The Hierarchical Product of Graphs
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
The Hierarchical Product of Graphs
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Introduction
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
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.
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Introduction
Our work
• Deterministic graphs
• Algebraic methods
• Far from ”real networks”
but a beautiful mathematical object !!!
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Introduction
Our work
• Deterministic graphs
• Algebraic methods
• Far from ”real networks”
but a beautiful mathematical object !!!
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
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.
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Graphs and matrices
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
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 ).
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
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)
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
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
The Hierarchical Product of Graphs
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
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.
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of 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 ′}
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
Definition and basic properties
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
Definition and basic properties
GN = Gu N· · · uG is the hierachical N-power of G
Example
The hierarchical powers K 22 , K 4
2 and K 52
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
Definition and basic properties
Example
The hierarchical power C 34
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
Definition and basic properties
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
Vertex hierarchy
Example
Tm = Km2 has 2m−k vertices of degree k = 1, . . . ,m − 1, and two
vertices of degree m
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
Vertex hierarchy
Example
Modularity and symmetry of Tm = Km2
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
Vertex hierarchy
Example
Modularity and symmetry of Tm = Km2
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
Vertex hierarchy
Example
Modularity and symmetry of Tm = Km2
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
Vertex hierarchy
Example
Modularity and symmetry of Tm = Km2
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
Vertex hierarchy
Example
Modularity and symmetry of Tm = Km2
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
Vertex hierarchy
Example
Modularity and symmetry of Tm = Km2
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
Metric parameters
Proof.
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
Metric parameters
Proof.
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
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!
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
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!
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
The hierarchical product
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
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)
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
Spectral properties of G u Km2
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
Spectral properties of G u Km2
G u K2
Example
The Petersen graph, hierarchically multiplied by K2
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
Spectral properties of G u Km2
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 )
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
Spectral properties of G u Km2
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
Spectral properties of G u Km2
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
Spectral properties of G u Km2
φH(x) = xnφG (x − 1x )
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
Spectral properties of G u Km2
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
Spectral properties of G u Km2
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)
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
Spectral properties of G u Km2
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
The Hierarchical Product of Graphs
Algebraic properties
Spectral properties of G u Km2
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
The Hierarchical Product of Graphs
Algebraic properties
Spectral properties of G u Km2
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
The Hierarchical Product of Graphs
Algebraic properties
Spectral properties of G u Km2
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
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of the binary hypertree Tm = Km2
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of the binary hypertree Tm = Km2
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)
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of the binary hypertree Tm = Km2
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)
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of the binary hypertree Tm = Km2
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of the binary hypertree Tm = Km2
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of the binary hypertree Tm = Km2
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)
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of the binary hypertree Tm = Km2
The distinct eigenvalues of the hypertree Tm for 0 ≤ m ≤ 6.
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of the binary hypertree Tm = Km2
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of the binary hypertree Tm = Km2
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of the binary hypertree Tm = Km2
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of the binary hypertree Tm = Km2
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of the binary hypertree Tm = Km2
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of a generic two-term product G2 u G1
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of a generic two-term product G2 u G1
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)
)
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of a generic two-term product G2 u G1
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
)• The characteristic polynomial of H is
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)
)
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of a generic two-term product G2 u G1
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
)
• The characteristic polynomial of H is
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)
)
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of a generic two-term product G2 u G1
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
)• The characteristic polynomial of H is
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)
)
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of a generic two-term product G2 u G1
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
)• The characteristic polynomial of H is
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)
)Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Algebraic properties
The spectrum of a generic two-term product G2 u G1
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Related works
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Related works
Hypertrees and generalized hypertrees
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Related works
Hypertrees and generalized hypertrees
Tm = Km2
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Related works
Hypertrees and generalized hypertrees
Eigenvalues of the hypertree Tm for 0 ≤ m ≤ 6.
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Related works
Hypertrees and generalized hypertrees
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Related works
Hypertrees and generalized hypertrees
Eigenvalues of T m3 for 0 ≤ m ≤ 3.
0
-1.414 1.414
-2.053 2.053
-2.523 2.523
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Related works
Generalization of the hierarchical product
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Related works
Generalization of the hierarchical product
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Related works
Generalization of the hierarchical product
Example
Two views of a generalized hierarchical product K 33 with
U1 = U2 = {0, 1}.
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
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
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
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).
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08
The Hierarchical Product of Graphs
Conclusions
Gracias !!!
Seminario “Vıctor Neumann-Lara”, IMUNAM, 8-4-08