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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(4),2009-2016
PublishedOnlineApril2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.114217
êålÃÎÒ.Ê.d4Šã
§§§{{{ddd
úô“‰ŒÆêƆOŽÅ‰ÆÆ§úô7u
ÂvFϵ2022c322F¶¹^Fϵ2022c416F¶uÙFϵ2022c424F
Á‡
‰½ãG´{üÕëÏã§RD(G)L«ãGHararyÝ§•¡•ãGêålÝ"
ãGêålÃÎÒ.Ê.dÝ½Â•RQ(G)=RT(G)+RD(G)§Ù¥RT(G)L«ãG
êålD4ÝéÝ"1Ü©•xäk½:êÚ½:ëÏÝ…k•ŒêålÃ
ÎÒ.Ê.dÌŒ »4Šã"1nÜ©•xäk½:êÚ½>ëÏÝ…k •Œêål
ÃÎÒ.Ê.dÌŒ»4Šã"
'…c
êålÃÎÒ.Ê.dÝ§ÌŒ»§ëÏÝ
TheExtremalGraphoftheReciprocal
DistanceSignlessLaplacianMatrix
MeijiaoCheng
CollegeofMathematicsandComputerScience,ZhejiangNormalUniversity,JinhuaZhejiang
Received:Mar.22
nd
,2021;accepted:Apr.16
th
,2022;published:Apr.24
th
,2022
Abstract
GraphGis a simpleundirected connectedgraph,RD(G)represents the Harary matrix
©ÙÚ^:§{d.êålÃÎÒ.Ê.d4Šã[J].A^êÆ?Ð,2022,11(4):2009-2016.
DOI:10.12677/aam.2022.114217
§{d
ofgraphG,whichisalsothereciprocaldistancematrixofgraphG.Thereciprocal
distancesignlessLaplacianmatrixofgraphGisdefinedasRQ(G)=RT(G) + RD(G),
whereRT(G)representsthereciprocaldistancetransitivitydiagonalmatrixofG.The
secondpartdescribestheextremalgraphswithmaximalspectralradiusoftheRQ(G)
amongallconnectedgraphsoffixedorderandfixedvertexconnectivity.Thethird
partcharacterizestheextremalgraphswithmaximalspectralradiusoftheRQ(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/
1.Úó
©•Ä¤kãÑ´{üÕëÏã.ãG´:8•V(G) ={v
1
,v
2
,...,v
n
},>8•E(G)
nëÏã.^v
i
v
j
L«:v
i
†v
j
3ãG¥>,dž¡:v
i
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j
ƒ…¡:v
i
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†
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i
v
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ƒ'é.eéun‡:ãG¥?¿ü‡:þ,K¡ ãG•ã,^K
n
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:v∈V(G),ãG¥†:v'é>êþ¡•:vÝ,P•d
G
