Æ ÁÍ ÁÕ À Í Ä∗ ÚÙÞ† ×ØÛ ÜÝÖ Ò ÅË 0157.5 ¯ v Æ G ¶ ¤Ç ¯Æ G

Æ ÁÍ ÁÕ À Í Ä∗ ÚÙÞ† ×ØÛ ÜÝÖ Ò ÅË 0157.5 ¯ v Æ G ¶ ¤Ç ¯Æ G

B 39 7 B 6 m
2016 h 11 |gg
Vol. 39 No. 6
ACTA MATHEMATICAE APPLICATAE SINICA
Nov., 2016
NJ_YIU[℄IqHYU[L
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MR(2000) oZMS 05C50
n\MS 0157.5
G
G
ω
P Kn−ω,ω
Kω
1
n
P Kn−ω,ω
Pn−ω
P Kn−ω,ω
he
G ? n %56 ("z-), pIFÆ? V (G) = {v , v , · · · , v }, Æ?
E(G). R6 G *?F v ∈ V (G), N (v) (? N (v)) F v Æ6 G *?#O
F8)?ÆF v ?O? d(v) = |N (v)|.  e "6 G *? e ∈/ E(G), b
| G + e Æ6 G * e #<:?62mAR|rFÆ (Æ)W , ^
F 2012 i 5 22 }92016 i 10 24 }9beh
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2016J01673) ~)v!;h* (2014M551831) 30Sd
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1
G
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2
n
FJu`D-mJ7=H.4=9V3H.N
929
G − W  G + W Æ6 G *uG W *?F () #<:?[66 G
*?rl8 C "6 G ?rlF4Æ C, < C *| JlFM6 G ?8
? ω(G) (? ω), ?6 G *:28?IFlG*rYBKw?
b|<$ [1].
A(G)  D(G) ^ ?6 G ?O#4OR 46 G ?AlA4K
w? L(G) = D(G) − A(G). Nz L(G) rlK? R(?4s$?\r_
?_8? 0. #t 0 ?$?rl)!x. L(G) ?)! µ (G) (6 G ?Al
A)!) <tL℄jN?
6
m
i
µ1 (G) ≥ µ2 (G) ≥ · · · ≥ µn−1 (G) ≥ µn (G) = 0.
aMCXWAlA)! µ (G) ? 0 6s+6#R}?6 G "I0
?x. µ (G) p0%(?6 G ?4I0O? α(G).
y ∈ R ?r n NUK y y *R}6 G *F v ?^K?!,
|p(?R6 G IF?rl)d! s y (?R6 G ?d! x ? L(G) R}
)! α(G) ?5C)!UK.UK(?6 G ? Fiedler UK,VR|r
v ∈ V (G), tL)[*
Fiedler[2]
n−1
n−1
n
v
[3]
α(G)xv = d(v)xv −
X
xu .
(1.1)
u∈N (v)
Nz x e = 0, p* e ?r n wrNUKI℄tLt α(G) ?Æ
T
X
x, L(G)x
x) =
α(G) = (x
(xvi − xvj )2 =
vi vj ∈E(G)
y T L(G)y
y
.
yT y
y ∈Rn \{0
0}, yT ee=0
min
(1.2)
 Royle Æ n,M6W α(G) ?6?a&6W α(G) ?6?
js0!2?&6:er?#s2K?GOk1xKrY6C?4I
0O?W!H*~
oTt4I0O?'w<$ de Abreu ?6
GO .
G
68 ω ?# n %I06q)?Æp* 2 ≤ ω ≤ n. B6
PK
(6 1) <w6 K ÆrFz,r-esU P
V<:?6N
z P K = P  P K = K . ÆnK8 ω ? n %I06 G *G>M
6:W4I0O?6?`D6 P K . 2R`D6 P K ?4I0O
?rYa&p<M'X
[4]
Godsil
[5−13]
[14]
n,ω
n−ω,ω
n−2,2
n
0,n
ω
n−ω
n
n,ω
n−ω,ω
Kw
n−ω,ω
vw
v1 vw +1 vw + 2
vn
v2
^ 1 A?5 P K
n−ω,ω
{ ~ f f 930
2
39
7
jElX
R6 G *?F v ∈ V (G), L (G) L(G) 0x uF v #R}?_NV
<:?44C ?R6 G *?F4Æ V ⊂ V (G), L (G) L(G) 0x^
u V *?#?F#R}?_NV<:?44 B  H ?Jl n %4
p* B L(P ) 0x uU P ?rlPF#R}?_NV#<:?
