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  • The Hierarchical Product of Graphs

    The Hierarchical Product of Graphs

    Lali Barrière Francesc Comellas

    Cristina Dalfó Miquel Àngel Fiol

    Universitat Politècnica de Catalunya - DMA4

    April 8, 2008

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

  • The Hierarchical Product of Graphs

    Outline

    1 Introduction

    2 Graphs and matrices

    3 The hierarchical product Definition and basic properties Vertex hierarchy Metric parameters

    4 Algebraic properties Spectral properties of G u Km2 The spectrum of the binary hypertree Tm = K

    m 2

    The spectrum of a generic two-term product G2 u G1 5 Related works

    Hypertrees and generalized hypertrees Generalization of the hierarchical product

    6 Conclusions

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

  • The Hierarchical Product of Graphs

    Introduction

    1 Introduction

    2 Graphs and matrices

    3 The hierarchical product Definition and basic properties Vertex hierarchy Metric parameters

    4 Algebraic properties Spectral properties of G u Km2 The spectrum of the binary hypertree Tm = K

    m 2

    The spectrum of a generic two-term product G2 u G1 5 Related works

    Hypertrees and generalized hypertrees Generalization of the hierarchical product

    6 Conclusions

    Seminario “V́ıctor 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 for scale-free networks. Phys. Rev. E 65 (2002) 056101.

    • E. Ravasz, A.-L. Barabási, Hierarchical organization in complex networks, Phys. Rev. E 67 (2003) 026112.

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

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

  • The Hierarchical Product of Graphs

    Introduction

    Previous work

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

    • D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs. Theory and Applications, 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, New York, 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 “V́ıctor Neumann-Lara”, IMUNAM, 8-4-08

  • The Hierarchical Product of Graphs

    Graphs and matrices

    1 Introduction

    2 Graphs and matrices

    3 The hierarchical product Definition and basic properties Vertex hierarchy Metric parameters

    4 Algebraic properties Spectral properties of G u Km2 The spectrum of the binary hypertree Tm = K

    m 2

    The spectrum of a generic two-term product G2 u G1 5 Related works

    Hypertrees and generalized hypertrees Generalization of the hierarchical product

    6 Conclusions

    Seminario “V́ıctor 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 M spM := set of roots of ΦM(x), called eigenvalues of M

    λ ∈ spM ⇒ dim ker(λI −M) ≥ 1 • Eigenvectors, eigenspaces

    v is a λ-eigenvector if Mv = λv λ ∈ spM, Eλ := set of λ-eigenvectors of M

    Eλ is a subspace of Rn

    Seminario “V́ıctor 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 ∼ j 0, 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 ∼ j 0, if i � j , i 6= j

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

    Seminario “V́ıctor 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 1 0 1 0

     L(G ) =  1 −1 0−1 2 −1

    0 −1 1

    

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

  • The Hierarchical Product of Graphs

    Graphs and matrices

    Example: G = P3

    A(G ) =

     0 1 01 0 1 0 1 0

    ⇒ ΦA(x) = det  x −1 0−1 x −1

    0 −1 x

     = x3−2x Eigenvalues and eigenvectors ΦA(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 “V́ıctor 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 1 0 1 x − 1

     = x3 − 4x2 + 3x Laplacian eigenvalues and eigenvectors QL(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 “V́ıctor 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 ) = {λm00 , λ

    m1 1 , . . . , λ

    md d }

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

    1 A symmetric ⇒ ∀λi ∈ R; A diagonalizes; λi ∈ Q⇒ λi ∈ Z 2 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 “V́ıctor 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 “V́ıctor Neumann-Lara”, IMUNAM, 8-4-08

  • The Hierarchical Product of Graphs

    Graphs and matrices

    Not so basic properties

    1 δ1 + · · ·+ δn

    n ≤ λ0 ≤ max

    i δi

    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 “V́ıctor 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

    2πi n ⇒ sp(Cn) = {ωr +ω−r = 2 cos 2πrn : 0 ≤ r ≤ n− 1}

    A(C4) =

     0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0

    ⇒ ΦC4 (x) = (x2 − 4) · x2 ω = e

    2πi 4 = i⇒ λ0 = ω4 + ω−4 = 2, λ1 = ω3 + ω−3 = 0,

    λ2 = ω + ω −1 = 0, λ3 = ω

    2 + ω−2 = −2

    • sp(Pn) = {2 cos πrn+1 : 1 ≤ r ≤ n}

    { sp(G ) = {λm00 , λ

    m1 1 , . . . , λ

    md d }

    sp(H) = {µk00 , µ k1 1 , . . . , µ

    kd′ d ′ }

    } ⇒

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

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

  • The Hierarchical Product of