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PureMathematics
n
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Æ
,2021,11(11),1933-1948
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2021.1111216
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ExtremalUnicyclicGraphsofthe
SmallDiameterwithRespectto
PermanentalSum
YinggangBai
SchoolofMathematicsandStatistics,QinghaiNationalitiesUniversity,XiningQinghai
Received:Oct.19
th
,2021;accepted:Nov.23
rd
,2021;published:Nov.30
th
,2021
Abstract
Let
G
beagraphwith
n
vertex,thepermanentalsumof
G
isthesumoftheabsolute
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f
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[J].
n
Ø
ê
Æ
,2021,11(11):1933-1948.
DOI:10.12677/pm.2021.1111216
x
=
f
valutesofthecofficients.Computingthepermanentalpolynomialsofgraphsis
]p
.
Inthispaper,wewilldeterminethegraphminimizingthepermanentalsumamong
allunicyclicgraphswithdiameter
3
and
4
,andthecorrespondingextremalbicyclic
graphsarealsodetermined.
Keywords
PermanentalPolynomial,PermanentalSum,HosoyaIndex,
UnicyclicGraph
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2021.11112161935
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DOI:10.12677/pm.2021.11112161936
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:
G
1
3
(
a,b
)
¥
,
a
,
b
þ
Œ
u
u
0,
a
+
b
+5 =
n
"
G
2
3
(
a,b,c
)
¥
a
,
b
,
c
–
ü
‡
ê
Œ
u
u
1 ,
a
+
b
+
c
+3 =
n
""
G
3
3
(
a,b
)
¥
a
,
b
–
˜
‡
ê
Œ
u
u
1,
a
+
b
+4 =
n
"
G
4
3
(
a,b
)
¥
a
,
b
–
˜
‡
ê
Œ
u
u
1,
a
+
b
+5 =
n
"
±
þ
a
,
b
,
c
þ
•
g
,
ê
,
n
≥
7
"
½
n
4.1
-
G
3
(
n
) =
{
G
1
3
(
a,b
)
,G
2
3
(
a,b,c
)
,G
3
3
(
a,b
)
,G
4
3
(
a,b
)
}
L
«
†
»
•
3
n
ü
ã
8
Ü
,
G
∈
G
3
(
n
),
K
PS
(
G
)
≥
3
n
−
4,
…
=
G
∼
=
G
1
3
(
a
+
b
+
c
−
1
,
0
,
1)(
„
ã
5)
ž
Ò
¤
á
"
y
²
(I)
é
G
2
3
(
a,b,c
)
A^
C
†
n
,
C
•
G
1
3
(
a
+
b,
0
,c
) ,
=
r
†
º:
v
3
ƒ
ë
¤
k
]
!
:
í
Ø
Ú
DOI:10.12677/pm.2021.11112161937
n
Ø
ê
Æ
x
=
f
º:
v
2
ƒ
ë
,
d
Ú
n
3.6
PS
(
G
1
3
(
a,b,c
)
≥
PS
(
G
1
3
(
a
+
b,
0
,c
))(1)
Figure5.
G
2
3
(
a
+
b
+
c
−
1
,
0
,
1)
,G
4
4
(1
,m
+
t
−
1))
ã
5.
G
2
3
(
a
+
b
+
c
−
1
,
0
,
1)
,G
4
4
(1
,m
+
t
−
1))
-
a
+
b
=
s,c
=
t,s
+
t
+ 3=
n,PS
(
G
1
3
(
a
+
b,
0
,c
))=
PS
(
G
1
3
(
s,
0
,t
)) (
s
≥
1
,t
≥
1),
d
Ú
n
3.8
PS
(
G
1
3
(
s,
0
,t
)) = 2
s
+2
t
+
st
+6
,PS
(
G
1
3
(
s
+1
,
0
,t
−
1)) = 2(
s
+1)+2(
t
−
1)+(
s
+1)(
t
−
1)+6
"
K
PS
(
G
1
3
(
s,
0
,t
))
−
PS
(
G
1
3
(
s
+1
,
0
,t
−
1)) =
PS
(
G
1
3
(
a
+
b,
0
,c
))
−
PS
(
G
1
3
(
a
+
b
+1
,
0
,c
−
1)) =
s
−
t
+1,
©
ü
«
œ
¹
"
œ
¹
˜
s
−
t
+1
≥
0
ž
,
PS
(
G
1
3
(
s,
0
,t
))
≥
PS
(
G
1
3
(
s
+1
,
0
,t
−
1))
"
PS
(
G
1
3
(
s
+
t
−
1
,
0
,
1))
≤
PS
(
G
1
3
(
s
+
t
−
2
,
0
,
2))
≤···≤
PS
(
G
1
3
(
s,
0
,t
))
"
¤
±
PS
(
G
1
3
(
a
+
b,
0
,c
)) =
PS
(
C
3
(
s,
0
,t
))
≥
PS
(
C
3
(
s
+
t
−
1
,
0
,
1)) = 3
n
−
4
"
œ
¹
s
−
t
+1
<
0
ž
,
PS
(
G
1
3
(
s,
0
,t
))
<PS
(
G
1
3
(
s
+1
,
0
,t
−
1))
"
PS
(
G
1
3
(1
,
0
,s
+
t
−
1))
<PS
(
G
1
3
(2
,
0
,s
+
t
−
2))
<
···
<PS
(
G
1
3
(
s,
0
,t
))
"
PS
(
G
1
3
(
a
+
b,
0
,c
)) =
PS
(
G
1
3
(
s,
0
,t
))
≥
PS
(
G
1
3
(1
,
0
,s
+
t
−
1)) = 3
n
−
4
"
:
PS
(
G
1
3
(
a
+
b,
0
,c
))
≥
3
n
−
4(2)
d
(1)(2)
G
∼
=
G
1
3
(
a,b,c
)
,PS
(
G
)
≥
3
n
−
4,
G
∼
=
G
1
3
(
a
+
b
+
c
−
1
,
0
,
1)
Ò
¤
á
"
(II)
é
G
3
3
(
a,b
)
A^
C
†
n
C
¤
G
3
3
(
a
+
b,
0),
=
r
†
º:
v
1
ƒ
ë
]
!
