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AdvancesinAppliedMathematics
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,2022,11(4),2009-2016
PublishedOnlineApril2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.114217
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TheExtremalGraphoftheReciprocal
DistanceSignlessLaplacianMatrix
MeijiaoCheng
CollegeofMathematicsandComputerScience,ZhejiangNormalUniversity,JinhuaZhejiang
Received:Mar.22
nd
,2021;accepted:Apr.16
th
,2022;published:Apr.24
th
,2022
Abstract
Graph
G
is a simpleundirected connectedgraph,
RD
(
G
)
represents the Harary matrix
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DOI:10.12677/aam.2022.114217
§
{
d
ofgraphG,whichisalsothereciprocaldistancematrixofgraph
G
.Thereciprocal
distancesignlessLaplacianmatrixofgraph
G
isdefinedas
RQ
(
G
)=
RT
(
G
) +
RD
(
G
)
,
where
RT
(
G
)
representsthereciprocaldistancetransitivitydiagonalmatrixof
G
.The
secondpartdescribestheextremalgraphswithmaximalspectralradiusofthe
RQ
(
G
)
amongallconnectedgraphsoffixedorderandfixedvertexconnectivity.Thethird
partcharacterizestheextremalgraphswithmaximalspectralradiusofthe
RQ
(
G
)
amongallconnectedgraphsoffixedorderandfixededgeconnectivity.
Keywords
ReciprocalDistanceSignlessLaplacianMatrix,SpectralRadius,Connectivity
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2022.1142172011
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?
¿
v
j
∈
V
(
K
n
2
),
k
1
d
G
0
(
v
1
,v
j
)
−
1
d
G
(
v
1
,v
j
)
=
1
2
;
3)
RTr
G
0
(
v
1
)
−
RTr
G
(
v
1
) =
n
2
−
(
n
1
−
1)
2
;
4)
é
u
?
¿
v
i
∈
V
(
K
n
1
)
\{
v
1
}
,
k
RTr
G
0
(
v
i
)
−
RTr
G
(
v
i
) =
−
1
2
;
5)
é
u
?
¿
v
j
∈
V
(
K
n
2
),
k
RTr
G
0
(
v
j
)
−
RTr
G
(
v
j
) =
1
2
.
Š
â
þ
ã
?
Ø
(
Ü
ª
(2),
k
x
T
RQ
(
G
0
)
x
−
x
T
RQ
(
G
)
x
=[
n
2
−
(
n
1
−
1)
2
x
2
1
−
(
n
1
−
1)
2
x
2
1
+
n
2
2
x
2
2
]
+[
n
2
x
1
x
2
−
(
n
1
−
1)
x
2
1
]
=
n
2
2
(
x
1
+
x
2
)
2
−
2(
n
1
−
1)
x
2
1
.
e
‡
ä
þ
ª
†
"
Œ
'
X
,
I
(
½
x
1
†
x
2
'
X
.
•
B
å
„
,
-
ρ
=
ρ
(
RQ
(
G
)).
Š
â
RQ
(
G
)
x
=
ρ
x
Œ
µ
ρx
1
= (
n
−
1
−
n
2
2
)
x
1
+[(
n
1
−
1)
x
1
+
n
2
2
x
2
+
rx
3
]
,
DOI:10.12677/aam.2022.1142172012
A^
ê
Æ
?
Ð
§
{
d
Ú
ρx
2
= (
n
−
1
−
n
1
2
)
x
2
+[
n
1
2
x
1
+(
n
2
−
1)
x
2
+
rx
3
]
.
k
(
ρ
−
n
−
n
1
+2+
n
1
+
n
2
2
)
x
1
=
rx
3
+
n
1
2
x
1
+
n
2
2
x
2
,
Ú
(
ρ
−
n
−
n
2
+2+
n
1
+
n
2
2
)
x
2
=
rx
3
+
n
1
2
x
1
+
n
2
2
x
2
.
Š
â
þ
ã
ü
‡ú
ª
Ò
m
>
ƒ
,
´
„
((
ρ
−
n
−
n
1
+2+
n
1
+
n
2
2
)
x
1
= (
ρ
−
n
−
n
2
+2+
n
1
+
n
2
2
)
x
2
.
