On the Moore–Penrose inverse of distance–regular graphs · Applications Generalized inverses...

On the Moore–Penroseinverse of

distance–regular graphs

E. Bendito1, A. Carmona1, A.M. Encinas1,

and M. Mitjana2

Dept. Matematica Aplicada 2I and 1III

Universitat Politecnica de Catalunya

Eurocomb’11. Budapest, 2011

–EuroComb’11, Budapest 2011–

IntroductionThe M–property

Distance–regular graphs

ApplicationsGeneralized inversesThe combinatorial Laplacian

M–matrix

I Matrices with non–positive off–diagonal andnon–negative diagonal entries

L = kI − A

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

A. Carmona M–P inverse of distance–regular graphs





M–matrix

I Matrices with non–positive off–diagonal andnon–negative diagonal entries

L = kI − A

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

I If k ≥ ρ(A), then L is an M–matrix






Where can we find M–matrices?

I Finite difference methods for solving PDEvértices en δ(F )

vértices en F








vértices en F

I Markov processes in probability and statistics








vértices en F

I Markov processes in probability and statistics

I The combinatorial Laplacian of a network

L =

c1 −c1 0

−c1 c1 + c2 −c20 −c2 c2







I Symmetric and irreducible M–matrices

non–singular 99K discrete Dirichlet problem 99K itsinverse corresponds with the Green operator associatedwith the boundary value problem

singular 99K discrete Poisson equation 99K itsMoore–Penrose inverse corresponds with the Greenoperator too






The Moore–Penrose inverse

A generalized inverse of a 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 A† satisfyingthe Penrose equations

AA†A = A (AA†)∗ = AA†

A†AA† = A† (A†A)∗ = A†A

A† is the Moore–Penrose inverse of 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 inversewhich is non–negative

Given an M–matrix, there exists a generalized inversewhich is also an M–matrix?

When the Moore –Penrose inverse of the combinatorialLaplacian of a network is an M–matrix?






Notations

Simple, finite and connected network Γ = (V,E, c)

Conductance c : V × V −→ [0,+∞) such that{c(x, x) = 0 for any x ∈ Vc(x, y) > 0 for x ∼ y

Real–valued functions on V C(V )

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

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






The combinatorial laplacian

L(u)(x) =∑y∈V

c(x, y)(u(x)− u(y)

), x ∈ V







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,diagonally dominant and singular M–matrix, L







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,diagonally dominant and singular M–matrix, L

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





Characterization

I L† is an M–matrix iff (L†)ij ≤ 0 for any i 6= j





Characterization


I If Γ has the M–property, then L† can be seen as thecombinatorial Laplacian of a new network Example: Cycle





Characterization



I Not all networks satisfy the M–property Example: Petersen





Characterization







Characterization



I Not all networks satisfy the M–property Example: Petersen





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

ai−1 ai+1

bi−1 ci+1

ki−1 ki ki+1

aici bi






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)

I k0 = 1 and ki =b0 · · · bi−1c1 · · · ci

, i = 1, . . . , D

I n = 1 + k + k2 + · · ·+ kD

I k > b1 ≥ · · · ≥ bD−1 ≥ 1

I 1 ≤ c2 ≤ · · · ≤ cD ≤ k

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






The intersection array ι(Γ)

I k0 = 1 and ki =b0 · · · bi−1c1 · · · ci

, i = 1, . . . , D

I n = 1 + k + k2 + · · ·+ kD

I k > b1 ≥ · · · ≥ bD−1 ≥ 1

I 1 ≤ c2 ≤ · · · ≤ cD ≤ k

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

Notation

> a1=λ

> c2=µ






Bipartite and Antipodal distance–regular graphs

(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 M–property

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

)






The M–property

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

)

I Γ has the M–property iff

D−1∑j=1

1

kjbj

( D∑i=j+1

ki

)2≤ n− 1

k






The M–property

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

)

I Γ has the M–property iff

D−1∑j=1

1

kjbj

( D∑i=j+1

ki

)2≤ n− 1

k

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

λ ≤ 3k − k2

n− 1− n =⇒ n < 3k






Small diameter

I If Γ has the M–property, then

D ≤ 3






Small diameter


D ≤ 3

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

3k < 1 + 3k ≤ 1 + k + k2 + k3 ≤ n






Small diameter


D ≤ 3

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

3k <1 + 3k ≤ 1 + k + k2 + k3 ≤ n

!!!!!






