# Combinatorial Matrix Theory and Spectral Graph Theory - PDF

Document Sample

```					                                       Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben
Combinatorial Matrix Theory and
Introduction
Spectral Graph Theory             IEPG

Minimum Rank
Basic properties
Trees
Leslie Hogben             Spectral Graph
Theory
Speciﬁc Matrices
Iowa State University and        Relationships

CdV Parameters
American Institute of Mathematics   µ(G ), ν(G )
ξ(G )
Forbidden minors
ICART 2008                mr: Recent Results
May 28, 2008              mr graph catalogs
Combinatorial
Matrix Theory and
Introduction                                     Spectral Graph
Theory

Inverse Eigenvalue Problem for a Graph (IEPG)     Leslie Hogben

Introduction
Minimum Rank Problem for a Graph
IEPG
Basic properties of minimum rank
Minimum Rank
Trees                                        Basic properties
Trees

Spectral Graph Theory                           Spectral Graph
Theory
Speciﬁc matrices A, L, |L|, A, L, |L|        Speciﬁc Matrices
Relationships

Relationships among A, L, |L|, A, L, |L|     CdV Parameters
µ(G ), ν(G )
ξ(G )
e
Colin de Verdi`re Type Parameters               Forbidden minors

µ(G ), ν(G )                                mr: Recent Results
mr graph catalogs
ξ(G )
Forbidden minors
Minimum Rank Problem: Recent Results
Minimum rank graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben
Combinatorial Matrix Theory
Introduction

Studies patterns of entries in a matrix rather than values   IEPG

In some applications, only the sign of the entry (or         Minimum Rank
Basic properties
whether it is nonzero) is known, not the numerical value     Trees

Spectral Graph
Uses graphs or digraphs to describe patterns                 Theory
Speciﬁc Matrices
Uses graph theory and combinatorics to obtain results        Relationships

CdV Parameters
about matrices                                               µ(G ), ν(G )
ξ(G )
Inverse Eigenvalue Problem of a Graph (IEPG)                 Forbidden minors

associates a family of matrices to a graph and studies       mr: Recent Results
mr graph catalogs
spectra
Combinatorial
Matrix Theory and
Spectral Graph
Theory
Algebraic Graph Theory                                           Leslie Hogben

Uses algebra and linear algebra to obtain results about    Introduction

graphs or digraphs                                         IEPG

Minimum Rank
Groups used extensively in the study of graphs             Basic properties
Trees
Spectral graph theory uses matrices and their              Spectral Graph
eigenvalues are used to obtain information about           Theory
Speciﬁc Matrices
graphs.                                                    Relationships

CdV Parameters
Speciﬁc matrices:                                          µ(G ), ν(G )
ξ(G )
adjacency, Laplacian, signless Laplacian matrices      Forbidden minors
Normalized versions of these matrices                  mr: Recent Results
mr graph catalogs
e
Colin de Verdi`re type parameters associate families of
matrices to a graph but still use the matrices to obtain
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben
Connections between these two approaches have yielded
Introduction
results in both directions.
IEPG

Terminology and notation                                   Minimum Rank
Basic properties
All matrices are real and symmetric                    Trees

Spectral Graph
Matrix B = [bij ]                                      Theory
Speciﬁc Matrices
σ(B) is the ordered spectrum (eigenvalues) of B,       Relationships

CdV Parameters
repeated according to multiplicity, in nondecreasing   µ(G ), ν(G )

order                                                  ξ(G )
Forbidden minors

All graphs are simple                                  mr: Recent Results
mr graph catalogs

Graph G = (V , E )
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben

Introduction

Inverse Eigenvalue Problem: What sets of real numbers         IEPG

Minimum Rank
β1 , . . . , βn are possible as the eigenvalues of a matrix   Basic properties
satisfying given properties of a matrix?                      Trees

Spectral Graph
Inverse Eigenvalue Problem of a Graph (IEPG): For a           Theory
Speciﬁc Matrices
given graph G , what eigenvalues are possible for a           Relationships

matrix B having nonzero oﬀ-diagonal entries                   CdV Parameters
µ(G ), ν(G )
determined by G?                                              ξ(G )
Forbidden minors

mr: Recent Results
mr graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
The graph G(B) = (V , E ) of n × n matrix B is         Theory

Leslie Hogben
V = {1, ..., n},
Introduction
E = {ij : bij = 0 and i = j}.
