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AdvancesinAppliedMathematics
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Ð
,2021,10(11),4047-4055
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1011430
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ResearchontheLocalAntimagic
ChromaticNumberBasedonGraph
Operations
DandanLiu
1
∗
,HongBian
1
†
,HaizhengYu
2
,LinaWei
1
1
SchoolofMathematicalSciences,XinjiangNormalUniversity,UrumqiXinjiang
2
CollegeofMathematicsandSystemSciences,XinjiangUniversity,UrumqiXinjiang
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,2021,10(11):
4047-4055.DOI:10.12677/aam.2021.1011430
4
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Received:Oct.23
rd
,2021;accepted:Nov.13
th
,2021;published:Nov.29
th
,2021
Abstract
Let
G
=(
V
(
G
)
,E
(
G
))
beasimpleconnectedgraphwith
|
V
(
G
)
|
=
n
and
|
E
(
G
)
|
=
m
.A
bijection
f
:
E
(
G
)
→{
1
,
2
,...,m
}
iscalledlocalantimagiclabelingifforanytwoadjacent
vertices
u
and
v
,
ω
(
u
)
6
=
ω
(
v
)
,where
ω
(
u
)=
P
e
∈
E
(
u
)
f
(
e
)
,and
E
(
u
)
isthesetofedges
incidentto
u
.Thusanylocalantimagiclabelinginducesapropervertexcoloring
of
G
,wherethevertex
u
isassignedthecolor
ω
(
u
)
.Thelocalantimagicchromatic
number
χ
la
(
G
)
istheminimumnumberofcolorstakenoverallcoloringsinducedby
localantimagiclabelingsof
G
.Inthispaper,westudytheexactvaluesofthelocal
antimagicchromaticnumbersofsomegraphsbasedgraphoperation,suchas
F
n
+
{
e
}
,
where
e
isapendantedgeaddingto
F
n
andthesub-dividedgraphs
P
m
(
S
n
)
and
P
m
(
S
l,q
)
ofsomespecialgraphs.
Keywords
AntimagicLabeling,LocalAntimagicLabeling,LocalAntimagicChromaticNumber,
Sub-Divided
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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+
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V
(
F
n
)+
{
v
}
,
>
8
•
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(
F
n
+
{
e
}
) =
E
(
F
n
)+
{
e
}
.
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±
²
w
χ
la
(
F
n
+
{
e
}
)
≥
χ
(
F
n
+
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}
)= 3.
e
5
,
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‡
V
f
:
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→{
1
,
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,...,
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n
+1
}
,
K
f
(
e
) = 3
n
+1;
f
(
u
i
v
i
) =
i,
1
≤
i
≤
n
;
f
(
xu
i
) = 2
n
+1
−
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1
≤
i
≤
n
;
DOI:10.12677/aam.2021.10114304049
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Ú
f
(
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1
≤
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•
:
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(
v
) = 3
n
+1;
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(
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) = 4
n
(
n
+1)+1;
ω
(
v
i
) = 3
n
+1
,
1
≤
i
≤
n
;
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ω
(
u
i
) = 2
n
+1
,
1
≤
i
≤
n.
²
w
/
,
f
´
ã
F
n
+
{
e
}
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‡
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‡
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,
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3
«
p
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ô
Ú
,
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d
χ
la
(
F
n
+
{
e
}
)
≤
3.
χ
la
(
F
n
+
{
e
}
) = 3.
X
F
3
+
{
e
}
Û
Ü
‡
Ž
X
Ú
„
ã
1.
Figure1.
Thelocalantimagiccoloring
of
F
3
+
{
e
}
ã
1.
F
3
+
{
e
}
Û
Ü
‡
Ž
X
Ú
œ
/
2
]
!
>
e
\
3
l
Ç
ã
F
n
?
¿˜
‡
Ý:
þ
.
Ø
”
˜
„
5
,
b
]
!
>
†
:
u
1
ƒ
ë
,
=
e
=
u
1
v
,
d
ž
Œ
±
º:
Šƒ
Ú
•
:
ω
(
v
) = 3
n
+1;
ω
(
u
1
) = 5
n
+2;
DOI:10.12677/aam.2021.10114304050
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4
ûû
ω
(
u
i
) = 2
n
+1
,
2
≤
i
≤
n
;
w
(
v
i
) = 3
n
+1
,
1
≤
i
≤
n
;
Ú
ω
(
x
) =
n
(4
n
+1)
,
1
≤
i
≤
n.