(v).ãG•ÝP•δ(G).
eãk:8V(G)={v
1
,v
2
,...,v
n
},>8E(G)={v
1
v
2
,v
2
v
3
,...,v
n−1
v
n
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n
L
«.ãG¥ü:åld
G
(v
i
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j
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i
Ú:v
j
m•á´•Ý,{P•d
ij
.
Plavsi´c<3©[1]¥0ãGHararyÝ•=êålÝRD(G) = (RD
ij
)
n×n
,½
奵
RD
ij
=
(
1
d
G
(v
i
,v
j
)
,ei6= j,
0,ÄK.
ãG¥:v
i
êD4ÝRTr
G
(v
i
)½Â•:v
i
ãG¥Øv
i
¤k:ålêƒ
Ú,•=
P
v
j
∈V(G)\{v
i
}
1
d
ij
.-n•RT(G)L«ãG:êD4ÝéÝ,Ù¥é
ƒRT
i,i
(G)½Â•:v
i
êD4ÝRTr
G
(v
i
).©[2]¥0ãêålÃÎÒ.Ê.d
ÝRQ(G)=RT(G) + RD(G).ãGéAêålÃÎÒ.Ê.dÝ¤kAŠ½Â•
ãGêålÃÎÒ.Ê.dÌ,P•σ(RQ(G))={λ
1
(RQ(G)),λ
2
(RQ(G)),...,λ
n
(RQ(G))},
Ù¥λ
1
(RQ(G))≥λ
2
(RQ(G))≥···≥λ
n
(RQ(G)).ãGêålÃÎÒ.Ê.dÌŒ»,=
ρ(RQ(G)) = λ
1
(RQ(G)).Cc5,êålÝÉ2•ïÄ.©[2,3]‰ÑãêålÃÎÒ
DOI:10.12677/aam.2022.1142172010A^êÆ?Ð
§{d
.Ê.dÝÌŒ »þe..Su3©[4]¥•xäkn‡:…©Ok½:ëÏÝ,>ëÏ
Ý,ÚêÚÕáêãa¥äk•ŒêålÌŒ»4Šã.Huang<3©[5]¥(½n‡
:…©Ok½šÝ,½šÝÜã,Úäk½•>êãa¥äk•ŒêålÌŒ
»4Šã.aq,©•ÄïÄäkn‡:…©Ok½:ëÏÝ,>ëÏÝãa¥äk•
ŒêålÃÎÒ.Ê.dÌŒ»4Šã.
2.äk½:ëÏÝêålÃÎÒ.Ê.d4Šã
‰½ãG=(V(G),E(G))´äkn‡:ëÏã,3ãGK?¿k−1‡:¤fã
ëÏ,Kk‡:¤fãØëÏ,K¡k•ãG:ëÏÝ,P•κ(G).eS´:8V(G)¥
˜‡š˜ýf8,S= V(G)\S.>8E(G)¥÷v˜‡à:3:8S¥,,˜‡à:3S¥>
8ܽ••>8,P•[S,S].eãG˜‡•>8¥kk^>,K¡T>•8•k->•.ãG
¤k>•¥•>ê•ãG>ëÏÝ,P•κ
0
(G).Šâ®k(JŒ•ãG:ëÏÝ,>ëÏ
Ý,•Ý÷vκ(G)≤κ
0
(G)≤δ(G).¤k:ëÏÝ•räkn‡:ã8ÜP•G
r
n
,¤k>
ëÏÝ•räkn‡:ã8ÜP•G
r
n
.´•,G
n−1
n
= G
n−1
n
= K
n
.
ÝRQ(G)L«ãGålêÃÎÒ.Ê.dÝ,éu?¿n‘•þx=
(x
1
,x
2
,...,x
n
)
T
,k
x
T
RQ(G)x=
n
X
i=1
RTr
G
(v
i
)x
i
2
+2
X
1≤i<j≤n
1
d
G
(v
i
,v
j
)
x
i
x
j
.(1)
@o,k
x
T
RQ(G
0
)x−x
T
RQ(G)x=
n
X
i=1
(RTr
G
0
(v
i
)−RTr
G
(v
i
))x
i
2
+2
X
1≤i<j≤n
(
1
d
G
0
(v
i
,v
j
)
−
1
d
G
(v
i
,v
j
)
)x
i
x
j
.
(2)
½n1ãG´nëÏã.ã
e
G= G+e,Ù¥e/∈E(G).džk
λ
i
(RQ(
e
G)) ≥λ
i
(RQ(G)),Ù¥1 ≤i≤n.(3)
y²éuãG¥?¿ü:uÚv,kd
e
G
(u,v) ≤d
G
(u,v),•=
1
d
e
G
(u,v)
≥
1
d
G
(u,v)
.ŠâãG†
ãG
0
ålêÃÎÒ.Ê.dÝƒm'XŒµRQ(
e
G) = RQ(G)+M,Ù¥
M=






