The Hierarchical Product of Graphs

Transcription

1 The Hierarchical Product of Graphs Lali Barrière Francesc Comellas Cristina Dalfó Miquel Àngel Fiol Universitat Politècnica de Catalunya - DMA4 April 8, 2008

2 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 K2 m The spectrum of the binary hypertree T m = K2 m The spectrum of a generic two-term product G 2 G 1 5 Related works Hypertrees and generalized hypertrees Generalization of the hierarchical product 6 Conclusions

3 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 K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 5 Related works Hypertrees and generalized hypertrees Generalization of the hierarchical product 6 Conclusions

4 Introduction Motivation Complex networks: randomness, heterogeneity, modularity M.E.J. Newman. The structure and function of complex networks. SIAM Rev. 45 (2003) Hierarchical networks: degree distribution, modularity S. Jung, S. Kim, B. Kahng. Geometric fractal growth model for scale-free networks. Phys. Rev. E 65 (2002) E. Ravasz, A.-L. Barabási, Hierarchical organization in complex networks, Phys. Rev. E 67 (2003) E. Ravasz, A. L. Somera, D. A. Mongru, Z. N. Oltvai, A.-L. Barabási, Hierarchical organization of modularity in metabolic networks, Science 297 (2002)

5 Introduction Our work Deterministic graphs Algebraic methods

6 Introduction Our work Deterministic graphs Algebraic methods Far from real networks but a beautiful mathematical object!!!

7 Introduction Previous work N. Biggs. Algebraic Graph Theory. Cambridge UP, Cambridge, D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs. Theory and Applications, Academic Press, New York, 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, A.J. Schwenk, Computing the characteristic polynomial of a graph, Lect. Notes Math. 406 (1974) J. R. Silvester, Determinants of block matrices, Maths Gazette 84 (2000)

8 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 K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 5 Related works Hypertrees and generalized hypertrees Generalization of the hierarchical product 6 Conclusions

9 Graphs and matrices Spectrum of a matrix M M n n matrix on R Characteristic polynomial of M Φ M (x) := det(xi M) Spectrum of M spm := set of roots of Φ M (x), called eigenvalues of M λ spm dim ker(λi M) 1 Eigenvectors, eigenspaces v is a λ-eigenvector if Mv = λv λ spm, E λ := set of λ-eigenvectors of M E λ is a subspace of R n

10 Graphs and matrices Adjacency matrix and Laplacian matrix G = (V, E), V = {1, 2,... n} Adjacency matrix of G: A(G) = (a i,j ) 1 i,j n a i,j = { 1, if i j 0, if i j tr(a) = 0, j a i,j = δ i (Ordinary) spectrum of G := spectrum of A(G). Laplacian matrix of G: δ i, if i = j L(G) = (l i,j ) 1 i,j n l i,j = 1, if i j 0, if i j, i j L(G) = diag(δ 1, δ 2,..., δ n ) A(G) Laplacian spectrum of G := spectrum of L(G).

11 Graphs and matrices Example: G = P A(G) = L(G) =

12 Graphs and matrices Example: G = P 3 A(G) = Φ A (x) = det x x x = x 3 2x Eigenvalues and eigenvectors Φ A (x) = (x 2) x (x + 2) λ 1 = 2, λ 2 = 0, λ 3 = 2 w 1 = ( 2, 2, ) w 2 = (1, 0, 1) w 3 = ( 2, 2, ) 1 2 3

13 Graphs and matrices Example: G = P 3 L(G) = Q L (x) = det x x x 1 = x 3 4x 2 + 3x Laplacian eigenvalues and eigenvectors Q L (x) = x (x 1) (x 3) µ 1 = 3, µ 2 = 1, µ 3 = 0 w 1 = (1, 2, 1) w 2 = (1, 0, 1) w 3 = (1, 1, 1)

14 Graphs and matrices Properties G = (V, E) graph A adjacency matrix Φ A (x) = Φ G (x) = det(xi A) characteristic polinomial sp(a) = sp(g) = {λ m 0 0, λm 1 1,..., λm d d } ev(a) = ev(g) = {λ 0 > λ 1 > > λ d } Basic properties 1 A symmetric λ i R; A diagonalizes; λ i Q λ i Z 2 G = G 1 G k connected comp. Φ G (x) = Π i Φ Gi (x) 3 G connected λ 0 = ρ(g) spectral radius of G i, λ i ρ(g) if m 1 ρ(g) 1 and there is a negative eigenvalue 4 w = (w 1,..., w n ) eigenvector of eigenvalue λ Aw = λw i, j i w j = λw i (assign weight w i to vertex i)