44V H L(P ) 0x^ uU P ?JlPF#R}?_NV#<
:?44
R[ M , τ (M ) ? M ?:W)! Φ(M ) = Φ(M ; x) = det(xI − M ) ? M
?)!TTÆ M = L(G), ?M[I℄ Φ(L(G); x)(6 G ?AlA)T
TÆ) ? Φ(G) Φ(G; x). tLJlzE.% B"6 G ?AlA)TT
Æ
iT 2.1 G = G u : vG ?I#6 G ?F u 6 G ?F v V<:?6
v
1
V1
1
n
n
n+1
n
n+1
n+2
[15]
n
n+2
1
2
1
2
Φ(L(G)) = Φ(L(G1 ))Φ(L(G2 )) − Φ(L(G1 ))Φ(Lv (G2 )) − Φ(L(G2 ))Φ(Lu (G1 )).
X
(1 [15, Lemma 8] RzE 2.1 ?a?C [YI℄<<:tLo?rÆ?'
iT 2.2 M =
??rl[?4p* A  B ^ ?
m × m  n × n 4 E ? m × n 4s+rl℄P?! 1 Æ (1, 1) C$
det(M ) = det(A) det(B) − det(A ) det(B ), * A  B ^ ? G A  B ?Cr_
CrNV<:?4
iT 2.3 R Φ(P ) = 0, Φ(B ) = 1, Φ(H ) = 1, A −E11
T
−E11
B
11
11
[7]
11
0
11
0
11
0
(1) xΦ(Bn ) = Φ(Pn+1 ) + Φ(Pn );
(2) xΦ(Hn ) = Φ(Pn+1 ) (n ≥ 1).
iT 2.4 G ?r n %6 G = G + e ?Æ6 G *+r[ e V<:6
6 G  G ?AlA)!</R 1 ≤ i ≤ n − 1, [16]
′
′
µi+1 (G′ ) ≤ µi (G) ≤ µi (G′ ).
zE 2.4, tL9XNz?
`W 2.5 G ?I06 v ? G *resF (O? 1 ?F). α(G) ≤ α(G − v).
tL"Æ0%(? Cauchy "ÆlKEp(?/KE .
iT 2.6 A ? n % Hermitian 4s6)! λ ≥ λ ≥ · · · ≥ λ . B ? A
?rl m (m ≤ n) %/46)! ρ ≥ ρ ≥ · · · ≥ ρ , R i = 1, 2, · · · , m, tL
"Æ)G
[16]
1
1
2
λn−m+i ≤ ρi ≤ λi .
zE 2.6, ' α(P ) = 4 sin
n
2 π
2n ,
I℄tL'X
m
2
n
FJu`D-mJ7=H.4=9V3H.N
931
iT 2.7  k > l ≥ 1, α(P ) > α(P ) s τ (B ) > τ (B ). 2τ (B ) = α(P ).
G ? n ≥ 2 %I06 v ?6 G ?rlF G (k ≥ l ≥ 1) (6 2) 6 G
ÆF v z,J-&^ ? k  l ?esU P = v(v )v · · · v v  Q = v(u )u · · · u u
V<:?p* v , v , · · · , v  u , u , · · · , u rY."P2?[IFR
6
m
l
k
l
k
2n+1
n
k,l
k+1
1
2
1
k
2
2 1
k
l+1
2 1
l
l
Gk+1,l−1 = Gk,l − u1 u2 + v1 u1 ,
(6 G
6 G 0x#r-V<:? (6 2).
k+1,l−1
k,l
v
G
ul
u2
u1
vk
vk -1
v2
v1
u3
u2
vk vk -1
v2
v1
u1
Gk +1,l -1
Gk ,l
^ 2 "
iT 2.8 G  G
? Fiedler UK
[17]
ul
v
G
k,l
(k ≥ l ≥ 1)
k+1,l−1
^ ?^#Kw?6 x ?6 G
k,l
α(Gk,l ) ≥ α(Gk+1,l−1 ).