²
w
/
,
f
´
ã
F
n
+
{
e
}
˜
‡
Û
Ü
‡
Ž
I
Ò
…
¦
^
4
«
p
É
ô
Ú
,
χ
la
(
F
n
+
{
e
}
)
≤
4.
X
F
3
+
{
e
}
Û
Ü
‡
Ž
X
Ú
„
ã
2.
Figure2.
Thelocalantimagiccoloringof
F
3
+
{
e
}
ã
2.
F
3
+
{
e
}
Û
Ü
‡
Ž
X
Ú
Ú
n
2[2]
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u
?
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‡
k
l
‡
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f
ä
T
,
k
χ
la
(
T
)
≥
l
+1.
Ú
n
3[9]
ã
SS
n,m
´
(
ã
>
¿
©
ã
.
é
u
n
>
2
Ú
m
>
3
?
¿
ê
,
ã
SS
n,m
Û
Ü
‡
Ž
X
Ú
ê
•
n
+1
≤
χ
la
(
SS
n,m
)
≤
n
+2.
d
Ú
n
3,
©
é
¦
(
ã
¿
©
ã
P
m
(
S
n
)
Û
Ü
‡
Ž
X
Ú
ê
•
χ
la
(
P
m
(
S
n
))=
n
+1
ã
.
½
n
4
ã
P
m
(
S
n
)
´
(
ã
S
n
m
g
>
¿
©
ã
,
e
÷
v
±
e
?
˜
^
‡
:
1)
m
= 1,
…
2
≤
n
≤
3;
2)
m
•
ó
ê
…
2
m
≤
n
≤
2
m
+1,
K
k
χ
la
(
P
m
(
S
n
)) =
n
+1
.
y
²
-
ã
P
m
(
S
n
)
:
8
•
V
(
P
m
(
S
n
)) =
{
u
0
}∪{
u
ki
: 1
≤
k
≤
m,
1
≤
i
≤
n
}∪{
u
0
i
: 1
≤
i
≤
n
}
,
>
8
•
E
(
P
m
(
S
n
))=
{
u
0
u
1
i
:1
≤
i
≤
n
}∪{
u
ki
u
k
+1
i
:1
≤
k
≤
m
−
1
,
1
≤
i
≤
n
}∪{
u
mi
u
0
i
:
1
≤
i
≤
n
}
.
ã
P
m
(
S
n
)
º:
ê
•
|
V
(
P
m
(
S
n
))
|
=
mn
+
n
+1,
>
ê
•
|
E
(
P
m
(
S
n
))
|
=
mn
+
n
.
½
Â
˜
‡
V
f
:
E
→{
1
,
2
,...,
|
E
(
P
m
(
S
n
))
|}
,
:
DOI:10.12677/aam.2021.10114304051
A^
ê
Æ
?
Ð
4
ûû
œ
/
1
m
=1
ž
,
k
χ
la
(
P
1
(
S
2
))
≤
3,
Ú
χ
la
(
P
1
(
S
3
))
≤
4(
§
‚
Û
Ü
‡
Ž
X
Ú
„
e
ã
3
(a)
!
(b)),
f
´
˜
‡
Û
Ü
‡
Ž
I
Ò
¦
2
≤
n
≤
3
ž
,
k
P
1
(
S
n
)
≤
n
+1.
Figure3.
(a)thelocalantimagiccoloringof
P
1
(
S
2
);(b)thelocalantimagic
coloringof
P
1
(
S
3
)
ã
3.
(a)
ã
P
1
(
S
2
)
Û
Ü
‡
Ž
X
Ú
¶
(b)
ã
P
1
(
S
3
)
Û
Ü
‡
Ž
X
Ú
œ
/
2
m
•
ó
êž
:
f
(
u
0
u
1
i
) =
i,
1
≤
i
≤
n
;
f
(
u
mi
u
0
i
) =
mn
+
i,
1
≤
i
≤
n
;
f
(
u
0
i
u
mi
) =
mn
+
n
+1
−
i,
1
≤
i
≤
n
;
f
(
u
ki
u
k
+1
i
) =
mn
+(1
−
k
)
n
+1
−
i,k
≡
1(
mod
2)
,
1
≤
i
≤
n
;
kn
+
i,k
≡
0(
mod
2)
,
1
≤
i
≤
n.