IEPG
Diagonal of B is ignored.                     Minimum Rank
Basic properties
Example:                                          Trees

G(B)
                                            Spectral Graph
Theory
2 −1  3 5                  1            2    Speciﬁc Matrices
 −1  0  0 0                                   Relationships

B=
 3

0 −3 0 
CdV Parameters
µ(G ), ν(G )
ξ(G )
5  0  0 0                  4            3    Forbidden minors

mr: Recent Results
mr graph catalogs
The family of matrices described by G is

S(G ) = {B : B T = B and G(B) = G }.
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben

Tools for IEPG
Introduction

IEPG
B is irreducible if and only if G(B) is connected
Minimum Rank
Any (symmetric) matrix is permutation similar to a         Basic properties
Trees
block diagonal matrix and the spectrum of B is the         Spectral Graph
Theory
union of the spectra of these blocks                       Speciﬁc Matrices
Relationships
The diagonal blocks correspond to the connected            CdV Parameters
components of G(B)                                         µ(G ), ν(G )
ξ(G )

It is customary to assume a graph is connected (at least   Forbidden minors

mr: Recent Results
until we cut it up)                                        mr graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben

B(i) is the principal submatrix obtained from B by   Introduction
deleting the i th row and column.                    IEPG

Eigenvalue interlacing: If β1 ≤ · · · ≤ βn are the   Minimum Rank
Basic properties
eigenvalues of B, and γ1 ≤ · · · ≤ γn−1 are the      Trees

eigenvalues of B(i), then                            Spectral Graph
Theory
Speciﬁc Matrices

β1 ≤ γ1 ≤ β2 ≤ · · · ≤ γn−1 ≤ βn         Relationships

CdV Parameters
µ(G ), ν(G )
[Parter 69], [Wiener 84] If G(B) is a tree and       ξ(G )
Forbidden minors
multB (β) ≥ 2, then there is k such that             mr: Recent Results
multB(k) (β) = multB (β) + 1                         mr graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
Ordered multiplicity lists                                                       Theory

Leslie Hogben
˘
If the distinct eigenvalues of B are β1 < · · · < βr with   ˘
Introduction
multiplicities m1 , . . . , mr , then (m1 , . . . , mr ) is called
IEPG
the ordered multiplicity list of B.
Minimum Rank
Determining the possible ordered multiplicity lists of                 Basic properties
Trees
matrices in S(G ) is the ordered multiplicity list problem             Spectral Graph
Theory
for G                                                                  Speciﬁc Matrices
Relationships
The IEPG of G can be solved by
CdV Parameters
solving the ordered multiplicity list problem for G               µ(G ), ν(G )
ξ(G )
proving that if ordered multiplicity list (m1 , . . . , mr ) is   Forbidden minors

possible, then for any real numbers γ1 < · · · < γr , there       mr: Recent Results
is B ∈ S(G ) having eigenvalues γ1 , . . . , γr with              mr graph catalogs

multiplicities m1 , . . . , mr .
If this case, IEPG for G is equivalent to the ordered
multiplicity list problem for G
Combinatorial
Matrix Theory and
Spectral Graph
Theory
[Fiedler 69], [Johnson, Leal Duarte, Saiago 03],
Leslie Hogben
[Barioli, Fallat 05]
The possible ordered multiplicity lists of matrices in S(T )   Introduction

have been determined for the following families of trees T :   IEPG

Minimum Rank
paths                                                     Basic properties
Trees
double paths                                              Spectral Graph
Theory
stars                                                     Speciﬁc Matrices
Relationships
generalized stars                                         CdV Parameters
µ(G ), ν(G )
double generalized stars                                  ξ(G )
Forbidden minors
For T in any of these families, IEPG for G is equivalent to    mr: Recent Results
the ordered multiplicity list problem for G                    mr graph catalogs

It was widely believed that the ordered multiplicity list
problem was always equivalent to IEPG for trees.
Combinatorial
Matrix Theory and
Spectral Graph
Example                                                              Theory

[Barioli, Fallat 03] For the tree TBF                             Leslie Hogben

Introduction

IEPG

Minimum Rank
Basic properties
Trees

Spectral Graph
Theory
Speciﬁc Matrices
Relationships
the ordered eigenvalue list for the adjacency matrix A is       CdV Parameters
µ(G ), ν(G )
√    √      √             √ √ √                     ξ(G )
(− 5, − 2, − 2, 0, 0, 0, 0, 2, 2, 5),                  Forbidden minors

mr: Recent Results

so the ordered multiplicity list (1, 2, 4, 2, 1) is possible.
mr graph catalogs

But if B ∈ S(TBF ) has the ﬁve distinct eigenvalues
˘      ˘      ˘     ˘    ˘
β1 < β2 < β3 < β4 < β5 with ordered multiplicity list
˘    ˘    ˘
˘1 + β5 = β2 + β4 .