,r# x 6= 0 x 6= 0, t"Æ?jk?
iT 2.9 u  v I06 G *?Jl"2?F H (k ≥ l ≥ 1) 6 G
^ ÆF u  v z,&? k  l ?esU P = vv · · · v v  Q = uu · · · u u V<:
?p* v , v , · · · , v  u , u , · · · , u rY."P2?[IFR x ?6 H ?
Fiedler UK
v1
[7]
u1
k,l
k
1
2
1
k
2
l
2 1
l
k,l
′
Hk+l
= Hk,l − vvk + u1 vk ,
x
2 1
′′
Hk+l
= Hk,l − uul + v1 ul .
α(H ) ≥ min {α(H ), α(H )}.
iT 2.10 G ? n %I06 v , · · · , v (s ≥ 2) ?6 G *?rY.
[PI?FsY7 N (v ) = · · · = N (v ). R G 6 G ÆF v , . . . , v + t
(0 ≤ t ≤
) -V<:?[6 α(G) 6= d(v ), α(G) = α(G ).
iT 2.11 µ ?6 G ?rlAlA)!pR}?)!UK? x. 
x = x , µ p6 G ?rlAlA)!pR}?)!UKp? x, p* G
Æ6 G * (6 G *[ uv ) uG (6 G * uv ) e = uv V<:?
6
iT 2.12 e = uv ?6 G ?r-s x ?6 G ? Fiedler UK x 6= x ,
α(G) > α(G − e).
{ Æ (1.2), I℄ α(G) = x L(G)xx > x L(G − e)xx ≥ α(G − e). v1 xu1
≥ 0,
′
k+l
k,l
′′
k+l
[12]
1
1
s
s
t
s(s−1)
2
1
1
s
t
[3]
u
v
′
′
u
T
T
v
{ ~ f f 932
3
39
7
pgQV
MI℄)rY&n :?rYb| G (2 ≤ ω ≤ n) 8 K
 ω :^ #Æ8 K ?\lFV^)?#I06q)?ÆNz G ∈ G ,
G rl<w6 K (pIF"\? v , v , · · · , v )  ω : T , T , · · · , T (|V (T )| ≥
|V (T )| ≥ · · · ≥ |V (T )| ≥ 1) ^ #ÆF v , v , · · · , v Vq)aN |V (T )| + |V (T )| +
· · · + |V (T )| = n. ,VR|r G ∈ G , I℄<t G ? G = K (T , T , · · · , T ).
?M[I℄p<t G = K (P , P , · · · , P ) ? G = K (l , l , · · · , l ),
p* l ≥ l ≥ · · · ≥ l ≥ 0 s l + l + · · · + l + ω = n. l l = 0, [q
K (l , · · · , l , 0, 0, · · · , 0) ? K (l , · · · , l ). Nz K (n − 2) = P K = P ,
+
n,ω
ω
+
n,ω
ω
ω
2
ω
2
1
ω
1
2
ω
l1 +1
ω
ω
2
1
2
1
ω
2
i−1
P Kn−ω,ω = Kω (n − ω) ∈
ω
ω
1
ω
G+
n,ω
lω +1
2
1
ω
l2 +1
1
ω
1
+
n,ω
ω
1
1
2
1
ω
2
ω
i
2
i−1
n−2,2
n
⊂ Gn,ω .
iT 3.1 6 ω ≥ 3, α(K (k, l)) > α(P K ), p* k ≥ l ≥ 1 s k + l + ω = n.
