link
DESCRIPTION
Link. Let G be a digraph and (r,s) is an arc. is called a link. positive link. is called a positive link. r. s. start. end. negative link. is called a negative link. s. r. start. end. link chain. : a sequece of links. is called a link chain. - PowerPoint PPT PresentationTRANSCRIPT
Link
1),;,( sr
Let G be a digraph and (r,s) is an arc
is called a link.
positive link)1;,( sr is called a positive link
r s
start end
negative link)1;,( sr is called a negative link
rs
start end
link chain
),,( 1 is called a link chain
i j
eg. path : a sequence of arcs
a chain from i to j
: a sequece of links
simple chain
simple chain : chain with distinct starts for
not simple chain
closed chain : i=j, simple closed chain :cycle
it’s links
simple closed path = circuit
Singed length
linksnegativeoflinkspositiveofs ##:)(
,,1
length of
Let
inlinkof#:
is called siged length of
be a chain
chain product
1)(
kji
k
kkaA
,,1
with
Let ,,1;, kji kkkk
is called a chain product of A associated
be a chain
a cycle product.
If the chain is a cycle, we obtain
V1 V2 V3 Vk
D is called linearly k-partite
i=1,…,k-1
Vi Vi+1
cyclically m-partite
linearly m-partite
it consists of loopless isolated vertices
A digraph is linearly 1-partite
r-cyclic A square matrix A is r-cyclic if G(A) is
cyclically r-partite or equivalently
0
0
0
00
~
1
,1
23
12
r
rr
P
A
A
A
A
A
permutation similar
r-cyclic matrix in the superdiagonal block form
block-shift matrix
mkkMM ,1
1k
2m
whenever
A block-shift matrix is a square matrix
0kM
in block form with square diagonal blocks s.t.
00
0
00
,1
23
12
mmM
M
M
M
block-shift matrix
matrix iff it’s digraph is linearly partite.
A square matrix of size greater than one
is permutationally similar to a block-shift
Union of digraphs
21 GandG
),( 212121 EEVVGG
),( 222 EVG
In case
digraphs : 111 ,EVG
is called union of
and
21 VV
is called disjoint union
Remark 3.1The disjoint union of two cyclically m-partite
digraphs is cyclically m-partite
V1 V2 V3 Vk
U1 U2 U3 Uk
Remark 3.2
rm
A linearly r-partite digraph is cyclically
m-partite for any
Example for Remark 3.2
84
73
1062
951
VV
VV
VVV
VVV
A linearly 10-partite digraph is cyclically
4-partite
10=4x2+2
Lemma 3.3The disjoint union of a cyclically m-partite
digraph and a linearly s-partite digraph is
is a cyclically m-partite digraph.
.int
,1.3Re
.
,2.3Re
:1
partitemcyclicallyisuniondisjothe
markBy
partitemcyclicallyispartiteslinearlythe
markBy
msthatAssumCase
Case 2 s<m
m
ss
V
UV
UV
UV
22
11
Lemma 3.4
srtsr ,maxinteger t
The disjoint union of a linearly r-partite
is a linearly t-partite digraph for any positive
digraph and a linearly s-partite digraph is
partitesrlinearlyais
VVVUUUGG
digraphpartitekrlinearlyais
VVUVUVUU
GGskFor
digraphpartitesrrlinearlyais
VVUVUVUV
GG
UUUG
VVVVG
srassumemay
partiteslinearlyG
partiterlinearlyG
rs
rksskskk
rsss
s
rs
)(
:)3(
.
:,1,,1)2(
.,max
:
:
:
)1(
:
:
212121
1111
21
12211
21
212
211
2
1
Lemma 2.7.6
iV
mjiji ,1,
every chain from a vertex of
Given G linearly m-partite ( resp. cyclically
mVVV ,,, 21 Then
m-partite) with ordered partition
jVto a vertex
of
is of signed length j-i
(resp. j-i (mod m))
ijsandijthen
linknegativeaisIf
ijsandijthen
linkpositiveaisIf
When
Vinyvertex
atoVinxvertexafromchainabeLet
ionconsiderat
underchaintheoflengththeoninductionBy
j
i
1)(1
,
1)(1
,
,1
.
.
ijijssthen
ijshypotheseinductionby
andVxthenlinknegativeaisIf
ijijssthen
ijshypotheseinductionby
andVxthenlinkpositiveaisIf
andofstartthebex
Let
kConsider
jkk
jkk
kkk
k
111)ˆ()(
1)ˆ(
,
111)ˆ()(
1)ˆ(
,
,,,ˆ,
,,,,
2
1
1
121
21
Remark
Let G be a digraph and each of whose cycles
has zero signed length then every strongly
connected component of G is a single vertex
without loop.G is acyclic
0)(2
.2
.
