Download - The Hierarchical Product of Graphs · 1 Introduction 2 Graphs and matrices 3 The hierarchical product De nition and basic properties Vertex hierarchy Metric parameters 4 Algebraic

The Hierarchical Product of Graphs


Lali BarriereFrancesc Comellas

Cristina DalfoMiquel Angel Fiol

Universitat Politecnica de Catalunya - DMA4

April 8, 2008

Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08


Outline

1 Introduction

2 Graphs and matrices

3 The hierarchical productDefinition and basic propertiesVertex hierarchyMetric parameters

4 Algebraic propertiesSpectral properties of G u Km

2

The spectrum of the binary hypertree Tm = Km2

The spectrum of a generic two-term product G2 u G1

5 Related worksHypertrees and generalized hypertreesGeneralization of the hierarchical product

6 Conclusions



Introduction

1 Introduction




2




6 Conclusions



Introduction

Motivation

Complex networks: randomness, heterogeneity, modularity

• M.E.J. Newman. The structure and function of complex networks.SIAM Rev. 45 (2003) 167–256.

Hierarchical networks: degree distribution, modularity

• S. Jung, S. Kim, B. Kahng. Geometric fractal growth model forscale-free networks. Phys. Rev. E 65 (2002) 056101.

• E. Ravasz, A.-L. Barabasi, Hierarchical organization in complexnetworks, Phys. Rev. E 67 (2003) 026112.

• E. Ravasz, A. L. Somera, D. A. Mongru, Z. N. Oltvai, A.-L.Barabasi, Hierarchical organization of modularity in metabolicnetworks, Science 297 (2002) 1551–1555.



Introduction

Our work

• Deterministic graphs

• Algebraic methods

• Far from ”real networks”

but a beautiful mathematical object !!!



Introduction

Previous work

• N. Biggs. Algebraic Graph Theory. Cambridge UP, Cambridge,1974.

• D. Cvetkovic, M. Doob, H. Sachs, Spectra of Graphs. Theory andApplications, Academic Press, New York, 1980.

• M.A. Fiol, M. Mitjana. The local spectra of regular line graphs.Discrete Math., submitted.

• C. D. Godsil. Algebraic Combinatorics. Chapman and Hall, NewYork, 1993.

• A.J. Schwenk, Computing the characteristic polynomial of a graph,Lect. Notes Math. 406 (1974) 153–172.

• J. R. Silvester, Determinants of block matrices, Maths Gazette 84(2000) 460–467.



Graphs and matrices

1 Introduction




2




6 Conclusions



Graphs and matrices

Spectrum of a matrix M

M n × n matrix on R

• Characteristic polynomial of M

ΦM(x) := det(xI −M)

• Spectrum of MspM := set of roots of ΦM(x), called eigenvalues of M

λ ∈ spM ⇒ dim ker(λI −M) ≥ 1

• Eigenvectors, eigenspacesv is a λ-eigenvector if Mv = λv

λ ∈ spM, Eλ := set of λ-eigenvectors of MEλ is a subspace of Rn



Graphs and matrices

Adjacency matrix and Laplacian matrix

G = (V ,E ), V = {1, 2, . . . n} ⇒• Adjacency matrix of G :

A(G ) = (ai ,j)1≤i ,j≤n ai ,j =

{1, if i ∼ j0, if i � j

tr(A) = 0,∑

j ai ,j = δi(Ordinary) spectrum of G := spectrum of A(G ).

• Laplacian matrix of G :

L(G ) = (`i ,j)1≤i ,j≤n `i ,j =

δi , if i = j−1, if i ∼ j0, if i � j , i 6= j

L(G ) = diag(δ1, δ2, . . . , δn)− A(G )Laplacian spectrum of G := spectrum of L(G ).