(1, 2, 4, 2, 1), then β
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben
First step in solving IEPG is determining maximum
Introduction
possibility multiplicity of an eigenvalue.
IEPG

Minimum Rank
maximum multiplicitiy M(G ) = max multB (β)         Basic properties
B∈S(G )           Trees

minimum rank mr(G ) = min rank(B)                   Spectral Graph
Theory
B∈S(G )
Speciﬁc Matrices

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

CdV Parameters
M(G ) + mr(G ) = |G |                               µ(G ), ν(G )
ξ(G )
Forbidden minors

The Minimum Rank Problem for a Graph is to determine    mr: Recent Results
mr graph catalogs
mr(G ) for any graph G .
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben
Examples:
Path: mr(Pn ) = n − 1.         Complete graph: mr(Kn ) = 1       Introduction

IEPG

Minimum Rank
                                                              Basic properties
? ∗ 0 ...      0 0                                        Trees
   ∗ ? ∗ ...      0 0                      1 1 ... 1          Spectral Graph
                                                              Theory
   0 ∗ ? ...      0 0                     1 1 ... 1 
                                                            Speciﬁc Matrices
   . . . ..
. . .          . . 
. .                     . . ..
. .      . 
. 
Relationships
   . . .    .     . .                     . .    . .         CdV Parameters

   0 0 0 ...      ? ∗                      1 1 ... 1
µ(G ), ν(G )
ξ(G )
Forbidden minors
0 0 0 ...      ∗ ?                                          mr: Recent Results
mr graph catalogs

∗ is nonzero, ? is indeﬁnite
Combinatorial
Matrix Theory and
Spectral Graph
Minimum rank                                                      Theory

Leslie Hogben
Let G have n vertices.
Introduction
It is easy to obtain a matrix B ∈ S(G ) with             IEPG
rank(B) = n − 1 (translate)                              Minimum Rank
Basic properties
It is easy to have full rank (use large diagonal)        Trees

Spectral Graph
If G is the disjoint union of graphs Gi then             Theory
Speciﬁc Matrices
Relationships
mr(G ) =   mr(Gi )                  CdV Parameters
µ(G ), ν(G )
ξ(G )
Only connected graphs are studied                        Forbidden minors

mr: Recent Results
If G is connected,                                       mr graph catalogs

mr(G ) = 0 iﬀ G is single vertex
mr(G ) = 1 iﬀ G = Kn , n ≥ 2
mr(G ) = n − 1 if and only if G is a path [Fiedler 69]
Combinatorial
Matrix Theory and
Minimum Rank Problem for Trees                                 Spectral Graph
Theory
Let T be a tree. ∆(T ) is the maximum of p − q such that        Leslie Hogben
there is a set of q vertices whose deletion leaves p paths.
Introduction

Theorem (Johnson, Leal Duarte 99)                             IEPG

Minimum Rank
Basic properties
|T | − mr(T ) = M(T ) = ∆(T )                   Trees

Spectral Graph
Theory
A related method for computing mr(T ) directly appeared       Speciﬁc Matrices
Relationships
earlier in [Nylen 96].                                        CdV Parameters
µ(G ), ν(G )
Numerous algorithms compute ∆(T ) by using                    ξ(G )
Forbidden minors
high degree (≥ 3) vertices.                                   mr: Recent Results
mr graph catalogs
The following algorithm works from the outside in. v is an
outer high degree vertex if at most one component of T − v
contains high degree vertices.
Delete each outer high degree vertex. Repeat as needed.
