Combinatorial Matrix Theory and - Index of - Iowa State University

CombinatorialMatrix Theory and

Spectral GraphTheory

Leslie Hogben

Introduction

IEPG

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

IEPG





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 ), ν(G )ξ(G )Forbidden minors

Minimum Rank Problem: Recent ResultsMinimum rank graph catalogs



Leslie Hogben

Introduction

IEPG





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

IEPG





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

IEPG





Connections between these two approaches have yieldedresults in both directions.

Terminology and notation

! All matrices are real and symmetric

! Matrix B = [bij ]

! σ(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

IEPG





! Inverse Eigenvalue Problem: What sets of real numbersβ1, . . . ,βn 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

IEPG





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:

B =

2 −1 3 5−1 0 0 0

3 0 −3 05 0 0 0

G(B)

1 2

34

The family of matrices described by G is

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



Leslie Hogben

Introduction

IEPG





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

IEPG





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

! Eigenvalue interlacing: If β1 ≤ · · · ≤ βn are theeigenvalues of B, and γ1 ≤ · · · ≤ γn−1 are theeigenvalues of B(i), then

β1 ≤ γ1 ≤ β2 ≤ · · · ≤ γn−1 ≤ βn

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



Leslie Hogben

Introduction

IEPG





Ordered multiplicity lists

! If the distinct eigenvalues of B are β1 < · · · < βr 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 γ1 < · · · < γr , thereis B ∈ S(G ) having eigenvalues γ1, . . . , γr withmultiplicities m1, . . . ,mr .

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



Leslie Hogben

Introduction

IEPG





[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

IEPG





Example[Barioli, Fallat 03] For the tree TBF

the ordered eigenvalue list for the adjacency matrix A is

(−√

5,−√

2,−√

2, 0, 0, 0, 0,√

2,√

2,√

5),

so the ordered multiplicity list (1, 2, 4, 2, 1) is possible.But if B ∈ S(TBF ) has the five distinct eigenvaluesβ1 < β2 < β3 < β4 < β5 with ordered multiplicity list(1, 2, 4, 2, 1), then β1 + β5 = β2 + β4.



Leslie Hogben

Introduction

IEPG





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

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

multB(β)

! 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

IEPG





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

IEPG





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

IEPG





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

IEPG





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

1

2

3

4 5 6

8 7



Leslie Hogben

Introduction

IEPG






1

2

3

45 6

7



Leslie Hogben

Introduction

IEPG






1

2

3

5 6

7



Leslie Hogben

Introduction

IEPG





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

1

2

3

5 6

7



Leslie Hogben

Introduction

IEPG





Adjacency matrix

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

{1 if i ∼ j0 if i "∼ j

! σ(A) = (α1, . . . ,αn)

Example W5

2

1

3

45

A =

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

(α1,α2,α3,α4,α5) = (−2, 1−√

5, 0, 0, 1 +√

5)



Leslie Hogben

Introduction

IEPG





! 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

IEPG





Laplacian matrix

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

! σ(L) = (λ1, . . . ,λn)

Example W5

2

1

3

45

L =

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

(λ1,λ2,λ3,λ4,λ5) = (0, 3, 3, 5, 5)



Leslie Hogben

Introduction

IEPG





! 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 , λ1(G ) = 0

! algebraic connectivity: λ2(G ), second smallesteigenvalue of L

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

! [Fiedler 73] λ2(G ) ≤ κv(G )

Example λ2(W5) = 3 = κv(W5)



Leslie Hogben

Introduction

IEPG





Signless Laplacian matrix

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

! σ(|L|) = (µ1, . . . , µn)

Example W5

2

1

3

45

|L| =

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+

√17

2 )



Leslie Hogben

Introduction

IEPG





Normalized adjacency matrix

! A =√

D−1A

√D−1

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

! σ(A) = (α1, . . . , αn)

Example W5

2

1

3

45

A =

0 12√

31

2√

31

2√

31

2√

31

2√

30 1

3 0 13

12√

313 0 1

3 01

2√

30 1

3 0 13

12√

313 0 1

3 0

(α1, α2, α3, α4, α5) = (−23 ,−1

3 , 0, 0, 1)



Leslie Hogben

Introduction

IEPG





Normalized Laplacian matrix

! L =√

D−1L

√D−1

! σ(L) = (λ1, . . . , λn)

