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–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

On the M–property for distance–regular graphs

E. Bendito A. Carmona A.M. Encinas M.Mitjana

Universitat Politecnica de Catalunya, Barcelona

margarida.mitjana@upc.edu

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

M-matrix

Matrices with non–positive off–diagonal and non–negativediagonal entries

L=kI-A

k > 0,A ≥ 0 and the diagonal entries of A are less or equal to k

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

M-matrix

Matrices with non–positive off–diagonal and non–negativediagonal entries

L=kI-A

k > 0,A ≥ 0 and the diagonal entries of A are less or equal to k

If k ≥ ρ(A), then L is called an M–matrix

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

Where can we find M-matrices?

Finite difference methods for solving PDE

Growth models in economics

Markov processes in probability and statistics

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

Where can we find M-matrices?

Finite difference methods for solving PDE

Growth models in economics

Markov processes in probability and statistics

...the combinatorial Laplacian of a k-regular graph

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

Where can we find M-matrices?

Finite difference methods for solving PDE

Growth models in economics

Markov processes in probability and statistics

...the combinatorial Laplacian of a k-regular graph

Symmetric and irreducible M–matrices

non–singular 99K discrete Dirichlet problem 99K its inversecorresponds with the Green operator associated with theboundary value problem

singular 99K discrete Poisson equation 99K its Moore–Penroseinverse corresponds with the Green operator too

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

Generalized inverses. The Moore–Penrose inverse

A generalized inverse of matrix A

exists for a class of matrices larger than the class ofnon–singular matrices

has some of the properties of the usual inverse

reduces to the usual inverse when A is non–singular

For every finite matrix A there is a unique matrix X satisfying thePenrose equations

AXA = A (AX )∗ = AXXAX = X (XA)∗ = XA

X is the Moore–Penrose inverse, and it is denoted by A†

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

Statement of the Problem

If the M–matrix is non–singular its inverse has all entriespositive

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

Statement of the Problem

If the M–matrix is non–singular its inverse has all entriespositive

If the M–matrix is singular, it has a generalized inverse whichis non–negative

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

Statement of the Problem

If the M–matrix is non–singular its inverse has all entriespositive

If the M–matrix is singular, it has a generalized inverse whichis non–negative

Given an M–matrix, there exists a generalized inverse whichis also an M–matrix?

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

The Combinatorial Laplacian

When the Moore–Penrose inverse of the combinatorial Laplacian ofa graph is an M-matrix?

L† is an M–matrix iff L†ij ≤ 0 for any xi , xj ∈ V with i 6= j

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

The Combinatorial Laplacian

When the Moore–Penrose inverse of the combinatorial Laplacian ofa graph is an M-matrix?

L† is an M–matrix iff L†ij ≤ 0 for any xi , xj ∈ V with i 6= jExample: Petersen

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

The Combinatorial Laplacian

When the Moore–Penrose inverse of the combinatorial Laplacian ofa graph is an M-matrix?

L† is an M–matrix iff L†ij ≤ 0 for any xi , xj ∈ V with i 6= j

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

The Combinatorial Laplacian

When the Moore–Penrose inverse of the combinatorial Laplacian ofa graph is an M-matrix?

L† is an M–matrix iff L†ij ≤ 0 for any xi , xj ∈ V with i 6= j

Example: Cycle

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

The Combinatorial Laplacian

When the Moore–Penrose inverse of the combinatorial Laplacian ofa graph is an M-matrix?

L† is an M–matrix iff L†ij ≤ 0 for any xi , xj ∈ V with i 6= j

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

The Combinatorial Laplacian

When the Moore–Penrose inverse of the combinatorial Laplacian ofa graph is an M-matrix?

L† is an M–matrix iff L†ij ≤ 0 for any xi , xj ∈ V with i 6= j

When the Moore–Penrose inverse of the combinatorial Lapla-cian of a graph is the combinatorial Laplacian of a network?

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

Network

Finite network: Γ = (V ,E , c)

The conductance: c : V × V −→ [0,+∞) such that

{

c(x , x) = 0 for any x ∈ V

c(x , y) > 0 for x ∼ y .