m
1
m
1,2
···m
1,n
m
2,1
m
2
···m
2,n
.
.
.
.
.
.
.
.
.
.
.
.
m
n,1
m
n,2
···m
n






…km
i
=
P
n
j=1,j6=i
m
i,j
.´„,ÝM´éÓ`Ý.kM•AŠλ
n
(M) Œu½u
".ŠâWeyl
0
s Øª[6], Œ•λ
i
(RQ(
e
G)) ≥λ
i
(RQ(G))+λ
n
(M).=kλ
i
(RQ(
e
G)) ≥λ
i
(RQ(G))
DOI:10.12677/aam.2022.1142172011A^êÆ?Ð
§{d
¤á,Ù¥1 ≤i≤n.2
½n2-nÚr´÷v1 ≤r≤n−2ê.KãK
r
∨(K
1
∪K
n−r−1
)´äkn‡:,
:ëÏÝ•rãaG
r
n
¥äk•ŒålêÃÎÒ.Ê.dÝÌŒ»ã.
y²bãG´ã8G∈G
r
n
¥äk•ŒêålÃÎÒ.Ê.dÌŒ»ã.d½n1Œ
•ãGÓuK
r
∨(K
n
1
∪K
n
2
),Ù¥n
1
+n
2
= n−r.Ø”˜„5,b1 ≤n
1
≤n
2
.
|^‡y{,bn
1
>1.-x´ÝRQ(G)éAuÌŒ»ρ(RQ(G))perron•þ,…k
kxk= 1¤á.dãG= K
r
∨(K
n
1
∪K
n
2
),Œ•ÝRQ(G)perron•þµ
x
T
= (x
1
,...,x
1
|{z}
n
1
,x
2
,...,x
2
|{z}
n
2
,x
3
,...,x
3
|{z}
r
).
-v
1
´ãGfãK
n
1
¥:.E#ãG
0
µ
G
0
= G−
[
v
i
∈V(K
n
1
)\{v
1
}
v
1
v
i
+
[
v
j
∈V(K
n
2
)
v
1
v
j
.
´„ãG
0
= K
r
∨(K
n
1
−1
∪K
n
2
+1
).dž,k
ρ(RD
α
(G
0
)−ρ(RD
α
(G)) ≥x
T
RD
α
(G
0
)x−x
T
RD
α
(G)x.
Šâª(2)Œ•,‡?ØãGãGÌŒ»Czœ¹,I?Ø∀v
i
,v
j
∈V(G),
1
d
G
0
(v
i
,v
j
)
−
1
d
G
(v
i
,v
j
)
ÚRTr
G
0
(v
i
)−RTr
G
(v
i
)Š.Øe?Ø,
1
d
G
0
(v
i
,v
j
)
−
1
d
G
(v
i
,v
j
)
ÚRTr
G
0
(v
i
)−RTr
G
(v
i
)Š
•".
1)éu?¿v
i
∈V(K
n
1
)\{v
1
},k
1
d
G
0
(v
1
,v
i
)
−
1
d
G
(v
1
,v
i
)
= −
1
2
;
2)éu?¿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)éu?¿v
i
∈V(K
n
1
)\{v
1
},kRTr
G
0
(v
i
)−RTr
G
(v
i
) = −
1
2
;
5)éu?¿v
j
∈V(K
n
2
),kRTr
G
0
(v
j
)−RTr
G
(v
j
) =
1
2
.
Šâþã?Ø(ܪ(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‡äþª†"Œ'X,I(½x
1
†x
2
'X.•Bå„,-ρ=ρ(RQ(G)).Šâ
RQ(G)x= ρx 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.2022.1142172012A^êÆ?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
.
Šâþãü‡úªÒm>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)
dPerron-Frobenius½n,éAuÝRQ(G)ÌŒ»perron•þ¤kƒÑŒu",=x
1
,x
2
,x
3
Œ
u".rx
3
+
n
1
2
x
1
+
n
2
2
x
2
>0, džŒ(ρ−n−n
1
+2+
n
1
+n
2
2
) >0 Ú(ρ−n−n
2
+2+
n
1
+n
2
2
) >0.
dn
2
≥n
1
,(ܪ(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
(2x
1
)
2
−2(n
1
−1)x
2
1
= 2(n
2
−n
1
+1)x
2
1
>0,
=kρ(RQ(G
0
))>ρ(RQ(G)),†bgñ.nþ¤ãŒn
1
=1,ãG=K
r
∨(K
1
∪K
n−r−1
)´
ãaG
r
n
¥äk•ŒålêÃÎÒ.Ê.dÝÌŒ»ã.2
3.äk½>ëÏÝêålÃÎÒ.Ê.d4Šã
e¡•x>ëÏÝ•rnãG¥äk•ŒålêÃÎÒ.Ê.dÝÌŒ»4Šã.
Ún3[7]-:8S´•¹ãG¥•Ý:š˜ýf8.e