15 Graphs and matrices An easy case G = K n A(K n ) = J I, where J = (1) sp(j) = {n 1, 0 n 1 } E n = (1, 1,... 1) E 0 E n sp(k n ) = {(n 1) 1, ( 1) n 1 } E n 1 = (1, 1,... 1) E 1 E n

16 Graphs and matrices Not so basic properties δ δ n 1 λ 0 max δ i n i G δ-regular λ 0 = δ and w 0 = (1, 1,..., 1) 2 D = diamg D d = ev(g) 1 3 G bipartite sp(g) symmetric (with respect to 0) 4 ω G clique number of G, χ G chromatic number of G ω G 1 λ 0 χ G 1 + λ 0 λ d 5 G regular, α G independence number of G n α G 1 + λ 0 λ d 6 There exist non-isomorphic cospectral graphs.

17 Graphs and matrices Spectrum of some graphs sp(k m,n ) = {± mn, 0 m+n 2 } ω = e 2πi n sp(c n ) = {ω r + ω r = 2 cos 2πr n : 0 r n 1} A(C 4 ) = Φ C 4 (x) = (x 2 4) x ω = e 2πi 4 = i λ 0 = ω 4 + ω 4 = 2, λ 1 = ω 3 + ω 3 = 0, λ 2 = ω + ω 1 = 0, λ 3 = ω 2 + ω 2 = 2 sp(p n ) = {2 cos πr n+1 { : 1 r n} sp(g) = {λ m 0 0, λm 1 1,..., λm d d } } sp(h) = {µ k 0 0, µk 1 1,..., µk d d } sp(g H) = {(λ i + µ j ) m i +k j : 0 i d, 0 j d }

18 Graphs and matrices Φ G (x) coefficients Φ G (x) = x n + c 1 x n c n 1 x + c n 1 c 1 = tr(a) = 0 A k = (ai,j k ), ak i,j = number of walks of length k from i to j; c := number of closed walks of length k c = tr(a k ) = i λk i In particular, tr(a 2 ) = i λ2 i = 2 m and tr(a 3 ) = i λ3 i = 6 t, where t = number of triangles. 2 c 2 = m 3 c 3 = 2 t

19 The hierarchical product 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 K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 5 Related works Hypertrees and generalized hypertrees Generalization of the hierarchical product 6 Conclusions

20 The hierarchical product Definition and basic properties Definition For i = 1,... N, G i graph rooted at 0 H = G N G 2 G 1 vertices x N... x 3 x 2 x 1, x i V i if x j y j in G j then x N... x j+1 x j x N... x j+1 y j Example The hierarchical products K 2 K 3 and K 3 K 2

21 The hierarchical product Definition and basic properties G N G 2 G 1 is a spanning subgraph of G N G 2 G 1 Example The hierarchical product P 4 P 3 P 2

22 The hierarchical product Definition and basic properties G N = G N G is the hierachical N-power of G Example The hierarchical powers K 2 2, K 4 2 and K 5 2

23 The hierarchical product Definition and basic properties Example The hierarchical power C 3 4

24 The hierarchical product Definition and basic properties Order and size n i = V i and m i = E i H = G N G 2 G 1 n H = n N n 3 n 2 n 1 N m H = m k n k+1... n N k=1 Properties of Associativity. G 3 G 2 G 1 = G 3 (G 2 G 1 ) = (G 3 G 2 ) G 1 Right-distributivity. (G 3 G 2 ) G 1 = (G 3 G 1 ) (G 2 G 1 ) Left-semi-distributivity. G 3 (G 2 G 1 ) = (G 3 G 2 ) n 3 G 1, where n 3 G 1 = K n3 G 1 is n 3 copies of G 1 G K 1 = K 1 G = G (G, ) is a monoid