Graphs and matrices

Example: G = P3

1 2 3

A(G ) =

0 1 01 0 10 1 0

L(G ) =

1 −1 0−1 2 −1

0 −1 1



Graphs and matrices

Example: G = P3

A(G ) =

0 1 01 0 10 1 0

⇒ ΦA(x) = det

x −1 0−1 x −1

0 −1 x

= x3−2x

Eigenvalues and eigenvectorsΦA(x) = (x −

√2) · x · (x +

√2)⇒ λ1 =

√2, λ2 = 0, λ3 = −

√2

w1 = (√

2, 2,√

2) 1 2 3

!2

!22

w2 = (1, 0,−1) 1 2 3

!1 10

w3 = (√

2,−2,√

2) 1 2 3

!2"

2"

2



Graphs and matrices

Example: G = P3

L(G ) =

1 −1 0−1 2 −1

0 −1 1

⇒QL(x) = det

x − 1 1 01 x − 2 10 1 x − 1

= x3 − 4x2 + 3x

Laplacian eigenvalues and eigenvectorsQL(x) = x · (x − 1) · (x − 3)⇒ µ1 = 3, µ2 = 1, µ3 = 0

w1 = (1,−2, 1)w2 = (1, 0,−1)w3 = (1, 1, 1)



Graphs and matrices

Properties

G = (V ,E ) graph ⇒• A adjacency matrix• ΦA(x) = ΦG (x) = det(xI − A) characteristic polinomial• sp(A) = sp(G ) = {λm0

0 , λm11 , . . . , λmd

d }• ev(A) = ev(G ) = {λ0 > λ1 > · · · > λd}

Basic properties

1 A symmetric ⇒ ∀λi ∈ R; A diagonalizes; λi ∈ Q⇒ λi ∈ Z2 G = G1 ∪ · · · ∪ Gk connected comp. ⇒ ΦG (x) = ΠiΦGi

(x)

3 G connected ⇒ λ0 = ρ(G ) spectral radius of G∀i , |λi | ≤ ρ(G )if m ≥ 1⇒ ρ(G ) ≥ 1 and there is a negative eigenvalue

4 w = (w1, . . . ,wn) eigenvector of eigenvalue λ⇒Aw = λw⇔ ∀i ,

∑j∼i wj = λwi

(assign weight wi to vertex i)Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08


Graphs and matrices

An easy case

G = Kn

A(Kn) = J − I , where J = (1)sp(J) = {n1, 0n−1}En = (1, 1, . . . 1)E0 ⊥ En

⇒

sp(Kn) = {(n − 1)1, (−1)n−1}En−1 = (1, 1, . . . 1)E−1 ⊥ En



Graphs and matrices

Not so basic properties

1δ1 + · · ·+ δn

n≤ λ0 ≤ max

iδi

G δ-regular ⇒ λ0 = δ and w0 = (1, 1, . . . , 1)

2 D = diamG ⇒ D ≤ d = |ev(G )| − 1

3 G bipartite ⇔ sp(G ) symmetric (with respect to 0)

4 ωG clique number of G , χG chromatic number of G ⇒ωG ≤ 1− λ0

λd≤ χG ≤ 1 + λ0

5 G regular, αG independence number of G ⇒αG ≤

n

1 + λ0−λd

6 There exist non-isomorphic cospectral graphs.



Graphs and matrices

Spectrum of some graphs

• sp(Km,n) = {±√

mn, 0m+n−2}• ω = e

2πin ⇒ sp(Cn) = {ωr +ω−r = 2 cos 2πr

n : 0 ≤ r ≤ n− 1}

A(C4) =

0 1 0 11 0 1 00 1 0 11 0 1 0

⇒ ΦC4 (x) = (x2 − 4) · x2

ω = e2πi

4 = i⇒ λ0 = ω4 + ω−4 = 2, λ1 = ω3 + ω−3 = 0,λ2 = ω + ω−1 = 0, λ3 = ω2 + ω−2 = −2

• sp(Pn) = {2 cos πrn+1 : 1 ≤ r ≤ n}

•

{sp(G ) = {λm0

0 , λm11 , . . . , λmd

d }sp(H) = {µk0

0 , µk11 , . . . , µ

kd′d ′ }

}⇒

sp(G2H) = {(λi + µj)mi +kj : 0 ≤ i ≤ d , 0 ≤ j ≤ d ′}



Graphs and matrices

ΦG (x) coefficients

ΦG (x) = xn + c1xn−1 + · · ·+ cn−1x + cn ⇒

1 c1 = tr(A) = 0

Ak = (aki ,j), ak

i ,j = number of walks of length k from i to j ;c := number of closed walks of length k ⇒