C(V) is the set of real–valued functions on V

> the functions in C(V ) 99K R|V |

> the endomorphisms of C(V ) 99K |V |–order square matrices

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

Network

The combinatorial Laplacian of Γ

L(u)(x) =∑

y∈V

c(x , y)(

u(x)− u(y))

, x ∈ V

> L is a positive semi–definite self–adjoint operator

> L has 0 as its lowest eigenvalue whose associatedeigenfunctions are constant

> L can be interpreted as an irreducible, symmetric, diagonallydominant and singular M–matrix, L

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

Definition

Γ has the M–property iff L† is a M–matrix

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Distance–regular graphs

Let Γ be a distance–regular graph with intersection array

ι(Γ) = {b0, b1, . . . , bD−1; c1, . . . , cD}

Γ is regular of degree k , and

k = b0, bD = c0 = 0, c1 = 1, ai + bi + ci = k

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Examples

n–Cycle: ι(Cn) ={

2, 1, . . . , 1; 1, . . . , 1, cD

}

The Heawood Graph:ι(Γ) = {3, 2, 2; 1, 1, 3}

The Petersen Graph:ι(Γ) = {3, 2; 1, 1}

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

The intersection array ι(Γ)

Properties

i) k0 = 1 and ki =b0 · · · bi−1

c1 · · · ci, i = 1, . . . ,D

ii) n = 1 + k + k2 + · · ·+ kD

iii) k > b1 ≥ · · · ≥ bD−1 ≥ 1

iv) 1 ≤ c2 ≤ · · · ≤ cD ≤ k

v) If i + j ≤ D, then ci ≤ bj and ki ≤ kj when, in addition, i ≤ j

Proposition

Notation

> a1=λ

> c2=µ

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Bipartite and Antipodal distance–regular graphs Γ

(i) Γ is bipartite iff ai = 0, i = 1, . . . ,D

(ii) Γ is antipodal iff bi = cD−i , i = 0, . . . ,D, i 6=⌊

D2

Distance–regular graphs with k ≥ 3 other than bipartite andantipodal are primitive

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

The Moore–Penrose inverse of a distance–regular graph

Lemma

Let Γ be a distance–regular graph.

L†ij =D−1∑

r=d(xi ,xj)

1

nkrbr

(

D∑

l=r+1

kl

)

−D−1∑

r=0

1

n2krbr

(

r∑

l=0

kl

)(

D∑

l=r+1

kl

)

for all i , j = 1, . . . , n

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

The M–property

Proposition

A distance–regular graph Γ has the M–property iff

D−1∑

j=1

1

kjbj

(

D∑

i=j+1

ki

)2≤

n − 1

k

Corollary

If Γ has the M–property and D ≥ 2, then

λ ≤ 3k −k2

n − 1− n

and hence n < 3k.

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Small diameter

Proposition

If Γ is a distance–regular graph with the M–property, then

D ≤ 3

Proof.

If D ≥ 4, then from property (v) of the parameters

3k < 1 + 3k ≤ 1 + k + k2 + k3 ≤ n

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Diameter two

Proposition

A strongly regular graph with parameters (n, k , λ, µ) has theM–property iff

µ ≥ k −k2

n − 1

Observation

Every antipodal strongly regular graph has the M–property

Petersen graph, (10, 3, 0, 1), does not have the M–property

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Half of the strongly regular graphs have the M–property

Proposition

If Γ is a primitive strongly regular graph, then either Γ or Γ has the

M–property.

