Sp ect ral G raph Theo ry Int ro d uctio n · Matrix Th eory and Sp ect ra l Gra p h Th eo ry Les...

CombinatorialMatrix Theory and

Spectral GraphTheory

Leslie Hogben

Introduction

Minimum RankBasic propertiesTrees

Spectral GraphTheorySpecific MatricesRelationships

CdV Parametersµ(G), ν(G)ξ(G)Forbidden minors

mr: Recent Resultsmr graph catalogs

Combinatorial Matrix Theory andSpectral Graph Theory

Leslie Hogben

Iowa State University andAmerican Institute of Mathematics

ICART 2008May 28, 2008

Leslie Hogben

Introduction

Inverse Eigenvalue Problem for a Graph (IEPG)

Minimum Rank Problem for a GraphBasic properties of minimum rankTrees

Spectral Graph Theory

Specific matrices A,L, |L|, !A, !L, "|L|Relationships among A,L, |L|, !A, !L, "|L|

Colin de Verdiere Type Parametersµ(G ), !(G )"(G )Forbidden minors

Minimum Rank Problem: Recent ResultsMinimum rank graph catalogs

Leslie Hogben

Introduction

Combinatorial Matrix Theory

! Studies patterns of entries in a matrix rather than values

! In some applications, only the sign of the entry (orwhether it is nonzero) is known, not the numerical value

! Uses graphs or digraphs to describe patterns

! Uses graph theory and combinatorics to obtain resultsabout matrices

! Inverse Eigenvalue Problem of a Graph (IEPG)associates a family of matrices to a graph and studiesspectra

Leslie Hogben

Introduction

Algebraic Graph Theory

! Uses algebra and linear algebra to obtain results aboutgraphs or digraphs

! Groups used extensively in the study of graphs

! Spectral graph theory uses matrices and theireigenvalues are used to obtain information aboutgraphs.

! Specific matrices:! adjacency, Laplacian, signless Laplacian matrices! Normalized versions of these matrices

! Colin de Verdiere type parameters associate families ofmatrices to a graph but still use the matrices to obtaininformation about the graph

Leslie Hogben

Introduction

Connections between these two approaches have yieldedresults in both directions.

Terminology and notation

! All matrices are real and symmetric

! Matrix B = [bij ]

! #(B) is the ordered spectrum (eigenvalues) of B,repeated according to multiplicity, in nondecreasingorder

! All graphs are simple

! Graph G = (V ,E )

Leslie Hogben

Introduction

! Inverse Eigenvalue Problem: What sets of real numbers$1, . . . ,$n are possible as the eigenvalues of a matrixsatisfying given properties of a matrix?

! Inverse Eigenvalue Problem of a Graph (IEPG): For agiven graph G , what eigenvalues are possible for amatrix B having nonzero off-diagonal entriesdetermined by G?

Leslie Hogben

Introduction

The graph G(B) = (V ,E ) of n ! n matrix B is

! V = {1, ..., n},! E = {ij : bij "= 0 and i "= j}.! Diagonal of B is ignored.

Example:

2 #1 3 5#1 0 0 0

3 0 #3 05 0 0 0

The family of matrices described by G is

S(G ) = {B : BT = B and G(B) = G}.

Leslie Hogben

Introduction

Tools for IEPG

! B is irreducible if and only if G(B) is connected

! Any (symmetric) matrix is permutation similar to ablock diagonal matrix and the spectrum of B is theunion of the spectra of these blocks

! The diagonal blocks correspond to the connectedcomponents of G(B)

! It is customary to assume a graph is connected (at leastuntil we cut it up)

Leslie Hogben

Introduction

! B(i) is the principal submatrix obtained from B bydeleting the i th row and column.