Combinatorial
Matrix Theory and
Spectral Graph
Theory
Example                                        Leslie Hogben

Compute mr(T ) by computing ∆(T ) = M(T ).   Introduction

IEPG

Minimum Rank
Basic properties
Trees

Spectral Graph
Theory
2                    Speciﬁc Matrices
Relationships

1                      3           CdV Parameters
5                µ(G ), ν(G )
4             6          ξ(G )
Forbidden minors

mr: Recent Results
mr graph catalogs

8      7
Combinatorial
Matrix Theory and
Spectral Graph
Theory
Example                                        Leslie Hogben

Compute mr(T ) by computing ∆(T ) = M(T ).   Introduction

IEPG

Minimum Rank
Basic properties
Trees

Spectral Graph
Theory
2                    Speciﬁc Matrices
Relationships
3        CdV Parameters
1
4       5       6        µ(G ), ν(G )
ξ(G )
Forbidden minors

mr: Recent Results
mr graph catalogs

8          7
Combinatorial
Matrix Theory and
Spectral Graph
Theory
Example                                        Leslie Hogben

Compute mr(T ) by computing ∆(T ) = M(T ).   Introduction

IEPG

Minimum Rank
Basic properties
Trees

Spectral Graph
Theory
2                    Speciﬁc Matrices
Relationships
3        CdV Parameters
1
5       6        µ(G ), ν(G )
4                        ξ(G )
Forbidden minors

mr: Recent Results
mr graph catalogs

7
Combinatorial
Matrix Theory and
Spectral Graph
Theory
Example                                        Leslie Hogben

Compute mr(T ) by computing ∆(T ) = M(T ).   Introduction

IEPG

Minimum Rank
Basic properties
Trees

Spectral Graph
Theory
2                      Speciﬁc Matrices
Relationships
3        CdV Parameters
1
5       6        µ(G ), ν(G )
ξ(G )
Forbidden minors

mr: Recent Results
mr graph catalogs

7
Combinatorial
Matrix Theory and
Spectral Graph
Example                                                     Theory

Leslie Hogben
Compute mr(T ) by computing ∆(T ) = M(T ):
Introduction
the six vertices {1, 2, 3, 5, 6, 7} were deleted   IEPG

there are 18 paths                                 Minimum Rank
Basic properties
M(T ) = ∆(T ) = 18 − 6 = 12                        Trees

Spectral Graph
mr(T ) = 35 − 12 = 23                              Theory
Speciﬁc Matrices
Relationships

CdV Parameters
µ(G ), ν(G )
ξ(G )
Forbidden minors
2
mr: Recent Results
3
1                                    mr graph catalogs
5       6

7
Combinatorial
Matrix Theory and
Spectral Graph
Theory

1   if i ∼ j   Introduction
A(G ) = A = [aij ] where aij =
0   if i ∼ j   IEPG

Minimum Rank
σ(A) = (α1 , . . . , αn )                          Basic properties
Trees

Example W5                                       
Spectral Graph
Theory
2                 3                 0 1 1 1 1           Speciﬁc Matrices
1 0 1 0 1          Relationships
                   CdV Parameters
1
A = 1 1 0 1 0
                   µ(G ), ν(G )

1 0 1 0 1          ξ(G )
Forbidden minors

5                   4
1 1 0 1 0           mr: Recent Results
mr graph catalogs
√           √
(α1 , α2 , α3 , α4 , α5 ) = (−2, 1 − 5, 0, 0, 1 + 5)
Combinatorial
Matrix Theory and
Spectral Graph
Theory
If two graphs have diﬀerent spectra (equivalently,
Leslie Hogben
diﬀerent characteristic polynomials) of the adjacency
matrix, then they are not isomorphic                     Introduction

IEPG
However, non-isomorphic graphs can be cospectral
Minimum Rank
Basic properties
Trees
Example
Spectral Graph
Theory
Speciﬁc Matrices

p(x) = x 6 − 7x 4 − 4x 3 + 7x 2 + 4x − 1                     Relationships

CdV Parameters
µ(G ), ν(G )
Spectrally determined graphs:                                ξ(G )
Forbidden minors
Complete graphs             Empty graphs                 mr: Recent Results
mr graph catalogs
Graphs with one edge        Graphs missing only 1 edge
Regular of degree 2         Regular of degree n-3
Kn,n,...,n                  mKn
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Laplacian matrix                                        Leslie Hogben

L(G ) = L = D − A(G ) where                     Introduction

D = diag(deg(1), . . . , deg(n)).               IEPG

Minimum Rank
σ(L) = (λ1 , . . . , λn )                       Basic properties
Trees

Example W5                                      
Spectral Graph
Theory
2                   3               4 −1 −1 −1 −1     Speciﬁc Matrices
Relationships
−1 3 −1 0 −1
                  CdV Parameters
1
L = −1 −1 3 −1 0 
                  µ(G ), ν(G )
ξ(G )
−1 0 −1 3 −1      Forbidden minors

5                   4
−1 −1 0 −1 3       mr: Recent Results
mr graph catalogs

(λ1 , λ2 , λ3 , λ4 , λ5 ) = (0, 3, 3, 5, 5)
Combinatorial
Matrix Theory and
Spectral Graph
Theory
isomorphic graphs must have the same Laplacian            Leslie Hogben
spectrum (i.e., Laplacian characteristic polynomial)
Introduction
non-isomorphic graphs can be Laplacian cospectral
IEPG
[Schwenk 73], [McKay 77] For almost all trees T there   Minimum Rank
is a non-somorphic tree T that has both the same        Basic properties
Trees

adjacency spectrum and the same Lapalcian spectrum      Spectral Graph
Theory
for any G , λ1 (G ) = 0                                 Speciﬁc Matrices
Relationships

algebraic connectivity: λ2 (G ), second smallest        CdV Parameters
µ(G ), ν(G )
eigenvalue of L                                         ξ(G )
Forbidden minors
vertex connectivity: κv (G ), minimum number of         mr: Recent Results
vertices in a cutset (G = Kn )                          mr graph catalogs

[Fiedler 73] λ2 (G ) ≤ κv (G )
Example λ2 (W5 ) = 3 = κv (W5 )