{ I℄l 6 3 ?|[Æ^ R K (k, l)  P K ?F,_|
ω
n−ω,ω
ω
vw
v1
uk
n−ω,ω
u1
u2
Kw
Kw
v2
w2
wl
vw
v1 uk
u2 u1 wl
w2 w1
v2
w1
PK (n - w , w )
Kw ( k , l )
^ 3 5 K (k, l) } P K
ω
n−ω,ω
>HE{
zE 2.1 <<
Φ(Kω (k, l)) = Φ(P Kk,ω )Φ(Pl ) − Φ(P Kk,ω )Φ(Bl−1 ) − Φ(Lv2 (P Kk,ω ))Φ(Pl ),
Φ(P Kn−ω,ω ) = Φ(P Kk,ω )Φ(Pl ) − Φ(P Kk,ω )Φ(Bl−1 ) − Φ(Lu1 (P Kk,ω ))Φ(Pl ).
,V
Φ(Kω (k, l)) − Φ(P Kn−ω,ω ) = Φ(Pl ) Φ(Lu1 (P Kk,ω )) − Φ(Lv2 (P Kk,ω )) .
(1C RzE 2.1 ?a[YI℄<<
[15]
Φ(Lu1 (P Kk,ω )) = Φ(Kω )Φ(Bk−1 ) − Φ(Kω )Φ(Hk−2 ) − Φ(Lv1 (Kω ))Φ(Bk−1 ).
2v: Φ(K ) = x(x − ω)
ω
ω−1
, Φ(Lv1 (Kω )) = Φ(Lv2 (Kω )) = (x − 1)(x − ω)ω−2 ,
ω−3
Φ(L{v1 ,v2 } (Kω )) = (x − 2)(x − ω)
(3.3)
R [4]). Q:zE 2.3 << xΦ(B
(
k−1 )
−
6
m
xΦ(Hk−2
FJu`D-mJ7=H.4=9V3H.N
) = Φ(P ). ,VzE 2.2, [* (3.3) <?
933
k
Φ(Lu1 (P Kk,ω )) =(x − ω)ω−1 (xΦ(Bk−1 ) − xΦ(Hk−2 )) − Φ(Lv1 (Kω ))Φ(Bk−1 )
=(x − ω)ω−1 Φ(Pk ) − Φ(Lv1 (Kω ))Φ(Bk−1 ),
C ?I℄
Φ(Lv2 (P Kk,ω )) =Φ(Lv2 (Kω ))Φ(Pk ) − Φ(Lv2 (Kω ))Φ(Bk−1 ) − Φ(L{v1 ,v2 } (Kω ))Φ(Pk )
=(x − ω)ω−3 [x2 − (ω + 2)x + ω + 2]Φ(Pk ) − Φ(Lv2 (Kω ))Φ(Bk−1 ).
x.
Φ(Kω (k, l)) − Φ(P Kn−ω,ω ) = (ω − 2)(ω + 1 − x)(x − ω)ω−3 Φ(Pl )Φ(Pk ).
R α = α(P K ), µ ≥ µ ≥ · · · ≥ µ ≥ µ = 0 ?6 K (k, l) ?AlA)
!zE 2.4 << α < α(P ) s α < α(P ). ,V ω + 1 − α > 0. Q:zE 2.4 <<
α≤µ
(P K
−u w )=µ
(K (k, l) − v w ) ≤ µ
(K (k, l)). x.
1
n−ω,ω
2
n−1
l
n−2
1
n−ω,ω
n
ω
k
l
n−2
2
ω
l
n−2
ω
Φ(Kω (k, l); α) − Φ(P Kn−ω,ω ; α) =α(α − µn−1 ) · · · (α − µ1 )
=(ω − 2)(ω + 1 − α)(α − ω)ω−3 Φ(Pl ; α)Φ(Pk ; α),
α(µn−1 − α) · · · (µ1 − α)
=(−1)4 α2 (ω − 2)(ω + 1 − α)(ω − α)ω−3
l−1
Y
(µi (Pl ) − α)
i=1
|
µ
k−1
Y
(µi (Pk ) − α) > 0,
i=1
{z
}|
>0
{z
}
>0
KT 3.2 R G (2 ≤ ω ≤ n) *?#6p4I06?:W!=r?6
PK
0:
{ R|r K (T , T , · · · , T ) ∈ G , R |V (T )| = l + 1, i = 1, 2, · · · , ω, p*
l ≥ l ≥ · · · ≥ l ≥ 0 s l + l + · · · + l = n − ω.