! Eigenvalue interlacing: If $1 $ · · · $ $n are theeigenvalues of B, and %1 $ · · · $ %n!1 are theeigenvalues of B(i), then

$1 $ %1 $ $2 $ · · · $ %n!1 $ $n

! [Parter 69], [Wiener 84] If G(B) is a tree andmultB($) % 2, then there is k such thatmultB(k)($) = multB($) + 1

Leslie Hogben

Introduction

Ordered multiplicity lists

! If the distinct eigenvalues of B are $1 < · · · < $r withmultiplicities m1, . . . ,mr , then (m1, . . . ,mr ) is calledthe ordered multiplicity list of B.

! Determining the possible ordered multiplicity lists ofmatrices in S(G ) is the ordered multiplicity list problemfor G

! The IEPG of G can be solved by! solving the ordered multiplicity list problem for G! proving that if ordered multiplicity list (m1, . . . ,mr ) is

possible, then for any real numbers %1 < · · · < %r , thereis B & S(G ) having eigenvalues %1, . . . , %r withmultiplicities m1, . . . ,mr .

! If this case, IEPG for G is equivalent to the orderedmultiplicity list problem for G

Leslie Hogben

Introduction

[Fiedler 69], [Johnson, Leal Duarte, Saiago 03],[Barioli, Fallat 05]The possible ordered multiplicity lists of matrices in S(T )have been determined for the following families of trees T :

! paths

! double paths

! stars

! generalized stars

! double generalized stars

For T in any of these families, IEPG for G is equivalent tothe ordered multiplicity list problem for G

It was widely believed that the ordered multiplicity listproblem was always equivalent to IEPG for trees.

Leslie Hogben

Introduction

Example[Barioli, Fallat 03] For the tree TBF

the ordered eigenvalue list for the adjacency matrix A is

2, 0, 0, 0, 0,'

so the ordered multiplicity list (1, 2, 4, 2, 1) is possible.But if B & S(TBF ) has the five distinct eigenvalues$1 < $2 < $3 < $4 < $5 with ordered multiplicity list(1, 2, 4, 2, 1), then $1 + $5 = $2 + $4.

Leslie Hogben

Introduction

First step in solving IEPG is determining maximumpossibility multiplicity of an eigenvalue.

! maximum multiplicitiy M(G ) = maxB"S(G)

multB($)

! minimum rank mr(G ) = minB"S(G)

rank(B)

! M(G ) is the maximum nullity of a matrix in S(G )

! M(G ) + mr(G ) = |G |

The Minimum Rank Problem for a Graph is to determinemr(G ) for any graph G .

Leslie Hogben

Introduction

Examples:Path: mr(Pn) = n # 1. Complete graph: mr(Kn) = 1

$$$$$$$%

? ( 0 . . . 0 0( ? ( . . . 0 00 ( ? . . . 0 0...

......

. . ....

...0 0 0 . . . ? (0 0 0 . . . ( ?

'''''''(

1 1 . . . 11 1 . . . 1...

.... . .

...1 1 . . . 1

( is nonzero, ? is indefinite

Leslie Hogben

Introduction

Minimum rank

Let G have n vertices.

! It is easy to obtain a matrix B & S(G ) withrank(B) = n # 1 (translate)

! It is easy to have full rank (use large diagonal)

! If G is the disjoint union of graphs Gi then

mr(G ) =)

mr(Gi )

! Only connected graphs are studied

! If G is connected,mr(G ) = 0 iff G is single vertexmr(G ) = 1 iff G = Kn, n % 2

! mr(G ) = n # 1 if and only if G is a path [Fiedler 69]

Leslie Hogben

Introduction

Minimum Rank Problem for Trees

Let T be a tree. ∆(T ) is the maximum of p # q such thatthere is a set of q vertices whose deletion leaves p paths.

Theorem (Johnson, Leal Duarte 99)

|T |#mr(T ) = M(T ) = ∆(T )

A related method for computing mr(T ) directly appearedearlier in [Nylen 96].

Numerous algorithms compute ∆(T ) by usinghigh degree (% 3) vertices.

The following algorithm works from the outside in. v is anouter high degree vertex if at most one component of T # vcontains high degree vertices.