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Signless Laplacian matrix                                         Leslie Hogben

|L|(G ) = |L| = D + A(G ) where                            Introduction

D = diag(deg(1), . . . , deg(n))                           IEPG

Minimum Rank
σ(|L|) = (µ1 , . . . , µn )                                Basic properties
Trees

Example W5                                                    Spectral Graph
Theory
2                  3                4      1   1     1    1     Speciﬁc Matrices
1      3   1     0    1    Relationships
                           CdV Parameters
1
|L| = 1
       1   3     1    0   µ(G ), ν(G )

1      0   1     3    1
ξ(G )
Forbidden minors

5                  4
1      1   0     1    3     mr: Recent Results
mr graph catalogs
√             √
(µ1 , µ2 , µ3 , µ4 , µ5 ) = (1, 9−2 17 , 3, 3, 9+2 17 )
Combinatorial
Matrix Theory and
Spectral Graph
Theory
√ −1 √ −1
A = D A D where                                                   Introduction
D = diag(deg(1), . . . , deg(n))                                  IEPG

σ(A) = (α1 , . . . , αn )                                       Minimum Rank
Basic properties
Trees

Example W5                                                          Spectral Graph
1      1       1     1       Theory
0      √
2 3
√
2 3
√
2 3
√
2 3     Speciﬁc Matrices
3          √                                 1 
213                 1
2                                                                     Relationships
0      3       0      3 
 1           1               1           CdV Parameters
1
A =  2√3
             3          0    3     0 

µ(G ), ν(G )
ξ(G )
 √1
0          1
0      1     Forbidden minors
2 3                     3          3     mr: Recent Results
1      1               1
5                4                √
2 3     3          0    3     0      mr graph catalogs

2
(α1 , α2 , α3 , α4 , α5 ) = (− 3 , − 1 , 0, 0, 1)
3
Combinatorial
Matrix Theory and
Spectral Graph
Normalized Laplacian matrix                                     Theory
√ −1 √ −1
L= D L D                                                 Leslie Hogben

σ(L) = (λ1 , . . . , λn )                            Introduction

IEPG
2                    3
Minimum Rank
Basic properties

Example W5                  1                              Trees

Spectral Graph
Theory
Speciﬁc Matrices
5                   4
                1          1         1      1
   Relationships

1             −2 3
√        −2 3
√       −2 3
√    −2 3
√       CdV Parameters
 √                                                   µ(G ), ν(G )
− 2 1 3          1          −1
3        0      −1 
3 
ξ(G )
 1                                                    Forbidden minors

L = − 2√3
                 −1
3         1         −1
3     0      mr: Recent Results
− √ 1
0          −1        1      −1 
mr graph catalogs
 2 3                         3                3 
1
− 2√ 3           −1
3         0         −1
3     1

(λ1 , λ2 , λ3 , λ4 , λ5 ) = (0, 1, 1, 4 , 5 )
3 3
Combinatorial
Matrix Theory and
Normalized Signless Laplacian matrix                  Spectral Graph
Theory
√ −1 √ −1
|L| = D |L| D                                      Leslie Hogben

σ(|L|) = (µ1 , . . . , µn )                    Introduction

IEPG
2                 3
Minimum Rank
Basic properties
Trees
Example W5                 1
Spectral Graph
Theory
Speciﬁc Matrices
5                 4
             1     1       1       1
   Relationships

1        2 3
√     √
2 3
√
2 3
√
2 3
CdV Parameters
 √                                   1 
µ(G ), ν(G )

213            1     1
3       0       3 
ξ(G )

 1             1              1              Forbidden minors

|L| =  2√3
               3     1        3       0 
     mr: Recent Results
 √1
0        1
1        1      mr graph catalogs

2 3                     3             3 
1        1              1
√
2 3       3     0        3       1

(µ1 , µ2 , µ3 , µ4 , µ5 ) = ( 1 , 2 , 1, 1, 2)
3 3
Combinatorial
Matrix Theory and
Spectral Graph
Theory
Relationships between A, L, |L|, A, L, |L| and their spectra     Leslie Hogben

Let G be a graph of order n.                                   Introduction

A+L=I                                                    IEPG

Minimum Rank
So αn−k+1 + λk = 1                                       Basic properties
Trees

|L| + L = 2I                                             Spectral Graph
Theory
So µn−k+1 + λk = 2                                       Speciﬁc Matrices
Relationships

CdV Parameters
Example W5 eigenvalue relationships                            µ(G ), ν(G )
ξ(G )
(α5 , α4 , α3 , α2 , α1 ) =   (1, 0, 0, − 1 , − 2 )
3     3
Forbidden minors

mr: Recent Results
4  5                  mr graph catalogs
(λ1 , λ2 , λ3 , λ4 , λ5 ) = (0, 1, 1,    3, 3)
2  1
(µ5 , µ4 , µ3 , µ2 , µ1 ) = (2, 1, 1,    3, 3)
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben

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

L + A = rI , so λk = r − αn−k+1        Minimum Rank
Basic properties

|L| − A = rI , so µk = r + αk          Trees

Spectral Graph
A = 1 A, so αk = 1 αk
r            r
Theory
Speciﬁc Matrices

L = 1 L, so λk = 1 λk
Relationships

r            r                     CdV Parameters
1                  1           µ(G ), ν(G )
|L| =   r |L|,   so µk =   r µk        ξ(G )
Forbidden minors

mr: Recent Results
mr graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
A, L, |L|, A, L, |L| and the incidence matrix                  Theory

Leslie Hogben
Let G be a graph with n vertices and m edges.
