On the M–property for distance–regular graphs€¦ · Distance–regular graphs Statement of...

–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

[email protected]

M.Mitjana On the M–property for 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-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





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









...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





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†





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







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?





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







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








Example: Cycle








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





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





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





Definition

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





PropertiesCharacterizationStrongly regular graphsDiameter three


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






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}






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






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






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






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.






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






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






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






Diameter three

Proposition

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

k2b1

(

b2c2 + (b2 + c3)2)

≤ c22c23 (n − 1)






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






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






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.






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






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






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.






Thank you!






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 −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








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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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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On the M–property for distance–regular graphs€¦ · Distance–regular graphs Statement of...

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