25 The hierarchical product Vertex hierarchy Degrees eminario Víctor Neumann-Lara, IMUNAM, If δ i = δ Gi (0), then N δ G (0) = i=1 δ i x = x N x N 1... x k , x k 0 k 1 δ H (x) = δ i + δ Gk (x k ) i=1 If G is δ-regular, the degrees of the vertices of G N follow an exponential distribution, P(k) = γ k, for some constant γ For k = 1,..., N 1, G N contains (n 1)n N k vertices with degree kδ and n vertices with degree Nδ

26 The hierarchical product Vertex hierarchy Example T m = K2 m has 2m k vertices of degree k = 1,..., m 1, and two vertices of degree m

27 The hierarchical product Vertex hierarchy Modularity H = G N G 2 G 1, z an appropriate string H zx k... x 1 = H[{zx k... x 1 x i V i, 1 i k}] H x N... x k z = H[{x N... x k z x i V i, k i N}] Lemma H zx k... x 1 = G k G 1, for any fixed z H x N... x k 0 = G N G k H x N... x k z = (n N n k )K 1, for any fixed z 0 H = H 0 Lemma (G N G 2 G 1 ) = N k=1 (G k G k 1 G 1 ) (K2 N) = N 1 k=0 K 2 k K2 N {{0, 10}} = K N 1 2 K N 1 2

28 The hierarchical product Vertex hierarchy Example Modularity and symmetry of T m = K m 2

29 The hierarchical product Vertex hierarchy Example Modularity and symmetry of T m = K m 2

30 The hierarchical product Vertex hierarchy Example Modularity and symmetry of T m = K m 2

31 The hierarchical product Vertex hierarchy Example Modularity and symmetry of T m = K m 2

32 The hierarchical product Vertex hierarchy Example Modularity and symmetry of T m = K m 2

33 The hierarchical product Vertex hierarchy Example Modularity and symmetry of T m = K m 2

34 The hierarchical product Metric parameters Eccentricity, radius and diameter H = G N G 2 G 1 ε i = ecc Gi (0), r GN = r N and D GN = D N ρ i shortest path routing of G i, i = 1,..., N Proposition {ρ i } i=1...n induce a shortest path routing ρ in H The eccentricity, radius and diameter of H are N N 1 ecc H (0) = ε i, r H = r N + ε i, D H = D N + 2 i=1 i=1 N 1 i=1 ε i

35 The hierarchical product Metric parameters Proof.

36 The hierarchical product Metric parameters Proof.

37 The hierarchical product Metric parameters Mean distance G graph of order n 1 Mean distance. d G = dist G (v, w) n(n 1) v w V Local mean distance. dg 0 = 1 dist G (0, v) n v V Proposition { d 00 H = G 2 G 1 H = d d 2 0 d H = 1 [ n 1 (n1 1)d 1 + n 1 (n 2 1)(d 2 + 2d1 0)]

38 The hierarchical product Metric parameters Mean distance G graph of order n 1 Mean distance. d G = dist G (v, w) n(n 1) v w V Local mean distance. dg 0 = 1 dist G (0, v) n v V Proposition { d 00 H = G 2 G 1 H = d d 2 0 d H = 1 [ n 1 (n1 1)d 1 + n 1 (n 2 1)(d 2 + 2d1 0)] Proof. Just compute!

39 The hierarchical product Metric parameters Corollary H = G N, d = d G, d 0 = d 0 G ecc N (0) = Nε, d 0 N = Nd 0 r N = r + (N 1)ε, D N = D + 2(N 1)ε ( ) (N 1)n N +1 1 n N 1 n 1 d 0 d N = d + 2 Asymptotically, d N N d + 2d 0 ( N n n 1 ) d + 2Nd 0 n Example G = K 2 ecc = r = D = 1, d 0 = 1/2 and d = 1 The metric parameters of T m = K2 m are ecc m (0) = m, d 0 m = m/2 r m = m, D m = 2m 1 d m = m2m 2 m 1 1 m 1

40 Algebraic properties 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 K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 5 Related works Hypertrees and generalized hypertrees Generalization of the hierarchical product 6 Conclusions

41 Algebraic properties Background Kronecker product A B = (a ij B) If A and B are square, A B and B A are permutation similar