Introduction
incidence matrix P = P(G ) is the n × m 0,1-matrix
IEPG
with rows indexed by the vertices of G and columns
Minimum Rank
indexed by the edges of G                             Basic properties
Trees
1 if e is incident with v
P = [pve ] where pve =                                Spectral Graph
0 otherwise                 Theory
Speciﬁc Matrices

|L| = PP T                                            Relationships

√ −1 √ −1  √ −1  √ −1                           CdV Parameters
|L| = D |L| D = ( D P)( D P)T                         µ(G ), ν(G )
ξ(G )
Forbidden minors
L = P P T where P is an oriented incidence matrix.
√ −1 √ −1       √ −1       √ −1                   mr: Recent Results
L = D L D = ( D P )( D P )T                           mr graph catalogs

L, |L|, L, |L| are all positive semideﬁnite
all eigenvalues λk , µk , λk , µk are nonnegative.
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben
The spectral radius of B is ρ(B) = max |β|
β∈σ(B)
Introduction

IEPG
A, |L|, A, |L| ≥ 0 (nonnegative)
Minimum Rank
Basic properties
Theorem (Perron-Forbenius)                               Trees

Spectral Graph
Let P ≥ 0 be irreducible. Then                           Theory
Speciﬁc Matrices
ρ(P) > 0,                                            Relationships

CdV Parameters
ρ(P) is an eigenvalue of P,                          µ(G ), ν(G )
ξ(G )
Forbidden minors
eigenvalue ρ(P) has a positive eigenvector, and      mr: Recent Results
mr graph catalogs
ρ(P) is a simple eigenvalue of P (multiplicity 1).
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben

Introduction
Let G = Kn be connected.
IEPG
σ(|L|) ⊆ [0, 2] and µn = 2 = ρ(|L|)                        Minimum Rank
Basic properties
σ(A) ⊆ [−1, 1] and αn = 1 = ρ(A)                           Trees

Spectral Graph
σ(L) ⊆ [0, 2] and λ1 = 0                                   Theory
Speciﬁc Matrices
0 < λ2 ≤ 2                                                 Relationships

n                                                         CdV Parameters
n−1 ≤ λn ≤ 2    and λn = 2 if and only if G is bipartite   µ(G ), ν(G )
ξ(G )
n
i=1 λi = n
Forbidden minors

mr: Recent Results
mr graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben

e
Colin de Verdi`re’s graph parameters                          Introduction

e
Colin de Verdi`re deﬁned new graph parameters µ(G )       IEPG

and ν(G )                                                 Minimum Rank
Basic properties
minor monotone                                       Trees

bound M from below                                   Spectral Graph
Theory
use the Strong Arnold Property                       Speciﬁc Matrices
Relationships
Unlike the speciﬁc matrices originally used in spectral   CdV Parameters
graph theory, these parameters involve families of        µ(G ), ν(G )
ξ(G )
matrices                                                  Forbidden minors

mr: Recent Results
Close connections with IEPG and minimum rank              mr graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
Theory

A minor of G is a graph obtained from G by a sequence of       Leslie Hogben

edge deletions, vertex deletions, and edge contractions.
