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A Spectral Study of the Manhattan Networks
Electronic Notes in Discrete Mathematics 29 (2007) 267–271 www.elsevier.com/locate/endm
A Spectral Study of the Manhattan Networks F. Comellas, C. Dalf´o, M. A. Fiol 1,2 Departament de Matem` atica Aplicada IV Universitat Polit`ecnica de Catalunya Barcelona, Spain
M. Mitjana 1,3 Departament de Matem` atica Aplicada I Universitat Polit`ecnica de Catalunya Barcelona, Spain
Abstract The multidimensional Manhattan networks are a family of digraphs with many appealing properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we fully determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dimensional) case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity. Keywords: Manhattan Network, Spectra, Eigenvalues, Line digraph.
1
Research supported by the Secretaria de Estado de Universidades e Investigaci´ on (Ministerio de Educaci´on y Ciencia), Spain, and the European Regional Development Fund (ERDF) under projects MTM2005-08990-C02-01 and TEC2005-03575. 2 Email: {comellas,dalfo,fiol}@ma4.upc.edu 3 Email: [email protected] 1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.07.045
268
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F. Comellas et al. / Electronic Notes in Discrete Mathematics 29 (2007) 267–271
The Multidimensional Manhattan Network
The study of a class of directed torus networks known as Manhattan (Street) Networks has received significant attention since they were introduced independently and in different contexts by Morillo et al. [6] and Maxemchuk [5] as an unidirectional regular mesh structure resembling locally the topology of the avenues and streets of Manhattan (or l’Eixample in downtown Barcelona). In this work we address the question of computing the spectrum of the ndimensional Manhattan networks, showing, among other things, that they contain the spectra of the hypercubes. The standard Manhattan (Street) Network is a 2-regular digraph. Every vertex is represented by a pair of integers u = (u1 , u2 ), with 0 ≤ ui ≤ Ni , for some even integers Ni , i = 1, 2 and vertex u has two outgoing arcs: one horizontal (u1 ± 1, u2 ); and the other vertical (u1 , u2 ± 1) (where the sign depends on the parity of the other component and the arithmetic must be understood modulo Ni ). The natural extension of Manhattan networks to higher dimensions was introduced in [3] and its standard definition goes as follows: Definition 1.1 Given n even positive integers N1 , N2 , . . . , Nn , the n-dim Manhattan network Mn = M (N1 , N2 , . . . , Nn ) is a digraph with vertex set V (Mn ) = ZN1 × ZN2 × . . . × ZNn . Thus, each of its vertices is represented by an n-vector u = (u1 , u2 , . . . , un ), with 0 ≤ ui ≤ Ni − 1, i = 1, 2, . . . , n. The arc set A(Mn ) is defined by the following adjacencies:
(u1 , . . . , ui , . . . , un ) → (u1 , . . . , ui + (−1)
j=i
uj
, . . . , un )
(1 ≤ i ≤ n).
Therefore, Mn is a n-regular digraph on N = ni=1 Ni vertices. In particular, when Ni = 2, 1 ≤ i ≤ n, the n-dimensional Manhattan network is isomorphic to the symmetric digraph Q∗n , with Qn being the hypercube of dimension n or n-cube. In [3], the authors provided some structural results concerning the symmetries of the Manhattan networks which lead to the following useful alternative definition. Definition 1.2 The vertex set of Mn = M (N1 , N2 , . . . , Nn ) is, as before, ZN1 × · · · × ZNn and the arcs are now: (u1 , . . . , ui , . . . , un ) → (−u1 , . . . , −ui−1 , ui + 1, −ui+1 , . . . , −un )(1 ≤ i ≤ n). Recall that, given a digraph G = (V, A) with n vertices and m arcs, its line digraph LG = (VL , AL ) has vertices representing the arcs of G, and its
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F. Comellas et al. / Electronic Notes in Discrete Mathematics 29 (2007) 267–271
(0,3)
(1,3)
(2,3)
(3,3)
(4,3)
(5,3)
(0,3)
(5,1)
(4,3)
(3,1)
(2,3)
(1,1)
(0,2)
(1,2)
(2,2)
(3,2)
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(5,3)
(4,1)
(3,3)
(2,1)
(1,3)
(0,0)
(1,0)
(2,0)
(3,0)
(4,0)
(5,0)
(0,0)
(1,0)
(2,0)
(3,0)
(4,0)
(5,0)
Fig. 1. The vertices in a Manhattan network M (6, 4) with their standard labels and with the new labels.(Directed dashed lines stand for the identification of parallel sides of the rectangle representing the torus surface).
adjacencies are naturally induced by the arc adjacencies in G. The spectrum of the line digraph LG has the same non-zero eigenvalues as G. In fact the characterisc polynomials satisfy xn pLG = xm pG , (see [1]). In our case, for any N1 , N2 , the 2-dimensional Manhattan network M2 is a line digraph, whereas for n ≥ 3 it is not. In fact, a direct consequence of this and the results in [4], is that the spectrum of M2 = (N1 , N2 ) has the eigenvalue 0 with (geometric) multiplicity at least N1 N2 /2.