42 Algebraic properties Background Kronecker product A B = (a ij B) If A and B are square, A B and B A are permutation similar Lemma H = G 2 G 1 A H = A 2 D 1 + I 2 A 1 = D1 A 2 + A 1 I 2 where D 1 = diag(1, 0,... 0)

43 Algebraic properties Background Kronecker product A B = (a ij B) If A and B are square, A B and B A are permutation similar Lemma H = G 2 G 1 A H = A 2 D 1 + I 2 A 1 = D1 A 2 + A 1 I 2 where D 1 = diag(1, 0,... 0) Example H = G K n, G of order N A H = D 1 A G + A Kn I N = A G I N I N I N 0 I N... I N I N 0

44 Algebraic properties Theorem (Silvester, 2000) R commutative subring of F n n, the set of all n n matrices over a field F (or a commutative ring), and M R m m. Then, det F M = det F (det R M) Corollary ( (Silvester, ) 2000) A B M = where A, B, C, D commute with each other. C D Then, det M = det(ad BC)

45 Algebraic properties Spectral properties of G K m 2 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 K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 5 Related works Hypertrees and generalized hypertrees Generalization of the hierarchical product 6 Conclusions

46 Algebraic properties Spectral properties of G K m 2 G K 2 Example The Petersen graph, hierarchically multiplied by K 2

47 Algebraic properties Spectral properties of G K m 2 G K 2 G graph of order n, A adjacency matrix of G and φ G characteristic polynomial of G The adjacency matrix of H = G K 2 is ( ) A In A H = I n 0 The characteristic polynomial of H( is xin A I φ H (x) = det(xi 2n A H ) = det n I n xi n = det((x 2 1)I n xa) = x n φ G (x 1 x ) ) =

48 Algebraic properties Spectral properties of G K m 2 φ H (x) = x n φ G (x 1 x ) Proposition H = G K 2 and spg = {λ m 0 0 < λ m 1 1 <... < λ m d sph = {λ m 0 00 < λm 1 01 <... < λm d 0d < λm 0 10 < λm 1 11 <... < λm d 1d } where λ 0i = f 0 (λ i ) = λ i λ 2 i +4 2, λ 1i = f 1 (λ i ) = λ i + λ 2 i +4 2 d

49 Algebraic properties Spectral properties of G K m 2 φ H (x) = x n φ G (x 1 x ) Proposition H = G K 2 and spg = {λ m 0 0 < λ m 1 1 <... < λ m d sph = {λ m 0 00 < λm 1 01 <... < λm d 0d < λm 0 10 < λm 1 11 <... < λm d 1d } where λ 0i = f 0 (λ i ) = λ i λ 2 i +4 2, λ 1i = f 1 (λ i ) = λ i + λ 2 i +4 2 Proof. λ sph φ H (λ) = λ n φ G (λ 1 λ ) = 0 λ 1 λ spg λ i spg λ 2 λ i λ 1 = 0 d

50 Algebraic properties Spectral properties of G K m 2 φ H (x) = x n φ G (x 1 x )

51 Algebraic properties Spectral properties of G K m 2 H m = G K m 2 H m = H m 1 K 2, m 1. The adjacency matrix of H m is ( ) Am 1 I A m = m 1 I m 1 0 where I m denotes the identity matrix of size n2 m (the same as A m ) H 0 = G, A 0 = A the adjacency matrix of G Example

52 Algebraic properties Spectral properties of G K m 2 Let {p i, q i } i 0 be the family of polynomials satisfying the recurrence equations p i = p 2 i 1 q2 i 1 q i = p i 1 q i 1 with initial conditions p 0 = x and q 0 = 1 Proposition For every m 0, the characteristic polynomial of H m = G K2 m ( ) φ m (x) = q m (x) n pm (x) φ 0 q m (x) is Lemma If p and q are arbitrary polynomials, then ( pin qa qi ) det n = det((p 2 q 2 )I qi n pi n pqa) n

53 The Hierarchical Product of Graphs Algebraic properties Spectral properties of G K m 2 ( ) Proof of φ m (x) = q m (x) n φ pm(x) 0 q m(x). By induction on m, using the Lemma

54 The Hierarchical Product of Graphs Algebraic properties Spectral properties of G K m 2 ( ) Proof of φ m (x) = q m (x) n φ pm(x) 0 q m(x). By induction on m, using the Lemma Case m = 0. Trivially from q 0 (x) = 1 and p 0 (x) = x.