d
þ
ã
I
Ò
O
Ž
Œ
•
,
d
ž
ˆ
º:
Šƒ
Ú
•
:
ω
(
u
0
) =
n
(
n
+1)
2
;
ω
(
u
0
i
) =
mn
+
i,
1
≤
i
≤
n
;
ω
(
u
ki
) =
mn
+1
,k
≡
1(
mod
2)
,
1
≤
i
≤
n
;
(
m
+2)
n
+1
,k
≡
0(
mod
2)
,
1
≤
i
≤
n.
k
≡
1(
mod
2)
ž
,
ω
(
u
ki
)=
ω
(
u
01
)=
mn
+ 1;
ω
(
u
0
)
7
½
†
,
‡
ω
(
u
0
i
)
ƒ
,2
≤
i
≤
n
:
n
=2
m
ž
,
ω
(
u
0
)=
ω
(
u
0
n
2
)=
n
(
n
+1)
2
;
n
=2
m
+1
ž
,
ω
(
u
0
)=
ω
(
u
0
n
)=
(2
m
+1)(2
m
+1+1)
2
.
²
w
/
,
f
´
˜
‡
Û
Ü
‡
Ž
I
Ò
…
¦
^
n
+1
«
p
É
ô
Ú
,
Ï
d
χ
la
(
P
m
(
S
n
))
≤
n
+1.
(
ã
¿
©
ã
ä
k
n
‡
“
f
:
Ú
˜
‡
¥
%
:
,
d
Ú
n
2
Œ
•
,
χ
la
(
P
m
(
S
n
))
≥
n
+1.
n
þ
¤
ã
,
÷
v
þ
ã
?
˜
^
‡
ž
,
k
χ
la
(
P
m
(
S
n
)) =
n
+1.
X
P
2
(
S
4
)
Û
Ü
‡
Ž
X
Ú
„
e
ã
4.
½
n
5
ã
P
m
(
S
l,q
)
´
V
(
ã
S
l,q
m
g
>
¿
©
ã
,
l
≥
2
Ú
q
≥
2
•
?
¿
êž
,
K
l
+
q
+1
≤
P
m
(
S
l,q
)
≤
l
+
q
+3.
DOI:10.12677/aam.2021.10114304052
A^
ê
Æ
?
Ð
4
ûû
Figure4.
Thelocalantimagiccoloringof
P
2
(
S
4
)
ã
4.
P
2
(
S
4
)
Û
Ü
‡
Ž
X
Ú
y
²
-
ã
P
m
(
S
l,q
)
:
8
•
V
(
P
m
(
S
l,q
)) =
{
u
1
0
,u
2
0
}∪{
u
ki
: 1
≤
i
≤
l,
1
≤
k
≤
m
}∪{
u
0
i
: 1
≤
i
≤
l
}∪{
u
ki
:
l
+2
≤
i
≤
l
+
q
+1
,
1
≤
k
≤
m
}∪{
u
0
i
:
l
+2
≤
i
≤
l
+
q
+1
}∪{
u
i
:1
≤
i
≤
m
}
,
>
8
•
E
(
P
m
(
S
l,q
))=
{
u
1
0
u
1
i
:1
≤
i
≤
l
}∪{
u
ki
u
k
+1
i
:1
≤
i
≤
l,
1
≤
k
≤
m
−
1
}∪{
u
mi
u
0
i
:1
≤
i
≤
l
}∪{
u
1
0
u
1
}∪{
u
i
u
i
+1
: 1
≤
i
≤
m
−
1
}∪{
u
m
u
2
0
}∪{
u
2
0
u
1
i
:
l
+2
≤
i
≤
l
+
q
+1
}∪{
u
ki
u
k
+1
i
:
l
+ 2
≤
i
≤
l
+
q
+ 1
,
1
≤
k
≤
m
−
1
}∪{
u
mi
u
0
i
:
l
+ 2
≤
i
≤
l
+
q
+ 1
}
.
ã
P
m
(
S
l,q
)
º:
ê
•
|
V
(
P
m
(
S
l,q
))
|
=(
m
+1)(
l
+
q
+1)+ 1,
>
ê
•
|
E
(
P
m
(
S
l,q
))
|
=(
m
+1)(
l
+
q
+1).