~ l = 0 s K (T , T , · · · , T ) P K "2qzE 2.8 <<R
l i (1 ≤ i ≤ n − 2), α(K (T , T , · · · , T )) ≥ α(K (i)), p* K (i) ~6 4 #
x ?6 K (i) ? Fiedler UK x 6= 0 x 6= 0 (aÆ (1.1) <<
x = x
= ··· = x
= x = 0. ,VzE 2.11 < α(K (i)) K ?rl
AlA)!"<g?x? α(K (i)) ≤ 1 ( [2]). ,VzE 2.8 <<
α(K (i)) > α(P K
). 'X)G
~ l 6= 0, zE 2.8 < α(K (T , T , · · · , T )) ≥ α(K (l , l , · · · , l )). ,r
n−1
> α.
+
n,ω
n−ω,ω
ω
1
2
1
1
ω
2
2
ω
2
1
2
ω
1
+
ω
vω+1
i
n−ω,ω
2
+
ω
ω
vn
vn−1
i
ω
ω
v1
+
n,ω
ω
+
ω
vn−1
+
ω
vn
ω
+
ω
+
ω
n−ω,ω
2
ω
1
2
ω
ω
1
2
ω
934
#zE 2.9  3.1 <<
{ ~ f f 39
7
α(Kω (l1 , l2 , · · · , lω )) ≥α(Kω (l1 + l3 + · · · + lω , l2 ))
>α(Kω (l1 + l2 + · · · + lω )) = α(P Kn−ω,ω ).
'X)G
Kw
vw
v1 vw +1 vw + 2
vn
vn -1
vi
v2
^ 4 5K
+
ω (i),
o( i = 1, ω + 1, · · · , n − 2.
KT 3.3 6 ω ≤ n − 1, min
n (ω + 1) − p(ω + 1)2 − 4
2
o
, α(P2(n−ω)−1 ) ≤ α(P Kn−ω,ω ) ≤ α(Pn−ω+2 ).
{ ω ≤ n − 1, zE 2.10 < α(P K ) = α(P S ), p* P S (
6 5) I#U P ?rlPF℄6?*\V<:?6,V9X 2.5 <<
n−ω,ω
n−ω,ω
n−ω,ω
n−ω
α(P Sn−ω,ω ) ≤ α(P Sn−ω,ω − {v3 , · · · , vω }) = α(Pn−ω+2 ).
vw
v1
v3
vw +1 vw + 2
vn
v2
^ 5 5 PS
Q:zE 2.6 << τ (L
n−ω,ω .
vω+1 (P Kn−ω,ω ))
≤ α(P Kn−ω,ω ),
p* v ~6 5 #
ω+1
Lvω+1 (P Kn−ω,ω ) = LV1 (P Kn−ω,ω ) ⊕ Bn−ω−1 ,
p* ⊕ 4? V
1
= {vω+1 , · · · , vn },
Φ(LV1 (P Kn−ω,ω )) = (x − ω)ω−2 [x2 − (ω + 1)x + 1],
n (ω + 1) − p(ω + 1)2 − 4
o
τ (Lvω+1 (P Kn−ω,ω )) = min
, τ (Bn−ω−1 ) .
2
m
FJu`D-mJ7=H.4=9V3H.N
935
,VzE 2.7 << τ (B ) = α(P
), 'X)G
iT 3.4 6 ω ≤ n − 1, R|v e ∈/ E(P K ), α(P K ) < α(P K + e).
{  ω = n−1, R|v e ∈/ E(P K ), α(P K ) = 1 < α(P K +e) =
2, ,V'X)GtL9V ω < n − 1 ?t/ e = v v , p* i, j = 1, 2, · · · , n s i < j.