c = tr(Ak) =∑

i λki

In particular, tr(A2) =∑

i λ2i = 2 ·m and

tr(A3) =∑

i λ3i = 6 · t, where t = number of triangles.

2 −c2 = m

3 −c3 = 2 · t



The hierarchical product

1 Introduction




2




6 Conclusions




Definition and basic properties

Definition

For i = 1,. . . N, Gi graph rooted at 0H = GN u · · · u G2 u G1

• vertices xN . . . x3x2x1, xi ∈ Vi

• if xj ∼ yj in Gj thenxN . . . xj+1xj0 . . . 0 ∼ xN . . . xj+1yj0 . . . 0

Example

The hierarchical products K2 u K3 and K3 u K2





GN u · · · u G2 u G1 is a spanning subgraph of GN 2 · · · 2 G2 2 G1

Example

The hierarchical product P4 u P3 u P2





GN = Gu N· · · uG is the hierachical N-power of G

Example

The hierarchical powers K 22 , K 4

2 and K 52





Example

The hierarchical power C 34





Order and size

ni = |Vi | and mi = |Ei |H = GN u · · · u G2 u G1

• nH = nN · · · n3n2n1

• mH =N∑

k=1

mknk+1 . . . nN

Properties of u

• Associativity. G3 uG2 uG1 = G3 u (G2 uG1) = (G3 uG2)uG1

• Right-distributivity. (G3 ∪ G2) u G1 = (G3 u G1) ∪ (G2 u G1)

• Left-semi-distributivity. G3 u (G2 ∪ G1) = (G3 u G2) ∪ n3G1,where n3G1 = Kn3 u G1 is n3 copies of G1

• G u K1 = K1 u G = G

(G,u) is a monoid




Vertex hierarchy

Degrees

• If δi = δGi(0), then

• δG (0) =N∑

i=1

δi

• x = xNxN−1 . . . xk00 . . . 0, xk 6= 0⇒

δH(x) =k−1∑i=1

δi + δGk(xk)

• If G is δ-regular, the degrees of the vertices of GN follow anexponential distribution, P(k) = γ−k , for some constant γ

For k = 1, . . . ,N − 1, GN contains (n − 1)nN−k vertices withdegree kδ and n vertices with degree Nδ




Vertex hierarchy

Example

Tm = Km2 has 2m−k vertices of degree k = 1, . . . ,m − 1, and two

vertices of degree m




Vertex hierarchy

Modularity

H = GN u · · · u G2 u G1, z an appropriate stringH〈zxk . . . x1〉 = H[{zxk . . . x1|xi ∈ Vi , 1 ≤ i ≤ k}]H〈xN . . . xkz〉 = H[{xN . . . xkz|xi ∈ Vi , k ≤ i ≤ N}]Lemma

• H〈zxk . . . x1〉 = Gk u · · · u G1, for any fixed z

• H〈xN . . . xk0〉 = GN u · · · u Gk

• H〈xN . . . xkz〉 = (nN · · · nk)K1, for any fixed z 6= 0

H∗ = H − 0

Lemma

• (GN u · · · u G2 u G1)∗ =⋃N

k=1(G ∗k u Gk−1 u · · · u G1)

• (KN2 )∗ =

⋃N−1k=0 K k

2

• KN2 − {{0, 10}} = KN−1

2

⋃KN−1

2




Vertex hierarchy

Example

Modularity and symmetry of Tm = Km2




Metric parameters

Eccentricity, radius and diameter

H = GN u · · · u G2 u G1

εi = eccGi(0), rGN

= rN and DGN= DN

ρi shortest path routing of Gi , i = 1, . . . ,N

Proposition

• {ρi}i=1...N induce a shortest path routing ρ in H

• The eccentricity, radius and diameter of H are

eccH(0) =N∑

i=1

εi , rH = rN +N−1∑i=1

εi , DH = DN + 2N−1∑i=1

εi




Metric parameters

Proof.