Conference Graph(4m + 1, 2m,m − 1,m)

The graphs Γ and Γ, both ofthem, have the M–property iffΓ is a conference graph

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Diameter three

Proposition

A distance–regular graph with D = 3 has the M–property iff

k2b1

(

b2c2 + (b2 + c3)2)

≤ c22c23 (n − 1)

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Bipartite distance–regular graph with D = 3

ι(Γ) = {k , k − 1, k − µ; 1, µ, k}

Proposition

Γ satisfies the M–property iff4k

5≤ µ ≤ k − 1

In particular, k ≥ 5

Corollary

When, 1 ≤ µ < k − 1, then either Γ or Γ3 has the M–property,

except when

k − 1 < 5µ < 4k

in which case none of them has the M–property

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Antipodal distance–regular graph

ι(Γ) = {k , tµ, 1; 1, µ, k}

Proposition

An antipodal distance–regular graph with D = 3 has the

M–property iff it is a Taylor graph T (k , µ) such that k ≥ 5 and

k + 3

2≤ µ < k

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Antipodal distance–regular graph

Corollary

If Γ is the Taylor graph T (k , µ) with 1 ≤ µ ≤ k − 2, then

either Γ or Γ2 has the M–property, except when

µ ∈ {m − 2,m − 1,m,m + 1} when k = 2m and

µ ∈ {m − 1,m,m + 1} when k = 2m + 1, in which case

none of them has the M–property.

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Bipartite and antipodal distance–regular graph with D = 3

ι(Γ) = {k , k − 1, 1; 1, k − 1, k}

k–Crown graphs:

They are Taylor graphs with µ = k − 1 and hence they

have the M–property iff k ≥ 5

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Primitive case

Lemma

k2 = k iff Γ is either C6, C7 or T (µ, k)

Proposition

If Γ satisfies the M–property, then

1 < c2 ≤ b1 < 2c2 b2 < c3 k3 ≤ k − 3

Moreover, k ≥ 6 and c2 < b1 when Γ is not a Taylor graph

Corollary

If Γ satisfies the M–property, then b2 < b1 < k and 1 < c2 < c3

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Primitive case

Lemma

k2 = k iff Γ is either C6, C7 or T (µ, k)

Proposition

If Γ satisfies the M–property, then

1 < c2 ≤ b1 < 2c2 b2 < c3 k3 ≤ k − 3

Moreover, k ≥ 6 and c2 < b1 when Γ is not a Taylor graph

Corollary

If Γ satisfies the M–property, then b2 < b1 < k and 1 < c2 < c3

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Facts

None of the Shilla graphs satisfy the M–property

None of the families of primitive distance–regular graphs withdiameter 3 satisfy the M–property

Conjecture

No primitive distance–regular graph with D = 3 satisfy the

M–property.

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Thank you!

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

The 5-Cycle has the M–property

The Moore–Penrose inverse of L is an M–matrix

L =

2 −1 0 0 −1−1 2 −1 0 00 −1 2 −1 00 0 −1 2 −1−1 0 0 −1 2

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

The 5-Cycle has the M–property

The Moore–Penrose inverse of L is an M–matrix

L =

2 −1 0 0 −1−1 2 −1 0 00 −1 2 −1 00 0 −1 2 −1−1 0 0 −1 2

L† =

2/5 0 −1/5 −1/5 00 2/5 0 −1/5 −1/5

−1/5 0 2/5 0 −1/5−1/5 −1/5 0 2/5 00 −1/5 −1/5 0 2/5

M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

The 5-Cycle has the M–property

The Moore–Penrose inverse of L is an M–matrix

L =

2 −1 0 0 −1−1 2 −1 0 00 −1 2 −1 00 0 −1 2 −1−1 0 0 −1 2

L† =

2/5 0 −1/5 −1/5 00 2/5 0 −1/5 −1/5

−1/5 0 2/5 0 −1/5−1/5 −1/5 0 2/5 00 −1/5 −1/5 0 2/5

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M.Mitjana On the M–property for distance–regular graphs

–IWONT 11, Brussels 2011–

IntroductionThe M–property

Distance–regular graphs

PropertiesCharacterizationStrongly regular graphsDiameter three

Petersen Graph does NOT have the M–property

The Moore–Penrose inverse of L is NOT an M–matrix

(L†)ii = 0.33(L†)ij = 0.03 if d(xi , xj ) = 1(L†)ij = −0.07 if d(xi , xj ) = 2

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M.Mitjana On the M–property for distance–regular graphs

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