,
ssoand
lengthofcircuitaistherethen
lengthof
walkclosedaisthereThen
verticesmoreortwo
withcomponentstrongahasG
notSuppose
Remark
00
*0
~ P
A
A is acyclic if and only if
Theorem 2.7.7 (i)
A digraph is linearly partite if and only if
each of cycles has zero signed length.
.
6.7.2,2:2
,1:1
""
lengthsignedzerohascycleeach
TheorembymIfCase
cycleemptyonlyhasGthenmIfCase
partitemlinearlybeGLet
ifonly
.
2
,,,
1
sin,
.
:
.
,
int""
2
221
1
11
21
graphundirected
connectedwithandpartitelinearlyisGthen
arcthisbeGletGinarcanbexxIf
partitelinearlyisGthen
xvertexglethebeGletVxGiven
graphundirected
connectedwithandpartitelinearlyiswhich
verticeskwithGofsubdigraphinducedvertexG
GGGGconstructtoGoing
connectedisdigraphgiventheofgraph
undirectedthethatassumemaywepartitelinearly
stillispartitelinearlyofuniondisjoSinceIf
k
n
p
k
pii
k
k
k
i
k
k
kp
k
VVupartition
orderedwithpartiteplinearlyisGClaim
iIfCase
VVuVV
partitionorderedwithpartiteplinearlyisGClaim
iIfCase
uGVbygenerated
subdigraphinducedvertexthebeGLet
kiandVvsomeforarcanisvu
tsGVVuassumemaywe
generalityloseWithoutGconstructtoNow
GofpartitionorderedthebeVVLet
GGGdconstructehavewethatSuppose
,,,
)1(:
1:2
,,,,,
:
1:1
)(
1),(
..)(\
,.
.,,
1
11
1
1
1
21
Theorem 2.7.7 (ii)
A digraph is cyclically m-partite but not
linearly partite if and only if each of its
cycles has signed length an integral
mutiple of m and it has at least one cycle
with nonzero signed length.
)(
)(mod0)(
6.7.2.,.
.
.
)(,
""
sm
ms
ThmByVxSayxtoxfromchaina
asconsideredbecaninxvertexanyTake
GincycleanyConsider
lengthsignednonzerowithcycleoneleast
athasGibypartitelinearlynotisGSince
ifOnly
j
1)()()(
)(),,,(
,
)1;,(
.),(
..1,,[
.,,
:
.,,)(
,,1
,,1)(mod)(
,,1),,(
,),,(
)(
.
,
int
1
1
11
1
1
1
ij
ji
ji
ji
ji
n
k
ikk
jji
ii
iik
sssBut
smsoandcycleaisthen
vtovfromchainaisthereSince
vvletandjithatassumeMay
Ginarcanisvv
tsijwithmjinotSuppose
VV
partionorderwithpartitemcyclicallyisGClaim
xxGV
withGofsubdigraphinducedvertexthebeGLet
kiforVsoandVxThen
kiformjsxVLet
kifor
andofendthebexLet
smlengthsignednonzerowithcycleaTake
connectedisGdigraphgiventheofgraphundirected
thethatassumemaywepartitemcyclically
stillispartitemcyclicallyofuniondisjoSince
Exercise
s1 (ii) G is linearly s-partite for a unique positive
Let G be linearly partite and
(i) Prove that if G is linearly s-partite then
= signed length of the longest chain
1s integers iff the undirected graph of G is
connected . Then
Corollary p.1
lengths of the cycles of G.
G is cyclically partite but not linearly partite
(2) G is cyclically m-partite iff m divides the
(1) The cyclic index of G = g.c.d of signed
cyclic index of G.
Corollary p.2
which G is d-cyclically partite is unique
ordered partition of vertex set of G w.r.t.
where d is the cyclic index of G.
(3) If the undirected graph of G is connected
then up to cyclic rearrangement, the
.
)(.)(7.7.2)(
.
..
,arg
.
)(7.7.2)(
Gofindexcyclicthedividesm
partitemcyclicallyisG
iCorandiiThmByii
Gofcyclestheof
lengthssignedofdcgisGofindexcyclicthe
msuchestltheisGofindexcyclictheSince
Gofcyclesthe
oflengthssignedofdivisorcommonaism
partitelinearlynotbutpartitemcyclicallyisG
iiThmByi
Theorem 2.7.10
g.c.d of signed lengths of closed chains
Let G be a digraph with connected undirected
of G. Then for vertex x
graph and k= g.c.d of signed lengths of cycles
containing x = k.
can not write cycles
Theorem 2.7.11
)()( BGAG and A, B have the
Let nMBA ,
same corresponding cycle products.