S,S



<δ(G),Kk
|S|>δ(G).
½n4-nÚr´÷v1 ≤r≤n−2ê,KãK
r
∨(K
1
∪K
n−r−1
)´ãaG
r
n
¥•
˜äk•ŒêålÃÎÒ.Ê.dÌŒ»ã.
y²bãG´ã8G∈G
r
n
¥äk•ŒêålÃÎÒ.Ê.dÌŒ»ã.eδ(G) <r,
K½kκ
0
(G) ≥r.kδ(G)≥r.eãG÷vδ(G) = r,K[{v},V(G)\{v}]´ãGr->•.d
½n1Œ•,pfãG[V(G)\{v}]´ã.džãG= K
r
∨(K
1
∪K
n−r−1
).
bδ(G)>r.eãGr->••

S,S

,Ù¥|S|=n
1
,|S|=n
2
.ãG
1
=G[S]†
G
2
=G[S]©OL«d:8SÚSpfã.d½n1Œ•ãG
1
ÚG
2
Ñ´ã.Ï•
δ(G) >r,kn
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
)´ãGålêÃÎÒ.Ê.dÝRD
α
(G)Perron•
þ,Ù¥x
i
´éAuãG¥:v
i
ƒ.Ø”˜„5,-x
1
= min
1≤i≤n
{x
i
},x
1
≤x
2
≤···≤x
n
1
.
ãG¥:v
1
†:8
S¥t‡:ƒ.´„t≤min{r,n
2
}.
e¡?Øt= r†t<rü«œ¹.
œ¹1:et=r.dž,˜‡à:•:v
1
,,˜‡à:3:8S¥¤k>¤ãG¥>
DOI:10.12677/aam.2022.1142172013A^êÆ?Ð
§{d
•

S,S

.dž,½kn
2
≥r+2.n
2
=r½n
2
=r+1ž,ãG
2
¥½kÝ•r:.†b
δ(G) >rgñ.3ãGÄ:þE#ãG
0
:
G
0
= G−
[
v
i
∈V(G
1
)\v
1
v
1
v
i
+
[
v
i
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1
)\{v
1
},v
j
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2
)
v
i
v
j
.
´„ãG
0
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r
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1
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).-:8A=V(G
1
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1
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j
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1
v
j
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8C=V(G
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i
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j
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1
d
G
0
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i
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j
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−
1
d
G
(v
i
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j
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ÚRTr
G
0
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i
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G
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i
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1)éu?¿v
i
∈A,k
1
d
G
0
(v
1
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i
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−
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d
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i
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2
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2)éu?¿v
i
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j
∈B,k
1
d
G
0
(v
i
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j
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−
1
d
G
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i
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j
)
=
1
2
;
3)éu?¿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
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−|A|
2
;
5)éu?¿v
i
∈A,kRTr
G
0
(v
i
)−RTr
G
(v
i
) = −
1
2
+
1
2
|B|+
2
3
|C|;
6)éu?¿v
j
∈B,kRTr
G
0
(v
j
)−RTr
G
(v
j
) =
1
2
|A|;
7)éu?¿v
k
∈C,kRTr
G
0
(v
k
)−RTr
G
(v
k
) =
2
3
|A|.
Šâþã?Ø(ܪ(2),k
ρ(RQ(G
0
)−ρ(RQ(G)) ≥x
T
RQ(G
0
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T
RQ(G)x
=


−
1
2
|A|x
2
1
+(−
1
2
+
1
2
|B|+
2
3
|C|)
X
v
i
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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
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x
1
x
i
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v
i
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j
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x
i
x
j
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X
v
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3
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W= {v∈V(G
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2
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|W|>0.-:8V
F
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1
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f
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F
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j
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-:8V
F
0
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F
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G
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v
1
v
f
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[
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j
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i
v
j
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v
i
v
j
.
w,ãG
00
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r
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1
∪K
n−r−1
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i
,v
j
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1
d
G
00
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i
,v
j
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−
1
d
G
(v
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j
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RTr
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DOI:10.12677/aam.2022.1142172014A^êÆ?Ð
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1)éu?¿v
f
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F
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1
d
G
00
(v
1
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f
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−
1
d
G
(v
1
,v
f
)
= −
1
2
;
2)éu?¿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
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G
(v
1
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2
;
4)éu?¿v
f
∈V
F
,ka
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)éu?¿v
f
0
∈V
F
0
,ka
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)éu?¿v
j
∈V(G
2
),ka
3
= RTr
G
00
(v
j
)−RTr
G
(v
j
) =
P
v
i
∈V(G
1
)\{v
1
}
(1−
1
d
G
(v
i
,v
j
)
).
Šâþã?Ø(ܪ(2),k
ρ(RD
α
(G
00
)−ρ(RD
α
(G))≥x
T
RD
α
(G
00
)x−x
T
RD
α
(G)x
=