Delete each outer high degree vertex. Repeat as needed.

Leslie Hogben

Introduction

ExampleCompute mr(T ) by computing ∆(T ) = M(T ).

Leslie Hogben

Introduction

Leslie Hogben

Introduction

Leslie Hogben

Introduction

Leslie Hogben

Introduction

ExampleCompute mr(T ) by computing ∆(T ) = M(T ):

! the six vertices {1, 2, 3, 5, 6, 7} were deleted

! there are 18 paths

! M(T ) = ∆(T ) = 18# 6 = 12

! mr(T ) = 35# 12 = 23

Leslie Hogben

Introduction

Adjacency matrix

! A(G ) = A = [aij ] where aij =

*1 if i ) j0 if i ") j

! #(A) = (&1, . . . ,&n)

Example W5

0 1 1 1 11 0 1 0 11 1 0 1 01 0 1 0 11 1 0 1 0

(&1,&2,&3,&4,&5) = (#2, 1#'

5, 0, 0, 1 +'

Leslie Hogben

Introduction

! If two graphs have different spectra (equivalently,different characteristic polynomials) of the adjacencymatrix, then they are not isomorphic

! However, non-isomorphic graphs can be cospectral

Example

p(x) = x6 # 7x4 # 4x3 + 7x2 + 4x # 1

Spectrally determined graphs:

! Complete graphs Empty graphs

! Graphs with one edge Graphs missing only 1 edge

! Regular of degree 2 Regular of degree n-3

! Kn,n,...,n mKn

Leslie Hogben

Introduction

Laplacian matrix

! L(G ) = L = D #A(G ) whereD = diag(deg(1), . . . , deg(n)).

! #(L) = ('1, . . . ,'n)

Example W5

4 #1 #1 #1 #1#1 3 #1 0 #1#1 #1 3 #1 0#1 0 #1 3 #1#1 #1 0 #1 3

('1,'2,'3,'4,'5) = (0, 3, 3, 5, 5)

Leslie Hogben

Introduction

! isomorphic graphs must have the same Laplacianspectrum (i.e., Laplacian characteristic polynomial)

! non-isomorphic graphs can be Laplacian cospectral

! [Schwenk 73], [McKay 77] For almost all trees T thereis a non-somorphic tree T # that has both the sameadjacency spectrum and the same Lapalcian spectrum

! for any G , '1(G ) = 0

! algebraic connectivity: '2(G ), second smallesteigenvalue of L

! vertex connectivity: (v(G ), minimum number ofvertices in a cutset (G "= Kn)

! [Fiedler 73] '2(G ) $ (v(G )

Example '2(W5) = 3 = (v(W5)

Leslie Hogben

Introduction

Signless Laplacian matrix

! |L|(G ) = |L| = D + A(G ) whereD = diag(deg(1), . . . , deg(n))

! #(|L|) = (µ1, . . . , µn)

Example W5

4 1 1 1 11 3 1 0 11 1 3 1 01 0 1 3 11 1 0 1 3

(µ1, µ2, µ3, µ4, µ5) = (1, 9!$

172 , 3, 3, 9+

Leslie Hogben

Introduction

Normalized adjacency matrix

! !A ='

whereD = diag(deg(1), . . . , deg(n))

! #( !A) = (!&1, . . . , !&n)

Example W5

$$$$$$%

3 0 13

313 0 1

3 0 13

313 0 1

''''''(

(!&1, !&2, !&3, !&4, !&5) = (#23 ,#1

3 , 0, 0, 1)

Leslie Hogben

Introduction

Normalized Laplacian matrix

! !L ='

! #( !L) = (!'1, . . . , !'n)

Example W5

$$$$$$%

1 # 12$

3 0 #13

3 1 #13 0

3 1 #13

3 0 #13 1

''''''(

(!'1, !'2, !'3, !'4, !'5) = (0, 1, 1, 43 , 5

Leslie Hogben

Introduction

Normalized Signless Laplacian matrix

! "|L| ='

D!1|L|

! #("|L|) = (!µ1, . . . , !µn)

Example W5

"|L| =

$$$$$$%

3 0 13

313 1 1

3 1 13

313 0 1

''''''(

(!µ1, !µ2, !µ3, !µ4, !µ5) = (13 , 2

3 , 1, 1, 2)

Leslie Hogben

Introduction

Relationships between A,L, |L|, !A, !L, "|L| and their spectra

Let G be a graph of order n.