:
Ü
í
Ø
Ú
º:
v
4
ë
,
d
Ú
n
3.6
PS
(
G
3
3
(
a,b
))
≥
PS
(
G
3
3
(
a
+
b,
0))
"
d
Ú
n
3.8
PS
(
G
2
3
(
a
+
b,
0)) = 3
n
−
3
"
¤
±
PS
(
G
2
3
(
a,b
))
≥
3
n
−
3(3)
(III)
é
G
4
3
(
a,b
)
A^
C
†
n
,
C
¤
G
4
3
(
a
+
b,
0)
=
r
†
º:
v
3
ƒ
ë
]
!
:
í
Ø
Ú
º:
v
4
ë
,
d
Ú
n
3.6
PS
(
G
4
3
(
a,b
))
≥
G
4
3
(
a
+
b,
0)
DOI:10.12677/pm.2021.11112161938
n
Ø
ê
Æ
x
=
f
d
Ú
n
3.8
PS
(
G
4
3
(
a
+
b,
0)) = 5
n
−
12
"
¤
±
PS
(
G
4
3
(
a,b
))
≥
5
n
−
12(4)
(IV)
d
Ú
n
3.8
PS
(
G
1
3
(
a,b
)) = 2(
a
+3)(
b
+2)+2
"
PS
(
G
1
3
(
a
+1
,b
−
1)) = 2(
a
+4)(
b
+1)+2
,PS
(
G
1
3
(
a,b
))
−
PS
(
G
1
3
(
a
+1
,b
−
1)) =
a
−
b
+2
"
©
ü
«
œ
¹
"
œ
¹
˜
a
−
b
+2
≥
0
ž
,
PS
(
G
1
3
(
a,b
))
≥
G
1
3
(
a
+1
,b
−
1)),
:
PS
(
G
1
3
(
a
+
b,
0))
≤
PS
(
G
1
3
(
a
+
b
−
1
,
1)
≤···≤
PS
(
G
1
3
(
a,b
))
"
¤
±
PS
(
G
1
3
(
a,b
))
≥
PS
(
G
1
3
(
a
+
b,
0)) = 4
n
−
6
"
œ
¹
a
−
b
+2
<
0
ž
,
PS
(
G
1
3
(
a,b
))
<G
1
3
(
a
+1
,b
−
1)),,
:
PS
(
G
1
3
(0
,a
+
b
))
≤
PS
(
G
1
3
(1
,a
+
b
−
1))
≤···≤
PS
(
G
1
3
(
a,b
))
"
¤
±
PS
(
G
1
3
(
a,b
))
≥
PS
(
G
1
3
(0
,a
+
b
)) = 6
n
−
18
"
du
6
n
−
18
>
4
n
−
6,
¤
±
G
1
3
(
a,b
))
≥
4
n
−
6(5)
G
∼
=
G
2
3
(
a
+
b
+
c
−
1
,
0
,
1)
ž
,
d
(
.