(4)
d
Perron-Frobenius
½
n
,
é
Au
Ý
RQ
(
G
)
Ì
Œ
»
perron
•
þ
¤
k
ƒ
Ñ
Œ
u
"
,
=
x
1
,x
2
,x
3
Œ
u
"
.
rx
3
+
n
1
2
x
1
+
n
2
2
x
2
>
0,
d
ž
Œ
(
ρ
−
n
−
n
1
+2+
n
1
+
n
2
2
)
>
0
Ú
(
ρ
−
n
−
n
2
+2+
n
1
+
n
2
2
)
>
0.
d
n
2
≥
n
1
,
(
Ü
ª
(4)
Œ
x
2
x
1
≥
1.
k
n
2
2
(
x
1
+
x
2
)
2
−
2(
n
1
−
1)
x
2
1
≥
n
2
2
(2
x
1
)
2
−
2(
n
1
−
1)
x
2
1
= 2(
n
2
−
n
1
+1)
x
2
1
>
0
,
=
k
ρ
(
RQ
(
G
0
))
>ρ
(
RQ
(
G
)),
†
b
g
ñ
.
n
þ
¤
ã
Œ
n
1
=1,
ã
G
=
K
r
∨
(
K
1
∪
K
n
−
r
−
1
)
´
ã
a
G
r
n
¥
ä
k
•
Œ
å
l
ê
Ã
Î
Ò
.
Ê
.
d
Ý
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»
ã
.
2
3.
ä
k
½
>
ë
Ï
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ê
å
l
Ã
Î
Ò
.
Ê
.
d
4
Š
ã
e
¡
•
x
>
ë
Ï
Ý
•
r
n
ã
G
¥
ä
k
•
Œ
å
l
ê
Ã
Î
Ò
.
Ê
.
d
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»
4
Š
ã
.
Ú
n
3[7]
-
:
8
S
´
•
¹
ã
G
¥
•
Ý:
š
˜
ý
f
8
.
e
S,S
<δ
(
G
),
K
k
|
S
|
>δ
(
G
).
½
n
4
-
n
Ú
r
´
÷
v
1
≤
r
≤
n
−
2
ê
,
K
ã
K
r
∨
(
K
1
∪
K
n
−
r
−
1
)
´
ã
a
G
r
n
¥
•
˜
ä
k
•
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ê
å
l
Ã
Î
Ò
.
Ê
.
d
Ì
Œ
»
ã
.
y
²
b
ã
G
´
ã
8
G
∈
G
r
n
¥
ä
k
•
Œ
ê
å
l
Ã
Î
Ò
.
Ê
.
d
Ì
Œ
»
ã
.
e
δ
(
G
)
<r
,
K
½
k
κ
0
(
G
)
≥
r
.
k
δ
(
G
)
≥
r
.
e
ã
G
÷
v
δ
(
G
) =
r
,
K
[
{
v
}
,V
(
G
)
\{
v
}
]
´
ã
G
r
-
>
•
.
d
½
n
1
Œ
•
,
p
f
ã
G
[
V
(
G
)
\{
v
}
]
´
ã
.
d
ž
ã
G
=
K
r
∨
(
K
1
∪
K
n
−
r
−
1
).
b
δ
(
G
)
>r
.
e
ã
G
r
-
>
•
•
S,S
,
Ù
¥
|
S
|
=
n
1
,
|
S
|
=
n
2
.
ã
G
1
=
G
[
S
]
†
G
2
=
G
[
S
]
©
OL
«
d
:
8
S
Ú
S
p
f
ã
.
d
½
n
1
Œ
•
ã
G
1
Ú
G
2
Ñ
´
ã
.
Ï
•
δ
(
G
)
>r
,
k
n
1
,n
2
>
1.
b
V
(
G
1
) =
{
v
1
,v
2
,...,v
n
1
}
,
V
(
G
2
) =
{
v
n
1
+1
,v
n
1
+2
,...,v
n
}
.
-
x
=
(
x
1
,x
2
...,x
n
1
,x
n
1
+1
,x
n
1
+2
,...,x
n
)
´
ã
G
å
l
ê
Ã
Î
Ò
.