! !A + !L = I

! So !&n!k+1 + !'k = 1

! "|L| + !L = 2I

! So !µn!k+1 + !'k = 2

Example W5 eigenvalue relationships

(!&5, !&4, !&3, !&2, !&1) = (1, 0, 0,#13 ,#2

(!'1, !'2, !'3, !'4, !'5) = (0, 1, 1, 43 , 5

(!µ5, !µ4, !µ3, !µ2, !µ1) = (2, 1, 1, 23 , 1

Leslie Hogben

Introduction

Let G be an r -regular graph of order n.

! L + A = rI , so 'k = r # &n!k+1

! |L|#A = rI , so µk = r + &k

! !A = 1r A, so !&k = 1

! !L = 1r L, so !'k = 1

! "|L| = 1r |L|, so !µk = 1

Leslie Hogben

Introduction

A,L, |L|, !A, !L, "|L| and the incidence matrix

Let G be a graph with n vertices and m edges.

! incidence matrix P = P(G ) is the n !m 0,1-matrixwith rows indexed by the vertices of G and columnsindexed by the edges of G

P = [pve ] where pve =

*1 if e is incident with v0 otherwise

! |L| = PPT

! "|L| ='

D!1|L|

D!1P)(

'D!1P)T

! L = P #P #T where P # is an oriented incidence matrix.

! !L ='

D!1P #)(

'D!1P #)T

! L, |L|, !L, "|L| are all positive semidefinite

! all eigenvalues 'k , µk , !'k , !µk are nonnegative.

Leslie Hogben

Introduction

The spectral radius of B is )(B) = max!""(B)

A, |L|, !A, "|L| % 0 (nonnegative)

Theorem (Perron-Forbenius)

Let P % 0 be irreducible. Then

! )(P) > 0,

! )(P) is an eigenvalue of P,

! eigenvalue )(P) has a positive eigenvector, and

! )(P) is a simple eigenvalue of P (multiplicity 1).

Leslie Hogben

Introduction

Let G "= Kn be connected.

! #("|L|) * [0, 2] and !µn = 2 = )("|L|)! #( !A) * [#1, 1] and !&n = 1 = )( !A)

! #( !L) * [0, 2] and !'1 = 0

! 0 < !'2 $ 2

! nn!1 $ !'n $ 2 and !'n = 2 if and only if G is bipartite

i=1!'i = n

Leslie Hogben

Introduction

Colin de Verdiere’s graph parameters

! Colin de Verdiere defined new graph parameters µ(G )and !(G )

! minor monotone! bound M from below! use the Strong Arnold Property

! Unlike the specific matrices originally used in spectralgraph theory, these parameters involve families ofmatrices

! Close connections with IEPG and minimum rank

Leslie Hogben

Introduction

A minor of G is a graph obtained from G by a sequence ofedge deletions, vertex deletions, and edge contractions.

A graph parameter * is minor monotone if for every minor Hof G , *(H) $ *(G ).

X fully annihilates B if

1. BX = 0

2. X has 0 where B has nonzero

3. all diagonal elements of X are 0

The matrix B has the Strong Arnold Property (SAP) if thezero matrix is the only symmetric matrix that fullyannihilates B.

Leslie Hogben

Introduction

See [van der Holst, Lovasz, Shrijver 99] for informationabout SAP

! SAP comes from manifold theory.

! RB = {C : rank C = rank B}.! SB = S(G(B)).

! B has SAP if and only if manifolds RB and SB intersecttransversally at B.