)
PS
(
G
) = 3
n
−
4
"
G
6
=
G
2
3
(
a
+
b
+
c
−
1
,
0
,
1)
ž
,
©
o
«
œ
¹
"
œ
¹
˜
G
∼
=
G
2
3
(
a,b,c
)
ž
,
d
(2)
PS
(
G
)
>
3
n
−
4
"
œ
¹
G
∼
=
G
3
3
(
a,b
)
ž
,
d
(3)
PS
(
G
)
≥
3
n
−
3
>
3
n
−
4
"
œ
¹
n
G
∼
=
G
4
3
(
a,b
)
ž
,
d
(4)
PS
(
G
)
≥
5
n
−
12
>
3
n
−
4
"
œ
¹
o
G
∼
=
G
1
3
(
a,b
)
ž
,
d
(5)
PS
(
G
)
≥
4
n
−
6
>
3
n
−
4
"
¤
±
G
∈
G
3
(
n
)(
n
≥
7),
K
PS
(
G
)
≥
3
n
−
4,
…
=
G
∼
=
G
2
3
(
a
+
b
+
c
−
1
,
0
,
1)
ž
Ò
¤
á
"
5.
†
»
•
4
ü
ã
(
„
ã
6)
[
È
Ú
4
Š
ï
Ä
ã
G
1
4
,G
2
4
,G
3
4
,G
4
4
G
5
4
,G
6
4
,G
7
4
,G
8
4
,G
9
4
¥
ë
ê
I
‡
÷
v
±
e
^
‡
:
G
1
4
¥
,
a
i
þ
Œ
u
u
1,
i
∈{
1
,
2
,...,k
}
"
G
2
4
¥
©
ü
«
œ
¹
:(1)
p
1
,
p
2
–
˜
‡
ê
Œ
u
u
1,
a
i
þ
Œ
u
1
"
i
∈{
1
,
2
,...,k
}
"
(2)
p
1
=
p
2
= 0,
a
i
–
k
ü
‡
ê
Œ
u
u
1,
i
∈{
1
,
2
,...,k
}
"
(3)
G
3
4
¥
p
1
=
p
2
= 0
,a
i
–
k
˜
‡
ê
Œ
u
u
1
,i
∈{
1
,
2
,...,k
}
"
DOI:10.12677/pm.2021.11112161939
n
Ø
ê
Æ
x
=
f
Figure6.
Diagramofasinglecirclewithadiameterof4
ã
6.
†
»
•
4
ü
ã
(4)
G
4
4
¥
÷
v
p
1
p
3
6
= 0
½
ö
p
2
p
4
6
= 0
"
(5)
G
5
4
¥
,
p
1
=
p
2
= 0
,a
i
–
k
˜
‡
ê
Œ
u
u
1,
i
∈{
1
,
2
,...,k
}
"
(6)
G
6
4
¥
,
p
1
p
3
6
= 0,
p
1
p
4
6
= 0,
p
2
p
4
6
= 0,
p
2
p
5
6
= 0
Ú
p
3
p
5
6
= 0
±
þ
œ
¹
–
˜
«
œ
¹
¤
á
"
(7)
G
7
4
,
G
8
4
,G
9
4
–
k
˜
‡
p
i
Œ
u
u
1,
i
∈{
1
,
2
,
3
}
"
±
þ
s
,
r
,
a
i
,
p
i
,
i
∈{
1
,
2
,...,k
}
þ
•
g
,
ê
, n
≥
8
"
Ú
n
5.1
PS
(
G
1
4
(
s,r,a
1
,a
2
,...,a
k
))
≥
6
n
−
14
"
y
²
(I)
k
≥
2
ž
"
é
ã
G
1
4
(
s,r,a
1
,a
2
,...,a
k
),
A^
C
†
ò
Ú
º:
u
2
,u
3
,...,u
k
ƒ
ë
]
!
:
í
Ø
Ú
º:
v
4
ë
"
-
m
=
r
+(1+
a
2
)+(1+
a
3
)+
...