½
˜
‡
V
f
:
E
→{
1
,
2
,...,
|
E
(
P
m
(
S
l,q
))
|}
,
f
(
u
m
u
2
0
) = (
m
+1)(
l
+
q
+1)
−
l
;
f
(
u
0
i
u
mi
) = (
m
+1)(
l
+
q
+1)+1
−
i,
1
≤
i
≤
l
;
f
(
u
0
i
u
mi
) = (
m
+1)(
l
+
q
+1)+1
−
i,l
+2
≤
i
≤
l
+
q
+1;
f
(
u
1
0
u
1
) =
(
(
m
−
1)(
l
+
q
+1)
2
+
l
+1
,
1
≡
m
(
mod
2);
m
(
l
+
q
+1)+
[1
−
(
m
−
1)](
l
+
q
+1)
2
−
l,
1
≡
m
−
1(
mod
2);
f
(
u
i
u
i
−
1
) =
(
(
m
−
k
)(
l
+
q
+1)
2
+
l
+1
,k
≡
m
(
mod
2);
m
(
l
+
q
+1)+
[
k
−
(
m
−
1)](
l
+
q
+1)
2
−
l,k
≡
m
−
1(
mod
2);
f
(
u
1
0
u
1
i
) =
(
(
m
−
1)(
l
+
q
+1)
2
+
i,
1
≡
m
(
mod
2)
,
1
≤
i
≤
l
;
m
(
l
+
q
+1)+
[1
−
(
m
−
1)](
l
+
q
+1)
2
+1
−
i,
1
≡
m
−
1(
mod
2)
,
1
≤
i
≤
l
;
f
(
u
ki
u
k
−
1
i
) =
(
(
m
−
k
)(
l
+
q
+1)
2
+
i,k
≡
m
(
mod
2)
,
1
≤
i
≤
l
;
m
(
l
+
q
+1)+
[
k
−
(
m
−
1)](
l
+
q
+1)
2
+1
−
i,k
≡
m
−
1(
mod
2)
,
1
≤
i
≤
l
;
f
(
u
2
0
u
1
i
) =
(
(
m
−
1)(
l
+
q
+1)
2
+
i,k
≡
m
(
mod
2)
,l
+2
≤
i
≤
l
+
q
+1;
m
(
l
+
q
+1)+
[1
−
(
m
−
1)](
l
+
q
+1)
2
+1
−
i,k
≡
m
−
1(
mod
2)
,l
+2
≤
i
≤
l
+
q
+1;
DOI:10.12677/aam.2021.10114304053
A^
ê
Æ
?
Ð
4
ûû
f
(
u
ki
u
k
−
1
i
) =
(
m
−
k
)(
l
+
q
+1)
2
+
i,k
≡
m
(
mod
2)
,l
+2
≤
i
≤
l
+
q
+1;
m
(
l
+
q
+1)+
[
k
−
(
m
−
1)](
l
+
q
+1)
2
+1
−
i,k
≡
m
−
1(
mod
2)
,l
+2
≤
i
≤
l
+
q
+1
.
d
±
þ
I
Ò
Œ
,
ˆ
º:
Šƒ
Ú
•
:
ω
(
u
0
i
) = (
m
+1)(
l
+
q
+1)+1
−
i,
1
≤
i
≤
l
;
ω
(
u
1
0
) =
m
(
l
+1)
2
+
l
+1
2
,k
≡
m
(
mod
2);
(
m
+1)(
l
+1)
2
+
l
+1
2
,k
≡
m
−
1(
mod
2);
ω
(
u
0
i
) = (
m
+1)(
l
+
q
+1)+1
−
i,l
+2
≤
i
≤
l
+
q
+1;
ω
(
u
2
0
) =
mq
2
+
q
2
+(
m
+1)(
l
+
q
+1)
,k
≡
m
(
mod
2);
(
m
+1)
q
2
+
q
2
+(
m
+1)(
l
+
q
+1)
,k
≡
m
−
1(
mod
2);
ω
(
u
ki
) =
(
m
+1)(
l
+
q
+1)+1
,k
≡
m
(
mod
2)
,
1
≤
i
≤
l
;
m
(
l
+
q
+1)+1
,k
≡
m
−
1(
mod
2)
,
1
≤
i
≤
l
;
ω
(
u
i
) =
(
m
+1)(
l
+
q
+1)+1
,k
≡
m
(
mod
2)
,
1
≤
i
≤
m
;
m
(
l
+
q
+1)+1
,k
≡
m
−
1(
mod
2)
,
1
≤
i
≤
m
;
ω
(
u
ki
) =
(
m
+1)(
l
+
q
+1)+1
,k
≡
m
(
mod
2)
,l
+2
≤
i
≤
l
+
q
+1;
m
(
l
+
q
+1)+1
,k
≡
m
−
1(
mod
2)
,l
+2
≤
i
≤
l
+
q
+1
.