! Transversal intersection allows perturbation.

Leslie Hogben

Introduction

µ(G ) = max{null(L)} such that

1. L is a generalized Laplacian matrix(L & S(G ) and off-diagonal entries $ 0)

2. L has exactly one negative eigenvalue (with multiplicityone)

3. L has SAP

Theorem (Colin de Verdiere 90)

! µ is minor monotone

! µ(G ) $ 1 if and only if G is a path

! µ(G ) $ 2 if and only if G is outerplanar

! µ(G ) $ 3 if and only if G is planar

Leslie Hogben

Introduction

! For any graph,µ(G ) $ M(G ).

! If G is not planar then 3 < µ(G ) $ M(G ) is sometimesuseful for small graphs

! To study minimum rank, generalized Laplacians andnumber of negative eigenvalues are not usually relevant

Leslie Hogben

Introduction

Example

! K2,2,2 is planar but not outer planar, so µ(K2,2,2) = 3(no generalized Laplacian of K2,2,2 has rank 2)

! mr(K2,2,2) = 2 and M(K2,2,2) = 4

K2,2,2

rankB =

$$$$$$%

0 1 #1 0 1 11 0 1 1 0 1#1 1 #2 #1 1 00 1 #1 0 1 11 0 1 1 0 11 1 0 1 1 2

''''''(= 2

Leslie Hogben

Introduction

!(G ) = max{null(B)} such that

1. B & S(G )

2. B is positive semi-definite

3. B has SAP

! [Colin de Verdiere] ! is minor monotone

! For any graph, !(G ) $ M(G )

Leslie Hogben

Introduction

To study minimum rank, positive semi-definite is not usuallyrelevant.

Example

! No positive semi-definite matrix in S(K2,3) has rank 2so !(K2,3) = 2

! mr(K2,3) = 2 and M(K2,3) = 3

3 4 5rankB =

0 0 1 1 10 0 1 1 11 1 0 0 01 1 0 0 01 1 0 0 0

''''(= 2

Leslie Hogben

Introduction

The new parameter "

! To study minimum rank,positive semi-definite, generalized Laplacians andnumber of negative eigenvalues usually are not relevant.

! Minor monotonicity is useful.

Definition"(G ) = max{null(B) : B & S(G ),B has SAP}

Example: "(K2,2,2) = 4 = M(K2,2,2) because the matrix Bhas SAP

Example: "(K2,3) = 3 = M(K2,3) because the matrix B hasSAP

Leslie Hogben

Introduction

For any graph G ,

! µ(G ) $ "(G )

! !(G ) $ "(G )

! "(G ) $ M(G )

! "(Pn) = 1 = M(Pn)

! "(Kn) = n # 1 = M(Kn)

! If T is a non-path tree, "(T ) = 2.

Leslie Hogben

Introduction

Theorem (Barioli, Fallat, Hogben 05)

" is minor monotone.

Forbidden minorsSince " is minor monotone, the graphs G such that "(G ) $ kcan be characterized by a finite set of forbidden minors.

"(G ) $ 1 if and only if G contains no K3 or K1,3 minor.

Leslie Hogben

Introduction

Theorem (Hogben, van der Holst 07)

"(G ) $ 2 if and only if G contains no minor in the T3 family.

T3 family

Leslie Hogben

Introduction

Minimum rank problem

Minimum rank is characterized for:

! trees [Nylen 96], [Johnson, Leal-Duarte 99]

! unicyclic graphs [Barioli, Fallat, Hogben 05]

! extreme minimum rank:mr(G ) = 0, 1, 2: [Barrett, van der Holst, Loewy 04]mr(G ) = |G |# 1, |G |# 2: [Fiedler 69],[Hogben, van der Holst 07], [Johnson, Loewy, Smith]

Reduction techniques for

! cut-set of order 1 [Barioli, Fallat, Hogben 04]and order 2 [van der Holst 08]

! joins [Barioli, Fallat 06]

Leslie Hogben

Introduction

Minimum rank graph catalogs

! Minimum rank of many families of graphs determinedat the 06 AIM Workshop.