Ê
.
d
Ý
RD
α
(
G
)
Perron
•
þ
,
Ù
¥
x
i
´
é
Au
ã
G
¥
:
v
i
ƒ
.
Ø
”
˜
„
5
,
-
x
1
= min
1
≤
i
≤
n
{
x
i
}
,
x
1
≤
x
2
≤···≤
x
n
1
.
ã
G
¥
:
v
1
†
:
8
S
¥
t
‡
:
ƒ
.
´
„
t
≤
min
{
r,n
2
}
.
e
¡
?
Ø
t
=
r
†
t<r
ü
«
œ
¹
.
œ
¹
1:
e
t
=
r
.
d
ž
,
˜
‡
à:
•
:
v
1
,
,
˜
‡
à:
3
:
8
S
¥
¤
k
>
¤
ã
G
¥
>
DOI:10.12677/aam.2022.1142172013
A^
ê
Æ
?
Ð
§
{
d
•
S,S
.
d
ž
,
½
k
n
2
≥
r
+2.
n
2
=
r
½
n
2
=
r
+1
ž
,
ã
G
2
¥
½
k
Ý
•
r
:
.
†
b
δ
(
G
)
>r
g
ñ
.
3
ã
G
Ä
:
þ
E
#
ã
G
0
:
G
0
=
G
−
[
v
i
∈
V
(
G
1
)
\
v
1
v
1
v
i
+
[
v
i
∈
V
(
G
1
)
\{
v
1
}
,v
j
∈
V
(
G
2
)
v
i
v
j
.
´
„
ã
G
0
=
K
r
∨
(
K
1
∪
K
n
−
r
−
1
).
-
:
8
A
=
V
(
G
1
)
\{
v
1
}
,
B
=
{
v
j
∈
V
(
G
2
):
v
1
v
j
∈
E
(
G
)
}
,
:
8
C
=
V
(
G
2
)
\
B
.
Š
â
ª
(2)
Œ
•
,
‡
?
Ø
ã
G
ã
G
0
å
l
ê
Ã
Î
Ò
.
Ê
.
d
Ì
Œ
»
C
z
œ
¹
,
•
I
?
Ø
é
u
¤
k
v
i
,v
j
∈
V
(
G
),
1
d
G
0
(
v
i
,v
j
)
−
1
d
G
(
v
i
,v
j
)
Ú
RTr
G
0
(
v
i
)
−
RTr
G
(
v
i
).
Š
â
ã
(
C
z
œ
¹
©
Û
Œ
µ
1)
é
u
?
¿
v
i
∈
A
,
k
1
d
G
0
(
v
1
,v
i
)
−
1
d
G
(
v
1
,v
i
)
=
−
1
2
;
2)
é
u
?
¿
v
i
∈
A
,
v
j
∈
B
,
k
1
d
G
0
(
v
i
,v
j
)
−
1
d
G
(
v
i
,v
j
)
=
1
2
;
3)
é
u
?
¿
v
i
∈
A
,
v
k
∈
C
,
k
1
d
G
0
(
v
i
,v
k
)
−
1
d
G
(
v
i
,v
k
)
=
2
3
;
4)
RTr
G
0
(
v
1
)
−
RTr
G
(
v
1
) =
−|
A
|
2
;
5)
é
u
?
¿
v
i
∈
A
,
k
RTr
G
0
(
v
i
)
−
RTr
G
(
v
i
) =
−
1
2
+
1
2
|
B
|
+
2
3
|
C
|
;
6)
é
u
?
¿
v
j
∈
B
,
k
RTr
G
0
(
v
j
)
−
RTr
G
(
v
j
) =
1
2
|
A
|
;
7)
é
u
?
¿
v
k
∈
C
,
k
RTr
G
0
(
v
k
)
−
RTr
G
(
v
k
) =
2
3
|
A
|
.
Š
â
þ
ã
?