I℄l 6 1 #?[ÆR6 P K ,_|
 i = 1 s j = ω + 2, · · · , n i = 2, 3, · · · , ω s j = n, 2v: P K + v v −
v
v
∈G
s P K + v v − v v P K "2qzE 2.4 K
E 3.2 <<
6
n−ω−1
2(n−ω)−1
n−ω,ω
n−ω,ω
1,n−1
n−ω,ω
1,n−1
1,n−1
i j
n−ω,ω
n−ω,ω
+
n,ω
ω+1 ω+2
n−ω,ω
i j
ω+1 ω+2
i j
n−ω,ω
α(P Kn−ω,ω + vi vj ) ≥ α(P Kn−ω,ω + vi vj − vω+1 vω+2 ) > α(P Kn−ω,ω ).
 i, j = ω + 1, · · · , n, C ?[Y<<
α(P Kn−ω,ω + vi vj ) ≥ α(P Kn−ω,ω + vi vj − vi+1 vi+2 ) > α(P Kn−ω,ω ).
 i = 2, 3, · · · , ω s j = ω + 2, · · · , n − 1, C ?[Y<<
α(P Kn−ω,ω + vi vj ) ≥ α(P Kn−ω,ω + vi vj − v1 vω+1 ) > α(P Kn−ω,ω ).
i = 2, 3, · · · , ω s j = ω + 1. x ?6 P K
6= x ( x
6= x ), zE 2.12 << α(P K
n−ω,ω
xvω+1
vi
(
n−ω,ω
vω+1
v1
+ vi vω+1
n−ω,ω
? Fiedler UK
+ vi vω+1 ) > α(P Kn−ω,ω )
α(P K + v v ) > α(P K + v v − v v ) = α(P K )), 'X)G
 x = x , x = x s α(P K + v v ) = α(P K ), zE 2.4 <
< α(P K ) ≤ µ (P K − v v ) = α(P ). Q:2v: x = x
s x = x , ,VzE 2.11 << α(P K + v v ) = α(P K ) p6
PK
−v v
= K ∪P
?rlAlA)! α(P K + v v ) =
α(P K
) = α(P
). ,VKE 3.3 << α(P
) ≤ α(P
), α(P
)>
α(P
) (sÆzE 2.7 ) ?ZS
).
iT 3.5 R|v 3 ≤ ω ≤ n, α(P K ) > α(P K
{ 6 ω = n, u α(P K ) = n > α(P K ) = 1. 6 3 ≤ ω < n, z
E 3.4  2.4 << α(P K
) < α(P K
+ e) ≤ α(P K
), p* e ∈
/
E(P K
). x.'X)G
KT 3.6 RÆ G (2 ≤ ω ≤ n) *?#6p4I0O?:W!6
PK
=r0:
{  ω = n, +rl<w6 K ∈ G  ω = n − 1, zE 3.4 <<'X)
G ω = 2, 2v: [12] sM# n %I06*p4I0O?:W!U P
=r0:'X)GtLI℄9V 2 < ω < n − 1 ?t/
G ∈ G s G P K "2qKE 3.2 <'X)G
G ∈ G s G ∈/ G . G Æ6 G * GrY< G ∈ G
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6
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FJu`D-mJ7=H.4=9V3H.N
937
The Minimum Algebraic Connectivity of
Graphs with a Given Clique Number
LI Jianxi
(School
of Mathematics and Statistics, Minnan Normal University, Zhangzhou 363000, China)
(E-mail:
[email protected])
GUO Jiming
(Department
of Mathematics, East China University of Science and Technology, Shanghai 200237, China)
(E-mail:
[email protected])
SHIU Wai Chee
(Department
of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China)
(E-mail:
Abstract
[email protected])
The algebraic connectivity of a graph G is the second smallest eigenvalue of
its Laplacian matrix. In this paper, it is shown that among all connected graphs with the
clique number ω, the minimum value of the algebraic connectivity is attained for a kite
graph P Kn−ω,ω , obtained by appending a complete graph Kω to an end vertex of a path
Pn−ω . Moreover, some properties for P Kn−ω,ω are discussed.
Key words
algebraic connectivity; clique number; kite graph
MR(2000) Subject Classification 05C50
Chinese Library Classification 0157.5