! On-line catalogs of minimum rank for small graphs andfamilies developed.

! The ISU group determined the order of all graphs oforder 7.

Minimum rank of families of graphshttp://aimath.org/pastworkshops/catalog2.html

Minimum rank of small of graphshttp://aimath.org/pastworkshops/catalog1.html

Leslie Hogben

Introduction

Thank You!

Sp ect ral G raph Theo ry Int ro d uctio n · Matrix Th eory and Sp ect ra l Gra p h Th eo ry Les...

Documents

Transcript of Sp ect ral G raph Theo ry Int ro d uctio n · Matrix Th eory and Sp ect ra l Gra p h Th eo ry Les...

Sony Xperia SP User Guide (Greek)

Bitblock Ciphers, Feistel, and SP Networks

at J-PARC E15ag.riken.jp/J-PARC/sakuma/presentation/ECT-201710/ECT-201710_v… · KbarNN bound state search at J-PARC E15 F. Sakuma, RIKEN . on behalf of the J-PARC . E15 collaboration.

EKSTRAK PROPOLIS LEBAH (Trigona sp) SEBAGAI PREVENTIF ...repository.ub.ac.id/8272/1/DIAH VITA WIJAYANTI.pdf · ii LEMBAR PENGESAHAN SKRIPSI Ekstrak Propolis Lebah (Trigona sp) sebagai

fddfaf dsafr egtery ry r

Abstract - International University of Japan · 2015. 10. 6. · Task－as・obj ect here h皿plies obj ect－relatedness on｛he part of the subj ects；in other words， it ref壱rs

PRODUÇÃO E CARACTERIZAÇÃO DE α-AMILASE POR Bacillus sp ... · PRODUÇÃO E CARACTERIZAÇÃO DE α-AMILASE POR Bacillus sp. ... 72 12 Estabilidade da ... 2007; Produção e caracterização

Introduction to Linear Mixed-E ect Models

MG SP ProgGuidever4.9

Moléculas poliatómicas: CH TEV · 2H 4, etileno TEV Cada C usa as orbitais sp 2 para fazer três ligações σ: duas ligações (sp 2) C 1 –(1s) H 1 e uma ligação (sp 2) C 1–(sp

ΜPS-SP 10-30kVA 3-1 With Bypass Isolation Transformer (Option)

FOTOS DA INSTITUIÇÃO CURSO DE PSICOLOGIA DA FAEF - GARÇA/SP.

Tagging the EMC e ect in N using the BAND and LAD ...crex.fmf.uni-lj.si/eep17/schmidt_bled.pdf · Tagging the EMC e ect in d(e;e0N) reactions using the BAND and LAD detectors at JLab12

23 22-21-20 19 Paradox SP User Manual GR

1. Basic characteristics of semiconductors - TUTbutler.cc.tut.fi/~trantala/opetus/files/SP1-72122.Semiconductor... · 1. Basic characteristics of semiconductors SP I, sp 2012 1 Something

α-Pironas de sp., um Fungo Endofítico Isolado de ...€¦ · 7.-Pironas de Nodulisporium sp., um Fungo Endofítico Isolado de Catharanthus roseus (Boa-noite) Introdução Os fungos

Synechocystis sp. PCC 6803 CELL IMAGES Committee …cbs/projects/2004_report_ko... · · 2008-09-25Synechocystis sp. PCC 6803 CELL IMAGES by Shylaja Kokoori ... Chapter 1 GOALS

EFEKTIVITAS EKSTRAK TIMUN LAUT (HOLOTHUROIDAE SP ...

SP-1792 User Manual Manual del Usuario

Split-panel jackknife estimation of xed-e ect modelserp/erp seminar pdfs/papersfall09... · 2009. 9. 2. · Split-panel jackknife estimation of xed-e ect models Geert Dhaene K.U.