+(1+
a
k
)(
m
≥
2
,m
+
s
≥
2)
,t
=
a
1
(
t
≥
1),
d
½
n
3.4
PS
(
G
1
4
(
s,r,a
1
,a
2
,...,a
k
))
≥
PS
(
G
1
4
(
s,m,t
))(6)
d
Ú
n
3.8
DOI:10.12677/pm.2021.11112161940
n
Ø
ê
Æ
x
=
f
PS
(
G
1
4
(
s,m,t
)) = (6
m
+8
t
+4
s
)+(6
tm
+2
ts
+2
ms
)+2
tms
+14
PS
(
G
1
4
(
s,m,t
))
−
PS
(
G
1
4
(
s
+1
,m
−
1
,t
)) = 2(
t
+1)(3+
s
−
m
)
−
2,
©
ü
«
œ
¹
"
œ
¹
˜
3+
s>m,PS
(
G
1
4
(
s,m,t
))
>PS
(
G
1
4
(
s
+1
,m
−
1
,t
)),
PS
(
G
1
4
(
s
+
m,
0
,t
))
<PS
(
G
1
4
(
s
+
m
−
1
,
1
,t
))
<
···
<PS
(
G
1
4
(
s,m,t
))
¤
±
PS
(
G
1
4
(
s,m,t
))
≥
PS
(
G
1
4
(
s
+
m,
0
,t
))
"
œ
¹
3+
s
≤
m,PS
(
G
1
4
(
s,m,t
))
<PS
(
G
1
4
(
s
+1
,m
−
1
,t
)),
PS
(
G
1
4
(0
,s
+
m,t
))
<PS
(
G
1
4
(1
,s
+
m
−
1
,t
))
<
···
<PS
(
G
1
4
(
s,m,t
))
¤
±
PS
(
G
1
4
(
s,m,t
))
≥
PS
(
G
1
4
(0
,s
+
m,t
)))
"
PS
(
G
1
4
(0
,s
+
m,t
)))
−
PS
(
G
1
4
(
s
+
m,
0
,t
)) = 2(
m
+
s
)(2
t
+1)
>
0
"
¤
±
PS
(
G
1
4
(
s,m,t
))
≥
PS
(
G
1
4
(
s
+
m,
0
,t
))(7)
(II)
k
= 1
ž
"
-
m
=
r,m
≥
0
"
d
(7)
ª
PS
(
G
1
4
(
s,m,t
))
≥
PS
(
G
1
4
(
s
+
m,
0
,t
))
"
PS
(
G
1
4
(
s
+
m,
0
,t
))
−
PS
(
G
1
4
(
s
+
m
+1
,
0
,t
−
1)) = 2(
m
+
s
+3
−
t
),
©
ü
«
œ
¹
"
œ
¹
˜
t
≥
m
+
s
+3
,PS
(
G
1
4
(
s
+
m,
0
,t
))
≤
PS
(
G
1
4
(
s
+
m
+1
,
0
,t
−
1)),
PS
(
G
1
4
(2
,
0
,s
+
m
−
2))
≤
PS
(
G
1
4
(3
,
0
,s
+
m
−
3))
≤···≤
PS
(
G
1
4
(
s
+
m,
0
,t
))
"
¤
±
PS
(
G
1
4
(
s
+
m,
0
,t
))
≥
PS
(
G
1
4
(2
,
0
,s
+
m
−
2)) = 12
n
−
62
"
œ
¹
t<m
+
s
+3
,PS
(
G
1
4
(
s
+
m,
0
,t
))
>PS
(
G
1
4
(
s
+
m
+1
,
0
,t
−
1)),
PS
(
G
1
4
(
s
+
m
−
1
,
0
,
1))
≤
PS
(
G
1
4
(
s
+
m
−
2
,
0
,
2))
≤···≤
PS
(
G
1
4
(
s
+
m,
0
,t
))
¤
±
PS
(
G
1
4
(
s
+
m,
0
,t
))
≥
PS
(
G
1
4
(
s
+
m
−
1
,
0
,
1)) = 6
n
−
14
"
du
12
n
−
62
>
6
n
−
14,
PS
(
G
1
4
(
s
+
m,
0
,t
)
≥
6
n
−
14(8)
(
Ü
(6)(7)
Ú
(8)
PS
(
G
1
4
(
s,r,a
1
,a
2
,...,a
k
))
≥
6
n
−
14
"
Ú
n
5.2
PS
(
G
2
4
(
p
1
,p
2
,r,a
1
,a
2
,
···
,a
k
))
≥
5
n
−
11
"
y
²
(I)
p
1
,p
2
–
˜
‡
Ø
•
0
"
(i)
k
≥
2
ž
,
é
G
2
4
(
p
1
,p
2
,r,a
1
,a
2
,
···
,a
k
),
A^
C
†
ò
Ú
ò
Ú
º:
v
1
ë
]
!
:
í
Ø
Ú
º:
v
2
ë
"
2
|
^
C
†
n
ò
Ú
º:
u
2
,u
3
,...,u
k
ƒ
ë
]
!
:
í
Ø
Ú
º:
v
3
ë
,
é
A
DOI:10.12677/pm.2021.11112161941
n
Ø
ê
Æ
x
=
f
ã
P
•
G
2
4
(
s,t,m
)
"
-
p
1
+
p
2
=
s,t
=
r
+(
a
2
+1)+(
a
3
+1)+
...