Metric parameters

Mean distance

G graph of order n

Mean distance. dG =1

n(n − 1)

∑v 6=w∈V

distG (v ,w)

Local mean distance. d0G =

1

n

∑v∈V

distG (0, v)

Proposition

H = G2uG1 ⇒{

d00H = d0

1 + d02

dH = 1n−1

[(n1 − 1)d1 + n1(n2 − 1)(d2 + 2d0

1 )]

Proof.Just compute!




Metric parameters

Corollary

H = GN , d = dG , d0 = d0G

• eccN(0) = Nε, d0N = Nd0

• rN = r + (N − 1)ε, DN = D + 2(N − 1)ε

• dN = d + 2(

(N−1)nN+1nN−1

− 1n−1

)d0

Asymptotically, dN ∼N

d + 2d0

(N − n

n − 1

)∼n

d + 2Nd0

Example

G = K2 ⇒ ecc = r = D = 1, d0 = 1/2 and d = 1The metric parameters of Tm = Km

2 are

• eccm(0) = m, d0m = m/2

• rm = m, Dm = 2m − 1

• dm = m2m

2m−1 − 1 ∼ m − 1



Algebraic properties

1 Introduction




2




6 Conclusions




Background

Kronecker product A⊗ B = (aijB)If A and B are square, A⊗ B and B⊗ A are permutation similar

LemmaH = G2 u G1 ⇒

AH = A2 ⊗D1 + I2 ⊗ A1∼= D1 ⊗ A2 + A1 ⊗ I2

where D1 = diag(1, 0, . . . 0)

Example

H = G u Kn, G of order N ⇒

AH = D1 ⊗ AG + AKn ⊗ IN =

AG IN · · · ININ 0 · · · IN...

......

IN IN · · · 0




Theorem (Silvester, 2000)

R commutative subring of F n×n, the set of all n × n matrices overa field F (or a commutative ring), and M ∈ Rm×m. Then,

detF M = detF (detR M)

Corollary (Silvester, 2000)

M =

(A BC D

)where A, B, C, D commute with each other.

Then,

det M = det(AD− BC)




Spectral properties of G u Km2

1 Introduction




2




6 Conclusions





G u K2

Example

The Petersen graph, hierarchically multiplied by K2





G u K2

G graph of order n,A adjacency matrix of G andφG characteristic polynomial of G

• The adjacency matrix of H = G u K2 is

AH =

(A InIn 0

)• The characteristic polynomial of H is

φH(x) = det(xI2n − AH) = det

(xIn − A −In−In xIn

)=

= det((x2 − 1)In − xA) = xnφG (x − 1x )





φH(x) = xnφG (x − 1x )

Proposition

H = G u K2 and spG = {λm00 < λm1

1 < . . . < λmdd } ⇒

spH = {λm000 < λm1

01 < . . . < λmd0d < λm0

10 < λm111 < . . . < λmd

1d }

where λ0i = f0(λi ) =λi−√λ2

i +4

2 , λ1i = f1(λi ) =λi +√λ2

i +4

2

Proof.

λ ∈ spH ⇔ φH(λ) = λnφG (λ− 1

λ) = 0⇔ λ− 1

λ∈ spG

λi ∈ spG ⇒ λ2 − λiλ− 1 = 0





φH(x) = xnφG (x − 1x )





Hm = G u Km2

Hm = Hm−1 u K2, m ≥ 1. The adjacency matrix of Hm is

Am =

(Am−1 Im−1

Im−1 0

)where Im denotes the identity matrix of size n2m (the same as Am)

H0 = G , A0 = A the adjacency matrix of G

Example





Let {pi , qi}i≥0 be the family of polynomials satisfying therecurrence equations

pi = p2i−1 − q2

i−1

qi = pi−1qi−1

with initial conditionsp0 = x and q0 = 1

Proposition

For every m ≥ 0, the characteristic polynomial of Hm = G u Km2 is

φm(x) = qm(x)nφ0

(pm(x)

qm(x)

)

LemmaIf p and q are arbitrary polynomials, then

det( pIn − qA −qIn−qIn pIn

)= det((p2 − q2)In − pqA)





Proof of φm(x) = qm(x)nφ0

(pm(x)qm(x)

).

