,S©O´+H,Ýa¿,Kéu?¿h∈H, kR
h
=R, L
h
=L,
S
h
= S. ?¿x,y,h∈H,k
x
0
∼y
0
⇔yx
−1
∈R⇔(yx
−1
)
h
∈R
h
= R⇔(x
0
)
ˇ
h
∼(y
0
)
ˇ
h
,
x
1
∼y
1
⇔yx
−1
∈L⇔(yx
−1
)
h
∈L
h
= L⇔(x
1
)
ˇ
h
∼(y
1
)
ˇ
h
,
x
0
∼y
1
⇔yx
−1
∈S⇔(yx
−1
)
h
∈S
h
= S⇔(x
0
)
ˇ
h
∼(y
1
)
ˇ
h
,
y
1
∼x
0
⇔yx
−1
∈S⇔(yx
−1
)
h
∈S
h
= S⇔(y
1
)
ˇ
h
∼(x
0
)
ˇ
h
,
Ïd
ˇ
h∈Aut(Γ), 
ˇ
H≤Aut(Γ),u´Γ´éóVp4ã.
e¡½n‰Ñ¡N+Ú2Âoê+þéóVp4ã5Ÿ.
½n3.3
DOI:10.12677/pm.2021.11122302072nØêÆ
¯õ
(1)Ø•32n¡N+þëÏ G-Œé¡éóVp 4ã, Ù¥H=D
2n
,G=
ˇ
H
ˆ
H≤Aut(Γ),
n´óê.
(2)Ø•32Âoê+þëÏG-Œé¡éóVp4ã,Ù¥H= Q
4n
,G=
ˇ
H
ˆ
H≤Aut(Γ).
y²(1)H=D
2n
=ha,b|a
n
=b
2
=1,b
−1
ab=a
−1
i, Γ=BiCay(H,R,L,S), Ù¥n´
óê.eΓ´G-Œé¡éóVp4ã,KΓ´˜‡Üã,=R=L=∅,¿…G
1
0
=
˜
H3
Γ(1
0
) =S
1
þD4,=S
1
= s
˜
H
1
= (s
˜
H
)
1
= (s
H
)
1
,1 6=s∈S, ?S= s
H
.duΓ´ëÏã,dÚ
n2.1k,hSi= hs
H
i=H.s= a
i
b
j
, 0 ≤i≤n−1, 0 ≤j≤1.XJj=0,@os∈hai. dhai
´HAf+,khSi≤hai<H,ù†ΓëÏ5gñ.Ïds= a
i
b´˜‡.
en´óê.dub
a
=a
−1
a
b
b=a
−2
b=a
n−2
b,XJs†b3H¥Ý,@oS =
{b,a
2
b,···,a
n−2
b},´hSi=ha
2
,bi=ha
2
i:hbi
∼
=
D
n
,ù†ëÏ5gñ.XJs†b3H
¥ØÝ,@oS={ab,a
3
b,···,a
n−1
b}.´hSi=ha
2
,abi= ha
2
i:habi
∼
=
D
n
, ù•†ëÏ5g
ñ.n•óêž, ΓØ•3.
(2)H=Q
4n
=ha,b|a
2n
=1,b
2
=a
n
,b
−1
ab=a
−1
i,Γ=BiCay(H,R,L,S).Ï•Γ´
G-Œé¡éóVp4ã, KΓ´˜‡Üã,d(1)y²ŒR=L=∅, S=s
H
.duΓ´
ëÏã,dÚn2.1k,hSi= hs
H
i=H.s=a
i
b
j
, 0≤i≤2n−1, 0≤j≤1.XJj=0, @o
s∈hai. dhai´HAf+,khSi≤hai<H, ù†ΓëÏ5gñ.Ïds= a
i
b´˜‡4
.