Example W5

2

1

3

45

L =

1 − 12√

3− 1

2√

3− 1

2√

3− 1

2√

3− 1

2√

31 −1

3 0 −13

− 12√

3−1

3 1 −13 0

− 12√

30 −1

3 1 −13

− 12√

3−1

3 0 −13 1

(λ1, λ2, λ3, λ4, λ5) = (0, 1, 1, 43 , 5

3)



Leslie Hogben

Introduction

IEPG





Normalized Signless Laplacian matrix

! |L| =√

D−1|L|

√D−1

! σ(|L|) = (µ1, . . . , µn)

Example W5

2

1

3

45

|L| =

1 12√

31

2√

31

2√

31

2√

31

2√

31 1

3 0 13

12√

313 1 1

3 01

2√

30 1

3 1 13

12√

313 0 1

3 1

(µ1, µ2, µ3, µ4, µ5) = (13 , 2

3 , 1, 1, 2)



Leslie Hogben

Introduction

IEPG





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

Let G be a graph of order n.

! A + L = I

! So αn−k+1 + λk = 1

! |L| + L = 2I

! So µn−k+1 + λk = 2

Example W5 eigenvalue relationships

(α5, α4, α3, α2, α1) = (1, 0, 0,−13 ,−2

3)

(λ1, λ2, λ3, λ4, λ5) = (0, 1, 1, 43 , 5

3)

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

3)



Leslie Hogben

Introduction

IEPG





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

! L + A = rI , so λk = r − αn−k+1

! |L|−A = rI , so µk = r + αk

! A = 1r A, so αk = 1

r αk

! L = 1r L, so λk = 1

r λk

! |L| = 1r |L|, so µk = 1

r µk



Leslie Hogben

Introduction

IEPG





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−1

= (√

D−1P)(

√D−1P)T

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

! L =√

D−1L

√D−1

= (√

D−1P ′)(

√D−1P ′)T

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

! all eigenvalues λk , µk , λk , µk are nonnegative.



Leslie Hogben

Introduction

IEPG





The spectral radius of B is ρ(B) = maxβ∈σ(B)

|β|

A, |L|, A, |L| ≥ 0 (nonnegative)

Theorem (Perron-Forbenius)

Let P ≥ 0 be irreducible. Then

! ρ(P) > 0,

! ρ(P) is an eigenvalue of P,

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

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



Leslie Hogben

Introduction

IEPG





Let G "= Kn be connected.

! σ(|L|) ⊆ [0, 2] and µn = 2 = ρ(|L|)! σ(A) ⊆ [−1, 1] and αn = 1 = ρ(A)

! σ(L) ⊆ [0, 2] and λ1 = 0

! 0 < λ2 ≤ 2

! nn−1 ≤ λn ≤ 2 and λn = 2 if and only if G is bipartite

!∑n

i=1 λi = n



Leslie Hogben

Introduction

IEPG





Colin de Verdiere’s graph parameters

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

IEPG





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

A graph parameter ζ is minor monotone if for every minor Hof G , ζ(H) ≤ ζ(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

IEPG





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

IEPG





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

IEPG





! 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

IEPG





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

1

2

3

4

6

5

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

IEPG





ν(G ) = max{null(B)} such that

1. B ∈ S(G )

2. B is positive semi-definite

3. B has SAP

! [Colin de Verdiere] ν is minor monotone

! For any graph, ν(G ) ≤ M(G )



Leslie Hogben

Introduction

IEPG





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

Example

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

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

K2,3

1

2

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

IEPG





The new parameter ξ

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

! Minor monotonicity is useful.

Definitionξ(G ) = max{null(B) : B ∈ S(G ),B has SAP}

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

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



Leslie Hogben

Introduction

IEPG





For any graph G ,

! µ(G ) ≤ ξ(G )

! ν(G ) ≤ ξ(G )

! ξ(G ) ≤ M(G )

! ξ(Pn) = 1 = M(Pn)

! ξ(Kn) = n − 1 = M(Kn)

! If T is a non-path tree, ξ(T ) = 2.



Leslie Hogben

Introduction

IEPG





Theorem (Barioli, Fallat, Hogben 05)

ξ is minor monotone.

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

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

K3

K1,3



Leslie Hogben

Introduction

IEPG





Theorem (Hogben, van der Holst 07)

ξ(G ) ≤ 2 if and only if G contains no minor in the T3 family.

T3 family

K2,3K

4

T3



Leslie Hogben

Introduction

IEPG





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

IEPG





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

http://aimath.org/pastworkshops/catalog2.html



Leslie Hogben

Introduction

IEPG





Thank You!

Combinatorial Matrix Theory and - Index of - Iowa State University

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