By induction on m, using the Lemma

• Case m = 0. Trivially from q0(x) = 1 and p0(x) = x .

• m ≥ 1. By induction on i , we prove thatφm = det(pi Im−i − qiAm−i )

• i = 0 : φm = det(xIm − Am) = det(p0Im − q0Am)• i − 1⇒ i : φm = det(pi−1Im−i+1 − qi−1Am−i+1) =

= det((p2i−1 − q2

i−1)Im−i − pi−1qi−1Am−i ) =

= det(pi Im−i − qiAm−i )

• The case i = m givesφm(x) = det(pm(x)I0 − qm(x)A0) =

= det(

qm(x)(

pm(x)qm(x) I0 − A0

))= qm(x)nφ0

(pm(x)qm(x)

)





Proof of φm(x) = qm(x)nφ0

(pm(x)qm(x)

).

By induction on m, using the Lemma

• Case m = 0. Trivially from q0(x) = 1 and p0(x) = x .

• m ≥ 1. By induction on i , we prove thatφm = det(pi Im−i − qiAm−i )

• i = 0 : φm = det(xIm − Am) = det(p0Im − q0Am)• i − 1⇒ i : φm = det(pi−1Im−i+1 − qi−1Am−i+1) =

= det((p2i−1 − q2

i−1)Im−i − pi−1qi−1Am−i ) =

= det(pi Im−i − qiAm−i )

• The case i = m givesφm(x) = det(pm(x)I0 − qm(x)A0) =

= det(

qm(x)(

pm(x)qm(x) I0 − A0

))= qm(x)nφ0

(pm(x)qm(x)

)Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08




1 Introduction




2




6 Conclusions





Tm = Km2

pi = p2i−1 − q2

i−1

qi = pi−1qi−1

p0 = x , q0 = 1

Corollary

• φTm(x) = pm(x)

• φT∗m(x) = qm(x)

Proof.G = K1 ⇒ φ0(x) = x ⇒ φTm(x) = qm(x)nφ0

(pm(x)qm(x)

)= pm(x)

T ∗m = Tm − 0 =⋃m−1

i=0 Ti ⇒ φT∗m(x) =∏m−1

i=0 pi (x) = qm(x)





Proposition

Tm, m ≥ 1, has distinct eigenvalues λm0 < λm

1 < · · · < λmn−1, with

n = 2m, satisfying the following recurrence relation:

λmn2

+k =λm−1

k +√

(λm−1k )2 + 4

2

λmn−k−1 = −λm

k

for m > 1 and k = n2 ,

n2 + 1, . . . , n − 1

Proof.

• λ0i = f0(λi ) =λi−√λ2

i +4

2 , λ1i = f1(λi ) =λi +√λ2

i +4

2

• Tm bipartite ⇒ its spectrum is symmetric with respect to 0

• spT0 = {01} ⇒ the multiplicity of every λm1 is 1





Properties of spTm

λi ∈ spG ⇒ λ2 − λiλ− 1 = 0

f0(x) =x −√

x2 + 4

2f1(x) =

x +√

x2 + 4

2

m = 0⇒ spT0 = {0}

m = 1⇒ λ0 = f0(0) = −1, λ1 = f1(0) = 1

m = 2⇒λ0 = f0(−1) = f0(f0(0)) = −1.618 λ1 = f0(1) = f0(f1(0)) = −0.618λ2 = f1(−1) = f1(f0(0)) = 0.618 λ3 = f1(1) = f1(f1(0)) = 1.618. . .

m fixed, i = im−1 . . . i1i0 ∈ Zm2 ⇒

⇒ λi = (fim−1 ◦ · · · ◦ fi1 ◦ fi0)(0)





The distinct eigenvalues of the hypertree Tm for 0 ≤ m ≤ 6.