(
a
k
+1)(
t
≥
2)
,a
1
=
m
(
m
≥
1) ,
d
Ú
n
3.4
Ú
Ú
n
3.7
PS
(
G
2
4
(
p
1
,p
2
,r,a
1
,a
2
,
···
,a
k
))
≥
PS
(
G
2
4
(
s,t,m
))(9)
d
Ú
n
3.8
PS
(
G
2
4
(
s,t,m
)) = (
m
+1)(
st
+2
s
+2
t
+4)+
s
+2
"
PS
(
G
2
4
(
s,t,m
))
−
PS
(
G
2
4
(
s,t
+1
,m
−
1)) = (
s
+2)(
t
+2
−
m
),
©
ü
«
œ
¹
"
œ
¹
˜
m
≤
t
+2
,PS
(
G
2
4
(
s,t,m
))
≥
PS
(
G
2
4
(
s,t
+1
,m
−
1)),
PS
(
G
2
4
(
s,t
+
m
−
1
,
1))
≤
PS
(
G
2
4
(
s,t
+
m
−
2
,
2))
≤···≤
PS
(
G
2
4
(
s,t,m
))
"
¤
±
PS
(
G
2
4
(
s,t,m
))
≥
PS
(
G
2
4
(
s,t
+
m
−
1
,
1))
"
œ
¹
m>t
+2
,PS
(
G
2
4
(
s,t,m
))
<PS
(
G
2
4
(
s,t
+1
,m
−
1)),
PS
(
G
2
4
(
s,
0
,t
+
m
))
<PS
(
G
2
4
(
s,
1
,t
+
m
−
1))
≤···≤
PS
(
G
2
4
(
s,t,m
))
"
¤
±
PS
(
G
2
4
(
s,t,m
))
≥
PS
(
G
2
4
(
s,t
+
m
−
1
,
1))(10)
(ii)
k
= 1
,t
≥
0
ž
"
d
Ú
n
3.8
PS
(
G
2
4
(
s,t,m
)) = (
m
+1)(
st
+2
s
+2
t
+4)+
s
+2
"
d
(10)
PS
(
G
2
4
(
s,t,m
))
≥
PS
(
G
2
4
(
s,t
+
m
−
1
,
1))
"
PS
(
G
2
4
(
s,t
+
m
−
1
,
1))
−
PS
(
G
2
4
(
s
−
1
,t
+
m,
1)) = 2(
m
+
t
−
s
+1)
−
1,
©
ü
«
œ
¹
"
œ
¹
˜
s<m
+
t
+1
,PS
(
G
2
4
(
s,t
+
m
−
1
,
1))
>PS
(
G
2
4
(
s
−
1
,t
+
m,
1)),
PS
(
G
2
4
(1
,s
+
t
+
m
−
2
,
1))
<PS
(
G
2
4
(2
,s
+
t
+
m
−
3
,
1))
<
···
<PS
(
G
2
4
(
s,t
+
m
−
1
,
1))
¤
±
PS
(
G
2
4
(
s,t
+
m
−
1
,
1))
≥
PS
(
G
2
4
(1
,s
+
t
+
m
−
2
,
1)) = 6
n
−
21
"
œ
¹
s
≥
m
+
t
+1
,PS
(
G
2
4
(
s,t
+
m
−
1
,
1))
<PS
(
G
2
4
(
s
−
1
,t
+
m,
1)),
PS
(
G
2
4
(
s
+
t
+
m
−
1
,
0
,
1))
<PS
(
G
2
4
(
s
+
t
+
m
−
2
,
1
,
1)))
<
···
<PS
(
G
2
4
(
s,t
+
m
−
1
,
1))
¤
±
PS
(
G
2
4
(
s,t
+
m
−
1
,
1))
≥
PS
(
G
2
4
(
s
+
t
+
m
−
1
,
0
,
1))) = 5
n
−
11
"
du
6
n
−
21
>
5
n
−
11,
(
Ü
(8)
Ú
(9)(10)
PS
(
G
2
4
(
s,t
+
m
−
1
,
1))
≥
5
n
−
11(11)
(II)
p
1
=
p
2
= 0
ž
,
(i)
k
≥
3
A^
C
†
ò
†
º:
u
3
,u
4
,...,u
k
ë
]
!
:
í
Ø
Ú
º:
v
3
ƒ
ë
,
ã
P
•
G
2
∗
4
(
m,s,t
),
-
r
+(
a
3
+1)+(
a
4
+1)+
...