55 The Hierarchical Product of Graphs Algebraic properties Spectral properties of G K m 2 ( ) Proof of φ m (x) = q m (x) n φ pm(x) 0 q m(x). By induction on m, using the Lemma Case m = 0. Trivially from q 0 (x) = 1 and p 0 (x) = x. m 1. By induction on i, we prove that φ m = det(p i I m i q i A m i ) i = 0 : φ m = det(xi m A m ) = det(p 0 I m q 0 A m ) i 1 i : φ m = det(p i 1 I m i+1 q i 1 A m i+1 ) = = det((p 2 i 1 q2 i 1 )I m i p i 1 q i 1 A m i ) = = det(p i I m i q i A m i )

56 The Hierarchical Product of Graphs Algebraic properties Spectral properties of G K m 2 ( ) Proof of φ m (x) = q m (x) n φ pm(x) 0 q m(x). By induction on m, using the Lemma Case m = 0. Trivially from q 0 (x) = 1 and p 0 (x) = x. m 1. By induction on i, we prove that φ m = det(p i I m i q i A m i ) i = 0 : φ m = det(xi m A m ) = det(p 0 I m q 0 A m ) i 1 i : φ m = det(p i 1 I m i+1 q i 1 A m i+1 ) = = det((p 2 i 1 q2 i 1 )I m i p i 1 q i 1 A m i ) = = det(p i I m i q i A m i ) The case i = m gives ( φ m (x)( = det(p m (x)i)) 0 q m (x)a 0 ) = ( ) = det q m (x) pm(x) q I m(x) 0 A 0 = q m (x) n φ pm(x) 0 q m(x)

57 Algebraic properties The spectrum of the binary hypertree T m = K m 2 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 K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 5 Related works Hypertrees and generalized hypertrees Generalization of the hierarchical product 6 Conclusions

58 Algebraic properties The spectrum of the binary hypertree T m = K m 2 T m = K m 2 Corollary φ Tm (x) = p m (x) φ T m (x) = q m (x) p i = p 2 i 1 q2 i 1 q i = p i 1 q i 1 p 0 = x, q 0 = 1

59 Algebraic properties The spectrum of the binary hypertree T m = K m 2 T m = K m 2 Corollary φ Tm (x) = p m (x) φ T m (x) = q m (x) p i = p 2 i 1 q2 i 1 q i = p i 1 q i 1 p 0 = x, q 0 = 1 Proof. ( ) G = K 1 φ 0 (x) = x φ Tm (x) = q m (x) n φ pm(x) 0 q m(x) = p m (x) Tm = T m 0 = m 1 i=0 T i φ T m (x) = m 1 i=0 p i(x) = q m (x)

60 Algebraic properties The spectrum of the binary hypertree T m = K m 2 Proposition T m, m 1, has distinct eigenvalues λ m 0 < λm 1 < < λm n 1, with n = 2 m, satisfying the following recurrence relation: λ m λm 1 k + n +k = 2 λ m n k 1 = λm k for m > 1 and k = n 2, n 2 + 1,..., n 1 (λ m 1 k )

61 Algebraic properties The spectrum of the binary hypertree T m = K m 2 Proposition T m, m 1, has distinct eigenvalues λ m 0 < λm 1 < < λm n 1, with n = 2 m, satisfying the following recurrence relation: λ m λm 1 k + n +k = 2 λ m n k 1 = λm k for m > 1 and k = n 2, n 2 + 1,..., n 1 Proof. (λ m 1 k ) λ 0i = f 0 (λ i ) = λ i λ 2 i +4 2, λ 1i = f 1 (λ i ) = λ i + λ 2 i +4 2 T m bipartite its spectrum is symmetric with respect to 0 spt 0 = {0 1 } the multiplicity of every λ m 1 is 1

62 Algebraic properties The spectrum of the binary hypertree T m = K m 2 Properties of spt m λ i spg λ 2 λ i λ 1 = 0 f 0 (x) = x x f 1 (x) = x + x m = 0 spt 0 = {0} m = 1 λ 0 = f 0 (0) = 1, λ 1 = f 1 (0) = 1 m = 2 λ 0 = f 0 ( 1) = f 0 (f 0 (0)) = λ 1 = f 0 (1) = f 0 (f 1 (0)) = λ 2 = f 1 ( 1) = f 1 (f 0 (0)) = λ 3 = f 1 (1) = f 1 (f 1 (0)) = m fixed, i = i m 1... i 1 i 0 Z m 2 λ i = (f im 1 f i1 f i0 )(0)

63 Algebraic properties The spectrum of the binary hypertree T m = K m 2 The distinct eigenvalues of the hypertree T m for 0 m 6.