Introduction

A graph parameter ζ is minor monotone if for every minor H   IEPG

of G , ζ(H) ≤ ζ(G ).                                         Minimum Rank
Basic properties
Trees
X fully annihilates B if                                     Spectral Graph
Theory
1. BX = 0                                                   Speciﬁc Matrices
Relationships
2. X has 0 where B has nonzero                              CdV Parameters
µ(G ), ν(G )
3. all diagonal elements of X are 0                         ξ(G )
Forbidden minors

mr: Recent Results
The matrix B has the Strong Arnold Property (SAP) if the     mr graph catalogs

zero matrix is the only symmetric matrix that fully
annihilates B.
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben

a
See [van der Holst, Lov´sz, Shrijver 99] for information     Introduction

Minimum Rank
SAP comes from manifold theory.                          Basic properties
Trees

RB = {C : rank C = rank B}.                              Spectral Graph
Theory

SB = S(G(B)).                                            Speciﬁc Matrices
Relationships

B has SAP if and only if manifolds RB and SB intersect   CdV Parameters
µ(G ), ν(G )
transversally at B.                                      ξ(G )
Forbidden minors

Transversal intersection allows perturbation.            mr: Recent Results
mr graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
Theory
µ(G ) = max{null(L)} such that
Leslie Hogben
1. L is a generalized Laplacian matrix
(L ∈ S(G ) and oﬀ-diagonal entries ≤ 0)                    Introduction

IEPG
2. L has exactly one negative eigenvalue (with multiplicity   Minimum Rank
one)                                                       Basic properties
Trees

3. L has SAP                                                  Spectral Graph
Theory
Speciﬁc Matrices
Relationships
e
Theorem (Colin de Verdi`re 90)                                 CdV Parameters
µ(G ), ν(G )
µ is minor monotone                                        ξ(G )
Forbidden minors

µ(G ) ≤ 1 if and only if G is a path                       mr: Recent Results
mr graph catalogs
µ(G ) ≤ 2 if and only if G is outerplanar
µ(G ) ≤ 3 if and only if G is planar
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben

Introduction

IEPG
For any graph,
Minimum Rank
µ(G ) ≤ M(G ).                        Basic properties
Trees

If G is not planar then 3 < µ(G ) ≤ M(G ) is sometimes    Spectral Graph
Theory
useful for small graphs                                   Speciﬁc Matrices
Relationships
To study minimum rank, generalized Laplacians and         CdV Parameters
number of negative eigenvalues are not usually relevant   µ(G ), ν(G )
ξ(G )
Forbidden minors

mr: Recent Results
mr graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben

Example                                                         Introduction
K2,2,2 is planar but not outer planar, so µ(K2,2,2 ) = 3   IEPG

(no generalized Laplacian of K2,2,2 has rank 2)            Minimum Rank
Basic properties
mr(K2,2,2 ) = 2 and M(K2,2,2 ) = 4                         Trees

                                    Spectral Graph
K2,2,2                  0 1 −1 0 1 1                        Theory

1                    1 0 1        1 0 1
Speciﬁc Matrices

                                    Relationships

6          2              −1 1 −2 −1 1 0                      CdV Parameters

rank B =                   =2
0 1 −1 0 1 1
µ(G ), ν(G )

                                    ξ(G )
5          3              1 0 1        1 0 1
Forbidden minors

4                                                          mr: Recent Results
1 1 0       1 1 2                   mr graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben

Introduction
ν(G ) = max{null(B)}   such that               IEPG

1. B ∈ S(G )                                  Minimum Rank
Basic properties
2. B is positive semi-deﬁnite                 Trees

Spectral Graph
3. B has SAP                                  Theory
Speciﬁc Matrices
Relationships
e
[Colin de Verdi`re] ν is minor monotone   CdV Parameters
µ(G ), ν(G )
For any graph, ν(G ) ≤ M(G )               ξ(G )
Forbidden minors

mr: Recent Results
mr graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
Theory

To study minimum rank, positive semi-deﬁnite is not usually     Leslie Hogben

relevant.                                                     Introduction

IEPG
Example
Minimum Rank
No positive semi-deﬁnite matrix in S(K2,3 ) has rank 2    Basic properties
Trees
so ν(K2,3 ) = 2                                           Spectral Graph
Theory
mr(K2,3 ) = 2 and M(K2,3 ) = 3                            Speciﬁc Matrices
Relationships
                
K2,3                  0 0 1      1   1                    CdV Parameters
µ(G ), ν(G )
1                      0 0 1      1   1                   ξ(G )
                                   Forbidden minors