+(
a
k
+1) =
m,a
1
=
s,a
2
=
t
(
s
≥
1
,t
≥
1
,m
≥
2),
d
DOI:10.12677/pm.2021.11112161942
n
Ø
ê
Æ
x
=
f
Ú
n
3.4
PS
(
G
2
4
(
p
1
,p
2
,r,a
1
,a
2
,
···
,a
k
))
≥
PS
(
G
2
∗
4
(
m,s,t
))(12)
d
Ú
n
3.8
PS
(
G
2
∗
4
(
m,s,t
) = 2(
m
+3)(
t
+1)(
s
+1)+2(
s
+1)+2(
t
+1)
"
PS
(
G
2
∗
4
(
m,s,t
)
−
PS
(
G
2
∗
4
(
m
+1
,s
−
1
,t
) = 2[(
t
+1)(
m
+3
−
s
)+1],
©
ü
«
œ
¹
"
œ
¹
˜
s
≤
m
+3
,PS
(
G
2
∗
4
(
m,s,t
)
>PS
(
G
2
∗
4
(
m
+1
,s
−
1
,t
),
PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
<PS
(
G
2
∗
4
(
m
+
s
−
2
,
2
,t
)
<
···
<PS
(
G
2
∗
4
(
m,s,t
)
¤
±
PS
(
G
2
∗
4
(
m,s,t
)
≥
PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
"
œ
¹
s>m
+3
,PS
(
G
2
∗
4
(
m,s,t
)
<PS
(
G
2
∗
4
(
m
+1
,s
−
1
,t
),
PS
(
G
2
∗
4
(2
,m
+
s
−
2
,t
)
<PS
(
G
2
∗
4
(3
,m
+
s
−
3
,t
)
<
···
<PS
(
G
2
∗
4
(
m,s,t
)
¤
±
PS
(
G
2
∗
4
(
m,s,t
)
≥
PS
(
G
2
∗
4
(2
,m
+
s
−
2
,t
)
"
PS
(
G
2
∗
4
(2
,m
+
s
−
2
,t
)
−
PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
) = 2(3
t
+4)(
m
+
s
−
3)
≥
0
"
¤
±
PS
(
G
2
∗
4
(
m,s,t
)
≥
PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
"
(ii)
k
= 2
ž
,
r
=
m
≥
0
,a
1
=
s,a
2
=
t
d
Ú
n
3.8
PS
(
G
2
∗
4
(
m,s,t
) = 2(
m
+3)(
t
+1)(
s
+1)+2(
s
+1)+2(
t
+1)
"
PS
(
G
2
∗
4
(
m,s,t
)
−
PS
(
G
2
∗
4
(
m
+1
,s
−
1
,t
) = 2[(
t
+1)(
m
+3
−
s
)+1],
©
ü
«
œ
¹
"
œ
¹
˜
s
≤
m
+3
,PS
(
G
2
∗
4
(
m,s,t
)
>PS
(
G
2
∗
4
(
m
+1
,s
−
1
,t
),
PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
<PS
(
G
2
∗
4
(
m
+
s
−
2
,
2
,t
)
<
···
<PS
(
G
2
∗
4
(
m,s,t
)
¤
±
PS
(
G
2
∗
4
(
m,s,t
)
≥
PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
"
œ
¹
s>m
+3
,PS
(
G
2
∗
4
(
m,s,t
)
<PS
(
G
2
∗
4
(
m
+1
,s
−
1
,t
),
PS
(
G
2
∗
4
(0
,m
+
s,t
)
<PS
(
G
2
∗
4
(1
,m
+
s
−
1
,t
)
<
···
<PS
(
G
2
∗
4
(
m,s,t
)
¤
±
PS
(
G
2
∗
4
(
m,s,t
)
≥
PS
(
G
2
∗
4
(0
,m
+
s,t
)
"
PS
(
G
2
∗
4
(0
,m
+
s,t
)
−
PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
) = 2(
t
+2)(
m
+
s
−
1)
≥
0
"
¤
±
PS
(
G
2
∗
4
(
m,s,t
)
≥
PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
"
¤
±
n
Ü
(i)
Ú
(ii)
PS
(
G
2
∗
4
(
m,s,t
)
≥
PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)(13)
PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
−
PS
(
G
2
∗
4
(
m
+
s,
1
,t
−
1) = 4(
m
+
s
−
t
)+10,
©
ü
«
œ
¹
"
œ
¹
˜
m
+
s
≥
t
−
2
,PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
>PS
(
G
2
∗
4
(
m
+
s,
1
,t
−
1),
PS
(
G
2
∗
4
(
m
+
s
−
2
,
1
,
1)
<PS
(
G
2
∗
4
(
m
+
s
−
3
,
1
,
2)
<
···
<PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
DOI:10.12677/pm.2021.11112161943
n
Ø
ê
Æ
x
=
f
¤
±
PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
≥
PS
(
G
2
∗
4
(
m
+
s
−
2
,
1
,
1) = 8
n
−
24(14)
œ
¹
m
+
s<t
−
2
,PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
<PS
(
G
2
∗
4
(
m
+
s,
1
,t
−
1),
PS
(
G
2
∗
4
(0
,
1
,m
+
s
−
1)
<PS
(
G
2
∗
4
(1
,
1
,m
+
s
−
2)
<
···
<PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
¤
±
PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
≥
PS
(
G
2
∗
4
(0
,
1
,m
+
s
−
1) = 14
n
−
66
"
Ï
•
14
n
−
66
>
8
n
−
24,
Š
â
(11) (12)(13)(14)
PS
(
G
2
∗
4
(
m
+
s
−
1
,
1
,t
)
≥
8
n
−
24
du
8
n
−
24
>
5
n
−
11,
(
Ü
(I) (II)
(
Ø
Ñ
PS
(
G
2
4
)
≥
5
n
−
11
"
Ú
n
5.3
PS
(
G
3
4
(
p
1
,p
2
,r,a
1
,a
2
,
···
,a
k
))
≥
6
n
−
15
"
y
²
(I)
p
1
,p
2
–
˜
‡
Ø
•
0 ,
Ï
L
C
z
n
,
ò
†
v
2
ƒ
ë
]
!