64 Algebraic properties The spectrum of the binary hypertree T m = K m 2 Proposition The asymptotic behaviors of the spectral radius ρ k = max 0 i n 1 { λ i } = λ , the second largest eigenvalue θ k = λ , and the minimum positive eigenvalue σ k = min 0 i n 1 { λ i } = λ of the hypertree T k are: ρ k 2k, θ k 2k, σ k 1/ 2k

65 Algebraic properties The spectrum of the binary hypertree T m = K m 2 Proof of ρ k 2k, θ k 2k, σ k 1/ 2k.

66 Algebraic properties The spectrum of the binary hypertree T m = K m 2 Proof of ρ k 2k, θ k 2k, σ k 1/ 2k. ρ k σ k = 1

67 Algebraic properties The spectrum of the binary hypertree T m = K m 2 Proof of ρ k 2k, θ k 2k, σ k 1/ 2k. ρ k σ k = 1 ρ k and θ k verify the recurrence λ k+1 = f 1 (λ k ) = 1 2 (λ k + λ 2 k + 4)

68 Algebraic properties The spectrum of the binary hypertree T m = K m 2 Proof of ρ k 2k, θ k 2k, σ k 1/ 2k. ρ k σ k = 1 ρ k and θ k verify the recurrence λ k+1 = f 1 (λ k ) = 1 2 (λ k + λ 2 k + 4) Assuming λ k αk β α(k + 1) β αkβ + α 2 k 2β α 2 (k + 1) β [(k + 1) β k β ] 1 2(k + 1) 1 2 [(k + 1) 1 2 k 1 2 ] = 2(k + 1) 1 2 (k + 1) k 1 2 1

69 Algebraic properties The spectrum of a generic two-term product G 2 G 1 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 K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 5 Related works Hypertrees and generalized hypertrees Generalization of the hierarchical product 6 Conclusions

70 Algebraic properties The spectrum of a generic two-term product G 2 G 1 Theorem Let G 1 and G 2 be two graphs on n i vertices, with adjacency matrix A i and characteristic polynomial φ i (x), i = 1, 2. Consider the graph G1 = G 1 0, with adjacency matrix A 1 and characteristic polynomial φ 1. Then the characteristic polynomial φ H (x) of the hierarchical product H = G 2 G 1 is: ( ) φ H (x) = φ 1(x) n φ1 2 (x) φ 2 φ 1 (x)

71 Algebraic properties The spectrum of a generic two-term product G 2 G 1 Proof of φ H (x) = φ 1(x) n 2 φ 2 ( φ1 (x) φ 1 (x) ).

72 Algebraic properties The spectrum of a generic two-term product G 2 G 1 Proof of φ H (x) = φ 1(x) n 2 φ 2 ( φ1 (x) φ 1 (x) ). The adjacency matrix of H is an n 1 n 1 block matrix, with blocks of size n 2 n 2 ( ) A2 B A H = D 1 A 2 + A 1 I 2 = B A 1 I ( 2 ) (δ) where B = I I 2

73 Algebraic properties The spectrum of a generic two-term product G 2 G 1 Proof of φ H (x) = φ 1(x) n 2 φ 2 ( φ1 (x) φ 1 (x) ). The adjacency matrix of H is an n 1 n 1 block matrix, with blocks of size n 2 n 2 ( ) A2 B A H = D 1 A 2 + A 1 I 2 = B A 1 I ( 2 ) (δ) where B = I I 2 The characteristic polynomial( of H is ) xi2 A φ H (x) = det(xi A H ) = det 2 B B (xi 1 A 1) I 2