Ø
(
Ü
ª
(2),
k
ρ
(
RQ
(
G
0
)
−
ρ
(
RQ
(
G
))
≥
x
T
RQ
(
G
0
)
x
−
x
T
RQ
(
G
)
x
=
−
1
2
|
A
|
x
2
1
+(
−
1
2
+
1
2
|
B
|
+
2
3
|
C
|
)
X
v
i
∈
A
x
2
i
+
1
2
|
A
|
X
v
j
∈
B
x
2
j
+
2
3
|
A
|
X
v
k
∈
C
x
2
k
+
−
X
v
i
∈
A
x
1
x
i
+
X
v
i
∈
A,v
j
∈
B
x
i
x
j
+
X
v
i
∈
A,v
k
∈
C
4
3
x
i
x
k
>
0
,
Š
â
x
1
= min
1
≤
i
≤
n
{
x
i
}
,
k
þ
ã
•
˜
‡
Ø
ª
¤
á
.
Ï
d
k
ρ
(
RD
α
(
G
0
))
>ρ
(
RD
α
(
G
))
†
b
g
ñ
.
œ
¹
2:
t<r
.
d
ž
Š
â
Ú
n
3
Ú
δ
(
G
)
>r
Œ
•
n
2
>r
+1.
-
:
8
W
=
{
v
∈
V
(
G
2
) :
vw/
∈
E
(
G
)
,w
∈
V
(
G
1
)
\{
v
1
}}
.
d
|
[
V
(
G
1
)
\{
v
1
}
,V
(
G
2
)]
|
=
r
−
t
,
n
2
>r
+1,
Œ
|
W
|
>
0.
-
:
8
V
F
=
{
v
2
,...,v
n
1
−
r
+
t
}⊆
V
(
G
1
)
\{
v
1
}
,
´
„
∀
v
f
∈
V
F
†
v
j
∈
V
(
G
2
)
Ø
ƒ
.
-
:
8
V
F
0
=
V
(
G
1
)
\{{
v
1
}∪
V
F
}
.
´
„
|
[
V
F
0
,V
(
G
2
)]
|
=
r
−
t
.
d
ž
E
#
ã
G
00
:
G
00
=
G
−
[
v
f
∈
V
(
F
)
v
1
v
f
+
[
v
i
∈
V
(
G
1
)
\{
v
1
}
,v
j
∈
V
(
G
2
)
,v
i
v
j
/
∈
E
(
G
)
v
i
v
j
.
w
,
ã
G
00
=
K
r
∨
(
K
1
∪
K
n
−
r
−
1
).
e
¡
?
Ø
∀
v
i
,v
j
∈
V
(
G
),
1
d
G
00
(
v
i
,v
j
)
−
1
d
G
(
v
i
,v
j
)
Ú
RTr
G
00
(
v
i
)
−
RTr
G
(
v
i
)
Š
.
Š
â
ã
(
C
z
œ
¹
©
Û
Œ
µ
DOI:10.12677/aam.2022.1142172014
A^
ê
Æ
?
Ð
§
{
d
1)
é
u
?
¿
v
f
∈
V
F
,
k
1
d
G
00
(
v
1
,v
f
)
−
1
d
G
(
v
1
,v
f
)
=
−
1
2
;
2)
é
u
?
¿
v
i
∈
V
(
G
1
)
\{
v
1
}
,
v
j
∈
V
(
G
2
),
k
1
d
G
00
(
v
i
,v
j
)
−
1
d
G
(
v
i
,v
j
)
= 1
−
1
d
G
(
v
i
,v
j
)
;
3)
RTr
G
00
(
v
1
)
−
RTr
G
(
v
1
) =
−|
F
|
2
;
4)
é
u
?
¿
v
f
∈
V
F
,
k
a
1
=
RTr
G
00
(
v
f
)
−
RTr
G
(
v
f
) =
−
1
2
+
P
v
j
∈
V
(
G
2
)
(1
−
1
d
G
(
v
f
,v
j
)
);
5)
é
u
?
¿
v
f
0
∈
V
F
0
,
k
a
2
=
RTr
G
00
(
v
f
0
)
−
RTr
G
(
v
f
0
) =
P
v
j
∈
V
(
G
2
)
(1
−
1
d
G
(
v
f
0
,v
j
)
);
6)
é
u
?
¿
v
j
∈
V
(
G
2
),
k
a
3
=
RTr
G
00
(
v
j
)
−
RTr
G
(
v
j
) =
P
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