Proposition

The asymptotic behaviors of

• the spectral radius ρk = max0≤i≤n−1{|λi|} = λ111...1,

• the second largest eigenvalue θk = λ111...10, and

• the minimum positive eigenvalueσk = min0≤i≤n−1{|λi|} = λ100...0

of the hypertree Tk are:

ρk ∼√

2k , θk ∼√

2k , σk ∼ 1/√

2k





Proof of ρk ∼√

2k , θk ∼√

2k , σk ∼ 1/√

2k .

• ρkσk = 1

• ρk and θk verify the recurrence

λk+1 = f1(λk) = 12 (λk +

√λ2

k + 4)

• Assuming λk ∼ αkβ

α(k + 1)β ∼ αkβ +√α2k2β + 4

2⇒

⇒ α2(k + 1)β[(k + 1)β − kβ] ∼ 1

2(k + 1)12 [(k + 1)

12 − k

12 ] =

2(k + 1)12

(k + 1)12 + k

12

→ 1





1 Introduction




2




6 Conclusions





TheoremLet G1 and G2 be two graphs on ni vertices, with adjacency matrixAi and characteristic polynomial φi (x), i = 1, 2.Consider the graph G ∗1 = G1 − 0, with adjacency matrix A∗1 andcharacteristic polynomial φ∗1.Then the characteristic polynomial φH(x) of the hierarchicalproduct H = G2 u G1 is:

φH(x) = φ∗1(x)n2φ2

(φ1(x)

φ∗1(x)

)





Proof of φH(x) = φ∗1(x)n2φ2

(φ1(x)φ∗1 (x)

).

• The adjacency matrix of H is an n1 × n1 block matrix, withblocks of size n2 × n2

AH = D1 ⊗ A2 + A1 ⊗ I2 =

(A2 BB> A∗1 ⊗ I2

)where B =

(I2

(δ)· · · · · · I2 0 0 · · · · · · 0


φH(x) = det(xI− AH) = det

(xI2 − A2 −B−B> (xI∗1 − A∗1)⊗ I2

)• Computing the determinant in Rn2×n2 :φH(x) = det([xI2 − A2]φ∗1(x)I2 + φ1(x)I2 − xI2φ

∗1(x)) =

= det(φ1(x)I2 − φ∗1(x)A2) = det(φ∗1(x)

[φ1(x)φ∗1 (x) I2 − A2

])=

= φ∗1(x)n2φ2

(φ1(x)φ∗1 (x)

)






(φ1(x)φ∗1 (x)

).


AH = D1 ⊗ A2 + A1 ⊗ I2 =

(A2 BB> A∗1 ⊗ I2

)where B =

(I2

(δ)· · · · · · I2 0 0 · · · · · · 0

)

• The characteristic polynomial of H is


(xI2 − A2 −B−B> (xI∗1 − A∗1)⊗ I2


∗1(x)) =

= det(φ1(x)I2 − φ∗1(x)A2) = det(φ∗1(x)

[φ1(x)φ∗1 (x) I2 − A2

])=

= φ∗1(x)n2φ2

(φ1(x)φ∗1 (x)

)






(φ1(x)φ∗1 (x)

).


AH = D1 ⊗ A2 + A1 ⊗ I2 =

(A2 BB> A∗1 ⊗ I2

)where B =

(I2

(δ)· · · · · · I2 0 0 · · · · · · 0



(xI2 − A2 −B−B> (xI∗1 − A∗1)⊗ I2

)

• Computing the determinant in Rn2×n2 :φH(x) = det([xI2 − A2]φ∗1(x)I2 + φ1(x)I2 − xI2φ

∗1(x)) =

= det(φ1(x)I2 − φ∗1(x)A2) = det(φ∗1(x)

[φ1(x)φ∗1 (x) I2 − A2

])=

= φ∗1(x)n2φ2

(φ1(x)φ∗1 (x)

)






(φ1(x)φ∗1 (x)

).