i
=
l
+
q
+1,
k
≡
m
−
1(
mod
2)
ž
,
k
ω
(
u
0
l
+
q
+1
)=
m
(
l
+
q
+1)+1
.
²
w
/
,
f
´
ã
P
m
(
S
l,q
)
˜
‡
Û
Ü
‡
Ž
I
Ò
…
¦
^
l+q+3
«
p
É
ô
Ú
,
P
m
(
S
l,q
)
≤
l
+
q
+ 3.
Š
â
Ú
n
2
Œ
•
l
+
q
+ 1=
χ
(
P
m
(
S
l,q
))
≤
χ
la
(
P
m
(
S
l,q
)).
n
þ
Œ
,
ã
P
m
(
S
l,q
)
Û
Ü
‡
Ž
X
Ú
ê
•
l
+
q
+1
≤
P
m
(
S
l,q
)
≤
l
+
q
+3.
X
P
2
(
S
3
,
4
)
Û
Ü
‡
Ž
X
Ú
„
ã
5.
Figure5.
Thelocalantimagiccoloringof
P
2
(
S
3
,
4
)
ã
5.
P
2
(
S
3
,
4
)
Û
Ü
‡
Ž
X
Ú
DOI:10.12677/aam.2021.10114304054
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7
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8
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,
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Æ
Ä
7
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8
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(2021D01C078);2020
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6
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©
z
[1]Hartsfield,N.andRingel,G.(1990)PearlsinGraphTheory.AcademicPress,INC.,Boston.
[2]Arumugam,S.,Premalatha,K.,Bacˇa,M.andSemaniˇcov´a-Feˇnovˇc´ıkov´a,A.(2017)LocalAn-
timagicVertexColoringofaGraph.
GraphsandCombinatorics
,
33
,275-285.
https://doi.org/10.1007/s00373-017-1758-7
[3]Bensmail, J.,Senhaji,M.and Lyngsie,K.S.(2017)On aCombinationof the1-2-3 Conjecture
andtheAntimagicLabellingConjecture.
DiscreteMathematicsandTheoreticalComputer
Science
,
19
,1-17.
[4]Lau, G.C.,Shiu,W.C.andNg,H.K. (2018) OnLocal Antimagic Chromatic Number ofCycle-
RelatedJoinGraphs.
DiscussionesMathematicaeGraphTheory
,
41
,133-152.
https://doi.org/10.7151/dmgt.2177
[5]Lau,G.C.,Shiu,W.C.andNg,H.K.(2018)OnLocalAntimagicChromaticNumberofCut-
Vertices.arXiv:1805.04801[math.CO]
[6]Lau,G.C.,Ng,H.K.andShiu,W.C.(2020)AffirmativeSolutionsonLocalAntimagicChro-
maticNumber.
GraphsandCombinatorics
,
36
,1337-1354.
https://doi.org/10.1007/s00373-020-02197-2
[7]Arumugam, S.,Lee, Y.C., Premalatha, K. and Wang, T.M. (2018) On Local Antimagic Vertex
ColoringforCoronaProductsofGraphs.arXiv:1808.04956[math.CO]
https://doi.org/10.1007/s00373-017-1758-7
[8]Nazula,N.H.,Slamin,S.andDafik,D.(2018)LocalAntimagicVertexColoringofUnicyclic
Graphs.
IndonesianJournalofCombinatorics
,
2
,30-34.
https://doi.org/10.19184/ijc.2018.2.1.4
[9]Dafik, D.,Agustin, I.H.,Marsidi andKurniawati,E.Y. (2020)Onthe Local Antimagic Vertex
ColoringofSub-DevidedSomeSpecialGraph.
JournalofPhysicsConferenceSeries
,
1538
,
ArticleID:012021.https://doi.org/10.1088/1742-6596/1538/1/012021
DOI:10.12677/aam.2021.10114304055
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