XJs†b3H¥Ý,@oS= s
H
= {b,ba
2
,···,ba
2n−2
},džhSi= hb,a
2
i<Q
4n
,ù†ë
Ï5gñ.XJs†b3H¥ØÝ,@oS={ba,ba
3
,···,ba
2n−1
},d žhSi=hab,a
2
i<Q
4n
,
ù†ëÏ5gñ.Ø•32Âoê+þëÏG-Œé¡éóVp4ã.
íØ1Γ=BiCay(H,R,L,S) ´2m¡N+þëÏG-Œé¡éóVp4ã, Ù¥
H= D
2m
,G=
ˇ
H
ˆ
H≤Aut(Γ),m´Ûê.K
Γ= BiCay(H,∅,∅,{b,ab,···,a
m−1
b}).
y²H=D
2m
=ha,b|a
m
=b
2
=1,b
−1
ab=a
−1
i, Γ=BiCay(H,R,L,S),Ù¥m´
Ûê.eΓ´ëÏG-Œé¡éóVp4ã,d½n3.3y²Œ•, R=L=∅, ¿…S=s
H
,
Ù¥s=a
i
b´˜‡.dum´Ûê,KH¥z˜‡цb3H¥Ý,@o
S= s
H
= {b,ab,···,a
n−1
b}.džhSi= ha,bi= D
2m
,÷vëÏ5.
e¡ù‡½n‰Ñ¡N+Ú2Âoê+þAÏéóVp4ã(.
½n3.4n´ê, K
(1)en>4 ´óê,KK
2n
,K
n,n
,K
n,n
−nK
2
´¡N+þéóVp4ã.
(2)e4 |n…n≥8,KK
2n
,K
n,n
,K
n,n
−nK
2
´2Âoê+þéóVp4ã.
y²(1) Ï•n´Œu4óê, =n=2k, k>2,K•3n¡N+D
2k
=ha,b|a
k
=
b
2
=1,b
−1
ab=a
−1
i(k>2),-H=D
2k
,EHþVp4ãBiCay(H,H\{1},H\{1},H),
DOI:10.12677/pm.2021.11122302073nØêÆ
¯õ
dVp4ãk2n‡:,¿…?¿˜‡:ц,2n-1 ‡:ƒ,Šâã½ÂŒ,
K
2n
∼
=
BiCay(H,H\{1},H\{1},H).duH\{1}=(H\{1})
−1
´H˜Ýa¿,Šâ½
n3.2,K
2n
´HþéóVp4ã.
éuþãH,EHþVp4ãBiCay(H,∅,∅,H).dVp4ã•2n‡:þÜã,
¿…?¿˜‡:Ñ=†,˜‡Ü¥¤k:ƒ,ŠâÜã½ÂŒ,K
n,n
∼
=
BiCay(H,∅,∅,H),Šâ½n3.2,K
n,n
´HþéóVp4ã.
EHþVp4ãBiCay(H,∅,∅,H\{1}),KK
n,n
−nK
2
∼
=
BiCay(H,∅,∅,H\{1}).Ï•
H\{1}= (H\{1})
−1
´H˜Ýa¿,Šâ½n3.2, K
n,n
−nK
2
´HþéóVp4ã.
(2)du4 |n…n≥8,u´n= 4l,l≥2,K•3n2Âoê+
Q
4l
= ha,b|a
2l
= 1,b
2
= a
l
,b
−1
ab= a
−1
i(l≥2),
-H= Q
4l
,EHþVp4ãBiCay(H,H\{1},H\{1},H), BiCay(H,∅,∅,H),
BiCay(H,∅,∅,H\{1}),Šâ(1)y²,kK
2n
∼
=
BiCay(H,H\{1},H\{1},H),
K
n,n
∼
=
BiCay(H,∅,∅,H), K
n,n
−nK
2
∼
=
BiCay(H,∅,∅,H\{1}),¿…§‚Ñ´HþéóVp
4ã,K
2n
,K
n,n
,K
n,n
−nK
2
´2Âoê+þéóVp4ã.
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(11961076).
ë•©z
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