−
|F|
2
x
2
1
+
X
v
f
∈V
F
a
1
x
2
f
+
X
v
f
0
∈V
F
0
a
2
x
2
f
0
+
X
v
j
∈V(G
2
)
a
3
x
2
j


+


−
X
v
f
∈V
F
x
1
x
f
+
X
v
i
∈G
1
\{v
1
},v
j
∈V(G
2
)
2(1−
1
d
G
(v
i
,v
j
)
)x
i
x
j


.
(5)
®•éu?¿v
f
∈V
F
†:v
j
∈G
2
؃,kd
G
(v
f
,v
j
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1
>
1
2
(n
2
−1).éu?
¿v
f
∈V
F
,v
j
∈V(G
2
),цd
G
(v
f
,v
j
) ≥2.…kn
2
>r+1.k
−
|F|
2
x
2
1
+
X
v
f
∈V
F
a
1
x
2
f
>
1
2
|F|(n
2
−2)x
2
1
>0,
Ú
X
v
i
∈G
1
\{v
1
},v
j
∈V(G
2
)
2(1−
1
d
G
(v
i
,v
j
)
)x
i
x
j
−
X
v
f
∈V
F
x
1
x
f
=
X
v
f
∈V
F
,v
j
∈V(G
2
)
2(1−
1
d
G
(v
f
,v
j
)
)x
f
x
j
+
X
v
f
0
∈V
F
0
,v
j
∈V(G
2
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2(1−
1
d
G
(v
f
0
,v
j
)
)x
f
0
x
j
−
X
v
f
∈V
F
x
1
x
f
≥
X
v
f
∈V
F
,v
j
∈V(G
2
)
x
f
x
j
+
X
v
f
0
∈V
F
0
,v
j
∈V(G
2
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2(1−
1
d
G
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f
0
,v
j
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f
0
x
j
−
X
v
f
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F
x
1
x
f
>0.
dª(5),Œρ(RQ(G
00
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α
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α
(G)).†ãG´ã8
G
r
n
¥ä
k•ŒêålÃÎÒ.Ê.dÌŒ»ãgñ.
nþ?Ø,Œ•ãG= K
r
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r
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DOI:10.12677/aam.2022.1142172015A^êÆ?Ð
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ãaêålÃÎÒ.Ê.d4Šãk–?˜Ú&?.
ë•©z
[1]Plavsi´c,D.,Nikoli´c,S.,Trinajsti´c,N.andMihali´c,Z.(1993)OntheHararyIndexforthe
CharacterizationofChemicalGraphs.JournalofMathematicalChemistry,12,235-250.
https://doi.org/10.1007/BF01164638
[2]Alhevaz,A.,Baghipur,M.andRamane,H.S.(2019)ComputingtheReciprocalDistance
SignlessLaplacianEigenvaluesandEnergyofGraphs.MatematicheLXXIV,I,49-73.
https://doi.org/10.2478/ausi-2018-0011
[3]Medina,L.andTrigo,M.(2021)UpperBoundsandLowerBoundsfortheSpectralRadius of
ReciprocalDistance,ReciprocalDistanceLaplacianandReciprocalDistanceSignlessLapla-
cianMatrices.LinearAlgebraandItsApplications,609,386-412.
https://doi.org/10.1016/j.laa.2020.09.024
[4]Su,L.,Li,H.,Shi,M.and Zhang,J. (2014)On the SpectralRadius of theReciprocal Distance
Matrix.AdvancesinMathematics(China),43,551-558.
[5]Huang,F.,Li,X.andWang,S.(2015)OnGraphswithMaximumHararySpectralRadius.
AppliedMathematicsandComputation,266,937-945.
https://doi.org/10.1016/j.amc.2015.05.146
[6]So,W.(1994)CommutativityandSpectraofHermitianMatrices.LinearAlgebraandIts
Applications,212/213,121-129.https://doi.org/10.1016/0024-3795(94)90399-9
[7]West, D.B. (2001)Introduction to GraphTheory. Second Edition, Prentice Hall, Upper Saddle
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DOI:10.12677/aam.2022.1142172016A^êÆ?Ð

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