:
í
Ø
Ú
º:
v
3
ƒ
ë
"
2
Ï
L
C
z
ò
u
2
,...,u
k
ƒ
ë
]
!
:
í
Ø
Ú
º:
v
4
ë
"
-
p
1
+
p
2
=
t,r
+(
a
2
+1)+(
a
3
+
1)+
...
+(
a
k
−
1
+1) =
m,a
1
=
s
£
t
≥
1
,s
≥
1
,m
≥
1
,m
+
t
+
s
+5 =
n
)
"
d
Ú
n
3.4
Ú
3.6
PS
(
G
3
4
(
p
1
,p
2
,r,a
1
,a
2
,
···
,a
k
))
≥
PS
(
G
1
4
(
s,m,t
))
"
d
Ú
n
3.8
PS
(
G
3
4
(
s,m,t
)) = (3
m
+3
t
+2
mt
+9)(
s
+1)+2
t
+3
"
é
G
3
4
(
s,m,t
)
A^
C
†
n
,
ò
†
º:
v
3
ë
]
!
:
í
K
Ú
º:
v
4
ë
é
A
ã
P
•
G
3
4
(
s,m
+
t,
0)
"
d
Ú
n
3.6
PS
(
G
3
4
(
s,m,t
))
≥
G
3
4
(
s,m
+
t,
0)
"
é
G
3
4
(
s,m,t
)
A^
C
†
n
,
ò
†
º:
v
4
ë
]
!
:
í
K
Ú
º:
v
3
ë
é
A
ã
P
•
G
3
4
(
s,
0
,m
+
t
)
"
d
Ú
n
3.6
PS
(
G
3
4
(
s,m,t
))
≥
G
3
4
(
s,
0
,m
+
t
)
"
d
Ú
n
3.8
G
3
4
(
s,
0
,m
+
t
)
−
G
3
4
(
s,m
+
t,
0) = 2(
m
+
t
)
>
0
"
¤
±
PS
(
G
3
4
(
s,m,t
))
≥
G
3
4
(
s,m
+
t,
0)
"
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DOI:10.12677/pm.2021.11112161944
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DOI:10.12677/pm.2021.11112161945
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DOI:10.12677/pm.2021.11112161946
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[1]Merris, R., Rebman,K.R.andWatkins, W.(1981) Permanental Polynomialsof Graphs.
Linear
AlgebraandItsApplications
,
38
,273-288.https://doi.org/10.1016/0024-3795(81)90026-4
[2]Kasum, D., Trinajsti´c, N.andGutman, I.(1981) ChemicalGraph Theory.III.OnPermanental
Polynomial.
CroaticaChemicaActa
,
54
,321-328.
[3]Cash,G.G.(2000)ThePermanentalPolynomial.
JournalofChemicalInformationandMod-
elingSciences
,
40
,1203-1206.https://doi.org/10.1021/ci000031d
[4]Cash,G.G.(2000)PermanentalPolynomials of SmallerFullerenes.
JournalofChemicalInfor-
mationandModelingSciences
,
40
,1207-1209.https://doi.org/10.1021/ci0000326
[5]Chen, R. (2004)A Note on the Relations between the Permanental and Characteristic Polyno-
mials of CoronoidHydrocarbons.
MATCHCommunicationsinMathematicalandinComputer
Chemistry
,
51
,137-148.
[6]Chou,Q.,Liang,H.andBai,F.(2011)RemarksontheRelationsbetweenthePermanental
andCharacteristicPolynomialsofFullerenes.
MATCHCommunicationsinMathematicaland
inComputerChemistry
,
66
,743-750.
[7]Dehmer,M.,
etal.
(2017)HighlyUniqueNetworkDescriptorsBasedontheRootsofthe
PermanentalPolynomial.
InformationSciences
,
408
,176-181.
https://doi.org/10.1016/j.ins.2017.04.041
[8]Gutman,I.andCash,G.G.(2002)RelationsbetweenthePermanentalandCharacteristic
Polynomials ofFullerenes andBenzenoid Hydro-Carbons.
MATCHCommunicationsinMath-
ematicalandinComputerChemistry
,
45
,55-70.
[9]Liang,H.,Tong,H.andBai,F.(2008)ComputingthePermanentalPolynomialof
C
60
in
Parallel.
MATCHCommunicationsinMathematicalandinComputerChemistry
,
60
,349-
358.