rank B = 1 1 0
           0   0 = 2
                   mr: Recent Results
3 4      5               1 1 0      0   0                   mr graph catalogs

2                       1 1 0      0   0
Combinatorial
Matrix Theory and
Spectral Graph
Theory

The new parameter ξ                                              Leslie Hogben

To study minimum rank,                                     Introduction

positive semi-deﬁnite, generalized Laplacians and          IEPG

number of negative eigenvalues usually are not relevant.   Minimum Rank
Basic properties

Minor monotonicity is useful.                              Trees

Spectral Graph
Theory
Deﬁnition                                                      Speciﬁc Matrices
Relationships
ξ(G ) = max{null(B) : B ∈ S(G ), B has SAP}                    CdV Parameters
µ(G ), ν(G )
ξ(G )
Example: ξ(K2,2,2 ) = 4 = M(K2,2,2 ) because the matrix B      Forbidden minors

has SAP                                                        mr: Recent Results
mr graph catalogs

Example: ξ(K2,3 ) = 3 = M(K2,3 ) because the matrix B has
SAP
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben

For any graph G ,                         Introduction

IEPG
µ(G ) ≤ ξ(G )                         Minimum Rank
Basic properties
ν(G ) ≤ ξ(G )                         Trees

ξ(G ) ≤ M(G )                         Spectral Graph
Theory
Speciﬁc Matrices
Relationships

ξ(Pn ) = 1 = M(Pn )                   CdV Parameters
µ(G ), ν(G )

ξ(Kn ) = n − 1 = M(Kn )               ξ(G )
Forbidden minors

If T is a non-path tree, ξ(T ) = 2.   mr: Recent Results
mr graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben

Theorem (Barioli, Fallat, Hogben 05)                          Introduction

ξ is minor monotone.                                          IEPG

Minimum Rank

Forbidden minors                                              Basic properties
Trees

Since ξ is minor monotone, the graphs G such that ξ(G ) ≤ k   Spectral Graph
Theory
can be characterized by a ﬁnite set of forbidden minors.      Speciﬁc Matrices
Relationships

ξ(G ) ≤ 1 if and only if G contains no K3 or K1,3 minor.      CdV Parameters
µ(G ), ν(G )
ξ(G )
Forbidden minors
K3            K 1,3
mr: Recent Results
mr graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
Theory
Theorem (Hogben, van der Holst 07)                                 Leslie Hogben

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

IEPG
T3 family                              Minimum Rank
Basic properties
Trees
K 2,3
K4                                                             Spectral Graph
Theory
Speciﬁc Matrices
Relationships

CdV Parameters
µ(G ), ν(G )
ξ(G )
T3                                                        Forbidden minors

mr: Recent Results
mr graph catalogs
Combinatorial
Matrix Theory and
Spectral Graph
Theory
Minimum rank problem
Leslie Hogben

Minimum rank is characterized for:                         Introduction

IEPG
trees [Nylen 96], [Johnson, Leal-Duarte 99]
Minimum Rank
unicyclic graphs [Barioli, Fallat, Hogben 05]          Basic properties
Trees

extreme minimum rank:                                  Spectral Graph
Theory
mr(G ) = 0, 1, 2: [Barrett, van der Holst, Loewy 04]   Speciﬁc Matrices

mr(G ) = |G | − 1, |G | − 2: [Fiedler 69],             Relationships

CdV Parameters
[Hogben, van der Holst 07], [Johnson, Loewy, Smith]    µ(G ), ν(G )
ξ(G )
Reduction techniques for                                   Forbidden minors

mr: Recent Results
cut-set of order 1 [Barioli, Fallat, Hogben 04]        mr graph catalogs

and order 2 [van der Holst 08]
joins [Barioli, Fallat 06]
Combinatorial
Matrix Theory and
Spectral Graph
Theory
Minimum rank graph catalogs                                   Leslie Hogben

Minimum rank of many families of graphs determined      Introduction
at the 06 AIM Workshop.                                 IEPG

On-line catalogs of minimum rank for small graphs and   Minimum Rank
Basic properties
families developed.                                     Trees

Spectral Graph
The ISU group determined the order of all graphs of     Theory
order 7.                                                Speciﬁc Matrices
Relationships

CdV Parameters
Minimum rank of families of graphs               µ(G ), ν(G )
ξ(G )

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

mr: Recent Results

Minimum rank of small of graphs                 mr graph catalogs

http://aimath.org/pastworkshops/catalog1.html
Combinatorial
Matrix Theory and
Spectral Graph
Theory

Leslie Hogben

Introduction

IEPG

Thank You!   Minimum Rank
Basic properties
Trees

Spectral Graph
Theory
Speciﬁc Matrices
Relationships

CdV Parameters
µ(G ), ν(G )
ξ(G )
Forbidden minors

mr: Recent Results
mr graph catalogs

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 21 posted: 5/14/2010 language: English pages: 49