Diameter two

I A s.r.g. with parameters (n, k, λ, µ) has the M–property iff

µ ≥ k − k2

n− 1






Diameter two


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






Diameter two


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

How many s.r.g. have the M -property?






Half of the s.r.g. have the M–property

I If Γ is primitive, then either Γ or Γ has the M–property






Half of the s.r.g. have the M–property

I If Γ is primitive, then either Γ or Γ has the M–property

Conference Graph(4m+ 1, 2m,m− 1,m)

Γ and Γ, have the M–property iff Γ is a conferencegraph






Diameter three

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

k2b1

(b2c2 + (b2 + c3)

2)≤ c22c23(n− 1)






Diameter three


k2b1

(b2c2 + (b2 + c3)

2)≤ c22c23(n− 1)

I If Γ is bipartite with ι(Γ) = {k, k − 1, k − µ; 1, µ, k}






Diameter three


k2b1

(b2c2 + (b2 + c3)

2)≤ c22c23(n− 1)


Γ satisfies the M–property iff4k

5≤ µ ≤ k − 1 =⇒ k ≥ 5






Diameter three


k2b1

(b2c2 + (b2 + c3)

2)≤ c22c23(n− 1)


Γ satisfies the M–property iff4k

5≤ µ ≤ k − 1 =⇒ k ≥ 5

When, 1 ≤ µ < k − 1, then either Γ or Γ3 has theM–property, except when k − 1 < 5µ < 4k in which casenone of them has the M–property






Diameter three

I If Γ is antipodal with ι(Γ) = {k, tµ, 1; 1, µ, k}






Diameter three


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

k + 3

2≤ µ < k






Diameter three


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

k + 3

2≤ µ < k

If Γ is T (k, µ) with 1 ≤ µ ≤ k − 2, then either Γ or Γ2

has the M–property, except for

> µ ∈ {m− 2,m− 1,m,m+ 1} when k = 2m

> µ ∈ {m− 1,m,m+ 1} when k = 2m+ 1

in which case NONE has the M–property






Diameter three

I If Γ is bipartite and antipodal with

ι(Γ) = {k, k − 1, 1; 1, k − 1, k}






Diameter three


ι(Γ) = {k, k − 1, 1; 1, k − 1, k} =⇒ Γ = T (k, k − 1)






Diameter three


ι(Γ) = {k, k − 1, 1; 1, k − 1, k} =⇒ Γ = T (k, k − 1)

Γ is a k–Crown graph






Diameter three


ι(Γ) = {k, k − 1, 1; 1, k − 1, k} =⇒ Γ = T (k, k − 1)

Γ is a k–Crown graph it has the M–property iff k ≥ 5






Diameter three


ι(Γ) = {k, k − 1, 1; 1, k − 1, k} =⇒ Γ = T (k, k − 1)

Γ is a k–Crown graph it has the M–property iff k ≥ 5

I If Γ is primitive and 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






Facts

None of the Shilla graphs satisfy the M–property

None of the known families of primitive distance–regulargraphs with diameter 3 satisfy the M–property






Facts

None of the Shilla graphs satisfy the M–property

None of the known families of primitive distance–regulargraphs with diameter 3 satisfy the M–property

Conjecture

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

M–property.






The 5–Cycle has the M–property

L =

2 −1 0 0 −1−1 2 −1 0 0

0 −1 2 −1 00 0 −1 2 −1−1 0 0 −1 2

L† =1

5

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

0 −1 −1 0 2

Back






Petersen Graph does NOT have the M–property

(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 Moore–Penrose inverse of distance–regular graphs · Applications Generalized inverses...

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