[10]Shi,Y.,Dehmer,M.,Li,X.andGutman,I.(2016)GraphPolynomials.CRCPress,Boca
Raton.https://doi.org/10.1201/9781315367996
[11]Wu,T.andLai,H.(2018)OnthePermanentalNullityandMatchingNumberofGraphs.
LinearandMultilinearAlgebra
,
66
,516-524.https://doi.org/10.1080/03081087.2017.1302403
[12]Yan,W.andZhang,F.(2004)OnthePermanentalPolynomialofSomeGraphs.
Journalof
MathematicalChemistry
,
35
,175-188.https://doi.org/10.1023/B:JOMC.0000033254.54822.f8
[13]Yu, G.andQu,H.(2018)TheCoefficientsoftheImmanantalPolynomial.
AppliedMathematics
andComputation
,
339
,38-44.https://doi.org/10.1016/j.amc.2018.06.057
[14]Zhang,H.and Li,W.(2012)Computingthe PermanentalPolynomials ofBipartiteGraphsby
PfaffianOrientation.
DiscreteAppliedMathematics
,
160
,2069-2074.
https://doi.org/10.1016/j.dam.2012.04.007
DOI:10.12677/pm.2021.11112161947
n
Ø
ê
Æ
x
=
f
[15]Xie,S.,
etal.
(2004)CapturingtheLabileFullerene[50]asC
50
CI
10
.
Science
,
304
,699.
https://doi.org/10.1126/science.1095567
[16]Tong,H.,Liang,H.andBai,F.(2006)PermanentalPolynomialsoftheLargerFullerenes.
MATCHCommunicationsinMathematicalandinComputerChemistry
,
56
,141-152.
[17]Wu,T.andSo,W.(2019)UnicyclicGraphswithSecondLargestandSecondSmallestPer-
mantalSums.
AppliedMathematicsandComputation
,
351
,168-175.
https://doi.org/10.1016/j.amc.2019.01.056
[18]Chou,Q.,Liang,H.andBai,F.(2015)ComputingthePermanentalPolynomialoftheHigh
LevelFullerene
C
70
withHighPrecision.
MATCHCommunicationsinMathematicalandin
ComputerChemistry
,
73
,327-336.
[19]Li,W.,Qin,Z.andZhang,H.(2016)ExtremalHexagonalChainswithRespecttotheCo-
efficientsSumofthePermanentalPolynomial.
AppliedMathematicsandComputation
,
291
,
30-38.https://doi.org/10.1016/j.amc.2016.06.025
[20]Li,S.andWei,W.(2018)ExtremalOctagonalChainswithRespecttotheCoefficientsSum
ofthePermanentalPolynomial.
AppliedMathematicsandComputation
,
328
,45-57.
https://doi.org/10.1016/j.amc.2018.01.033
[21]Wu,T.andLai,H.(2018)OnthePermanentalSumofGraphs.
AppliedMathematicsand
Computation
,
331
,334-340.https://doi.org/10.1016/j.amc.2018.03.026
[22]Li,W.,Qin,Z.andWang,Y.(2020)EnumerationofPermanentalSumsofLattice.
Applied
MathematicsandComputation
,
370
,ArticleID:124914.
https://doi.org/10.1016/j.amc.2019.124914
[23]Wu, T.and L
j
,H. (2019)TheExtremalPermanentalSumforaQuasi-TreeGraph.
Complexity
,
2019
,ArticleID:4387650.https://doi.org/10.1155/2019/4387650
[24]Wu, T., Ren, S. and Das, K.(2019) SomeExtremal Graphs withRespect to PermanentalSum.
BulletinoftheMalaysianMathematicalSciencesSociety
,
42
,2947-2961.
https://doi.org/10.1007/s40840-018-0642-9
[25]Wu,T.andDas,K.(2020)OnthePermanentalSumofBicyclicGraphs.
Computationaland
AppliedMathematics
,
39
,ArticleNo.72.https://doi.org/10.1007/s40314-020-1108-x
[26]Hosoya,H. (1971)TopologicalIndex,a Newly ProposedQuantityCharacterizing the Topolog-
ical NatureofStructuralIsomersofSaturatedHydrocarbons.
BulletinoftheChemicalSociety
ofJapan
,
44
,2332-2339.https://doi.org/10.1246/bcsj.44.2332
[27]Feng,L.,Li,Z.,Liu,W.,Lu,L.andStevanovi´c,D.(2020)MinimalHarary IndexofUnicyclic
GraphswithDiameteratMost4.
AppliedMathematicsandComputation
,
381
,ArticleID:
125315.https://doi.org/10.1016/j.amc.2020.125315
[28]Wu,T.Z.andSo,W.(2019)UnicyclicGraphswithSecondLargestandSecondSmallest
PermanentalSums.
AppliedMathematicsandComputation
,
351
,168-175.
https://doi.org/10.1016/j.amc.2019.01.056
DOI:10.12677/pm.2021.11112161948
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