AH = D1 ⊗ A2 + A1 ⊗ I2 =

(A2 BB> A∗1 ⊗ I2

)where B =

(I2

(δ)· · · · · · I2 0 0 · · · · · · 0



(xI2 − A2 −B−B> (xI∗1 − A∗1)⊗ I2


∗1(x)) =

= det(φ1(x)I2 − φ∗1(x)A2) = det(φ∗1(x)

[φ1(x)φ∗1 (x) I2 − A2

])=

= φ∗1(x)n2φ2

(φ1(x)φ∗1 (x)

)Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08




Corollary

G1 walk-regular ⇒ φH(x) =

(φ′1(x)

n1

)n2

φ2

(n1φ1(x)

φ′1(x)

)Proof.φ∗1(x) = 1

n1φ′1(x)

Corollary

G graph of order n2 = N and characteristic polynomial φG ⇒ thecharacteristic polynomial of H = G u Kn is

φH(x) = (x + 1)N(n−2)(x − n + 2)NφG

((x + 1)(x − n + 1)

(x − n + 2)

)Proof.Kn is walk-regular, φKn = (x − n + 1)(x + 1)n−1 andφ′Kn

= (x + 1)n−1 + (n − 1)(x − n + 1)(x + 1)n−2



Related works

1 Introduction




2




6 Conclusions



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Hypertrees and generalized hypertrees

1 Introduction




2




6 Conclusions



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Tm = Km2



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Eigenvalues of the hypertree Tm for 0 ≤ m ≤ 6.



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The generalized hypertree

T mr = Pm

r

Example000

001

002

100

101

102

110

111

112

120

121

122

200

201

202

210

211

212

220

221

222

010

011

012

020

021

022



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Eigenvalues of T m3 for 0 ≤ m ≤ 3.

0

-1.414 1.414

-2.053 2.053

-2.523 2.523



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Generalization of the hierarchical product

1 Introduction




2




6 Conclusions



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Definition of the generalized hierarchical product

Gi = (Vi ,Ei ), ∅ 6= Ui ⊆ Vi , i = 1, 2, . . . ,N − 1

H = GN u GN−1(UN−1) u · · · u G1(U1) is the graph:

• vertices VN × · · ·V2 × V1

• if xj ∼ yj in Gj and ui ∈ Ui , i = 1, 2, . . . , j − 1 thenxN . . . xj+1xjuj−1 . . . u1 ∼ xN . . . xj+1yjuj−1 . . . u1

Example

• For every i , Ui = Vi ⇒GN u GN−1(UN−1) u · · · u G1(U1) = GN2GN−12 · · ·2G1

• For every i , Ui = {0} ⇒GN u GN−1(UN−1) u · · · u G1(U1) = GN u GN−1 u · · · u G1



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Example

Two views of a generalized hierarchical product K 33 with

U1 = U2 = {0, 1}.



Conclusions

Summary

1 Definition of the hierarchical product of graphs

2 Spectral properties

3 The particular case of Tm

4 Properties of T mr , spT m

r and⋃m

spT mr

5 Definition and properties of the generalized hierarchicalproduct



Conclusions

Publications

• The hierarchical product of graphs, L. Barriere, F. Comellas,C. Dalfo, M. A. Fiol, Discrete Applied Mathematics, toappear.

• On the spectra of hypertrees, BCDF, Linear Algebra and itsApplications, 428(7):1499–1510

• On the hierarchical product of graphs and the generalizedbinomial tree, BCDF, Linear and Multilinear Algebra,submitted (September 2007).

• The generalized hierarchical product of graphs, L. Barriere, C.Dalfo, M. A. Fiol, M. Mitjana, Journal of Graph Theory,submitted (March, 2008).



Conclusions

Gracias !!!