A and B are diagonally similar
1
1
1
1
1
1
1
1
1
1
)(
)(
,,1);,(,,,
)(
)()(
00
,1
sin""
kji
jik
ji
kjjii
kji
kkkk
ijij
jijiij
k
kk
k
kk
k
kk
k
kkkk
k
kk
ddB
ddb
dbdaA
kforjiwhere
AGofcycleaConsider
BGAG
bathen
njidbdathen
BDDAthatsuch
DmatrixdiagonalgularnonthatSuppose
)()(
1,
]
1,:2
1,:1
,,1[
,,1:
,,1
1
1
1
1
1111
1
11
1
11
11
1
11
1
1
BAhenceand
ddddclaimBy
ddddddthen
isandthenjsIfCase
ddddthen
jsandthenisIfCase
kfor
kforddddClaim
ssletand
kforofstartthebesLet
kss
kji
ssssji
kkkkk
ssji
kkkkk
ssji
kk
kk
k
kk
kkkk
k
kk
kk
k
kk
kk
k
kk
DgularnoneeryInductivel
Hinarcanisijifa
db
Hinarcanisjiifb
da
dLet
jidandkid
thatsuchichoosekjdIf
Hinarcanisiifa
b
Hinarcanisiifb
a
d
byivertextovertexfromcedis
ativertexfordeDeterdLettreeais
graphundirectedingcorrespondthethatsucharcs
nwithAGofHsubdigraphaTakeconnected
isAGofgraphundirectedthethatassumeMay
ji
iji
ij
iij
j
i
i
i
i
i
i
sinmindet
),(
),(
1),(),1(
,)1,(
)1,(
),1(
11tan
min.1.
1)(.
)(
1
1
1
1
1
jijiij
ijijij
ijijij
ij
ij
jijiij
dbda
bddathen
dAdaAaAand
BbBAsoandAGincycleais
thenandjiofncompositiothebeLet
Hinitojfromchainaistherethen
HinnotbutAGinarcanisjithatAssume
HinarcanisjioraifproblemNo
njifordbdaClaim
1
1
ˆ1
ˆ
ˆ
1
)()()(
)()()()(
,ˆ),(
ˆ
)(),(
.),(0,[
,1:
Theorem p.1
2,, mZmMA n Consider Let
AeA miD 2
~(b)
(a) A is m-cyclic
(c) All cycles of G(A) have signed lengths
an integral multiple of m.(d) All circuits of G(A) have lengths
an integral multiple of m.
Theorem p.2
Ae mi2
AeA mi2
~
(f)
(e)
)()( 2 tctcAe
Am
i
(g) A and
have the same peripheral spectrum.
Theorem p.3
When G(A) has at least one cycle with
Then
nonzero signed length, (a)-(c) are equivalent
)()()( cba
When A is irreducible, (a)-(d) are equivalent.
When A is nonnegative irreducible, then
(a)-(g) are equivalent.
)()(
)(1
)()()(
)()()(
~
)()(
)()(
)(2
)(2
211.7.2
2
AGincycleanyforsm
AGincycleanyfore
AGincycleanyforAeA
AGincycleanyforAeA
AeA
cb
knownba
mis
mis
miThmby
miD
equivalentaredaHence
d
AGof
lengthscircuitofdivisorcommonaism
AGoflengthscircuitofdcgm
AGofindexcyclicm
partitemcyclicallyisAGaThen
connectedstronglyisAGeirreduciblisAWhen
equivalentarecaHence
cbathatprovehaveWe
ac
iiThmbylengthsigned
nonzerowithcycleoneleastathasAGWhen
)()(
)(
)(
)(..
)(
)()(
.)(,
)()(
)()()(
)()(
)(7.7.2,
)(
.)()(
)()()()()(
int
int2
2
2
var)(
1,,1,0)(
:
)()(
)()()()()(
,
2
equivalentaregaHence
gfeba
cyclicmisAhm
egeranism
h
egeranis
h
m
mrotationaunder
iantinisAofspectrumperipheraltheg
hieA
Aofspectrumperipheralthe
ThmFrobeniusApply
AofityimprimitivofindexthehLet
agthatshowtoNow
gfebathatclearisit
eirreduciblenonnegativisAWhen
h
i
Spectral Index
nnnnn
A atatatattc
12
21
1)(
0ia
nMA and
Let
)(Aks
: = g.c.d of i at
The spectral index of A =
00:)( 21 ns aaaifAk
Combinatorial Spectral Theory of Nonnegative Matrices