74 Algebraic properties The spectrum of a generic two-term product G 2 G 1 Proof of φ H (x) = φ 1(x) n 2 φ 2 ( φ1 (x) φ 1 (x) ). The adjacency matrix of H is an n 1 n 1 block matrix, with blocks of size n 2 n 2 ( ) A2 B A H = D 1 A 2 + A 1 I 2 = B A 1 I ( 2 ) (δ) where B = I I 2 The characteristic polynomial( of H is ) xi2 A φ H (x) = det(xi A H ) = det 2 B B (xi 1 A 1) I 2 Computing the determinant in R n 2 n 2 : φ H (x) = det([xi 2 A 2 ]φ 1 (x)i 2 + ( φ 1 (x)i 2 [ xi 2 φ 1 (x)) ]) = = det(φ 1 (x)i 2 φ 1 (x)a 2) = det φ 1 (x) φ1 (x) φ (x)i 2 A 2 = ( ) 1 = φ 1 (x)n 2 φ φ1 (x) 2 φ 1 (x)

75 Algebraic properties The spectrum of a generic two-term product G 2 G 1 Corollary ( ) φ G 1 walk-regular φ H (x) = 1 (x) n2 ( ) n1 φ 1 (x) φ 2 φ 1 (x) Proof. φ 1 (x) = 1 n 1 φ 1 (x) n 1 Corollary Ggraph of order n 2 = N and characteristic polynomial φ G the characteristic polynomial of H = G K( n is ) (x + 1)(x n + 1) φ H (x) = (x + 1) N(n 2) (x n + 2) N φ G (x n + 2) Proof. K n is walk-regular, φ Kn = (x n + 1)(x + 1) n 1 and φ K n = (x + 1) n 1 + (n 1)(x n + 1)(x + 1) n 2

76 Related works 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 K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 5 Related works Hypertrees and generalized hypertrees Generalization of the hierarchical product 6 Conclusions

77 Related works Hypertrees and generalized hypertrees 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 K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 5 Related works Hypertrees and generalized hypertrees Generalization of the hierarchical product 6 Conclusions

78 Related works Hypertrees and generalized hypertrees T m = K m 2

79 Related works Hypertrees and generalized hypertrees Eigenvalues of the hypertree T m for 0 m 6.

80 Related works Hypertrees and generalized hypertrees The generalized hypertree T m r = P m r Example

81 Related works Hypertrees and generalized hypertrees Eigenvalues of T m 3 for 0 m

82 Related works Generalization of the hierarchical product 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 K m 2 The spectrum of the binary hypertree T m = K m 2 The spectrum of a generic two-term product G 2 G 1 5 Related works Hypertrees and generalized hypertrees Generalization of the hierarchical product 6 Conclusions

83 Related works Generalization of the hierarchical product Definition of the generalized hierarchical product G i = (V i, E i ), U i V i, i = 1, 2,..., N 1 H = G N G N 1 (U N 1 ) G 1 (U 1 ) is the graph: vertices V N V 2 V 1 if x j y j in G j and u i U i, i = 1, 2,..., j 1 then x N... x j+1 x j u j 1... u 1 x N... x j+1 y j u j 1... u 1 Example For every i, U i = V i G N G N 1 (U N 1 ) G 1 (U 1 ) = G N G N 1 G 1 For every i, U i = {0} G N G N 1 (U N 1 ) G 1 (U 1 ) = G N G N 1 G 1

84 Related works Generalization of the hierarchical product Example Two views of a generalized hierarchical product K 3 3 with U 1 = U 2 = {0, 1}.

85 Conclusions Summary 1 Definition of the hierarchical product of graphs 2 Spectral properties 3 The particular case of T m 4 Properties of Tr m, sptr m and m spt m r 5 Definition and properties of the generalized hierarchical product

86 Conclusions Publications The hierarchical product of graphs, L. Barrière, F. Comellas, C. Dalfó, M. A. Fiol, Discrete Applied Mathematics, to appear. On the spectra of hypertrees, BCDF, Linear Algebra and its Applications, 428(7): On the hierarchical product of graphs and the generalized binomial tree, BCDF, Linear and Multilinear Algebra, submitted (September 2007). The generalized hierarchical product of graphs, L. Barrière, C. Dalfó, M. A. Fiol, M. Mitjana, Journal of Graph Theory, submitted (March, 2008).

87 Conclusions Gracias!!!