2
The spectrum
We first recall that the eigenvalues of the directed CN are the N roots of the 2π unity, λk = ei N k , 0 ≤ k ≤ N − 1. Moreover, the spectrum of the n-cube n Qn is sp Qn = {(n − 2k)(k ) : k = 0, 1, . . . , n}, where the superscripts denote multiplicities; see for instance [2]. The following proposition shows that the spectrum of a n-dimensional Manhattan network contains the spectrum of the n-cube Qn . Proposition 2.1 The spectrum of the n-dimensional Manhattan network Mn = M (N1 , . . . , Nn ) contains all the eigenvalues (including multiplicities) of the n-cube Qn : sp Qn ⊆ sp Mn , with equality when Ni = 2, 1 ≤ i ≤ n. Furthermore, for every subset Ik ⊂ {1, 2, . . . , n} of cardinality k, 0 ≤ k ≤ n, the vector v with components vu = i∈Ik (−1)ui is an eigenvector with the eigenvalue n − 2k. The spectrum of the 2-dimensional Manhattan network M2 can be computed in terms of the eigenvalues λj of the directed cycles CNj . Proposition 2.2 The spectrum of the 2-dimensional Manhattan network
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F. Comellas et al. / Electronic Notes in Discrete Mathematics 29 (2007) 267–271
M2 = M (N1 , N2 ) is (the superscript denotes multiplicity) N1 N2 sp M2 = 0 2 , ± 2 cos 4kN11π + 2 cos 4kN22π 0 ≤ ki <
Ni ,i 2
= 1, 2 .
Proof (Sketch) Let A be the adjacency matrix of the 2-dimensional Manhattan network M2 = M (N1 , N2 ). Let λ1 and λ2 be eigenvalues of the cycles CN1 and CN2 , respectively. If we consider the entries of an eigenvector as charges in each vertex, the following equalities hold: ui+1 = λ1 ui , 0 ≤ i ≤ N1 − 1, vj+1 = λ2 vj , 0 ≤ j ≤ N2 − 1. We define a λ-eigenvector w in M2 , whose components are of the form: w(i,j) = α ui vj + β u−i v−j + γ ui v−j + δ u−i vj , or some constants α, β, γ, δ to be determined. Taking into account that vertex (i, j) in M2 is adjacent to vertices (i + 1, −j) and (−i, j + 1), and (Aw)(i,j) =
w(i ,j ) = λ w(i,j) ,
(i ,j )←(i,j)
it turns out that
⎛
⎞⎛ ⎞ ⎛ ⎞ 0 0 λ1 λ2 α α ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 λ12 λ11 ⎟ ⎜β ⎟ ⎜β ⎟ ⎜ ⎟⎜ ⎟ = λ⎜ ⎟. ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜λ1 λ1 0 0 ⎟ ⎜ γ ⎟ ⎜γ ⎟ 2 ⎝ ⎠⎝ ⎠ ⎝ ⎠ 1 λ2 λ1 0 0 δ δ
Then λ is an eigenvalue of the above of its character matrix and the zeros −2 2 + λ + λ . 2 istic polynomial are λ = 0 and λ = ± λ21 + λ−2 1 2 2 ˜ n , is obtained from the hypercube Qn by adding The folded (n + 1)-cube, Q ˜ n is a (n+1)-regular an edge between each pair of antipodal vertices. Hence, Q n graph on 2 vertices, with diameter D = (n + 1)/2 and distinct eigenvalues ˜ n = {n + 1 > n − 3 > n − 7 > · · · }. ev Q The graph that we call the conjugate n-cube, Qn , has the same vertex set as the hipercube, Zn2 , and two vertices are adjacent whenever the corresponding strings differ in all components but one. Thus, Qn is an n-regular graph which, coincides either with the hypercube Qn , or the two folded hypercube, ˜ n−1 depending on the dimension. ˜ n−1 ∪ Q Q The following result reduces the computation of the eigenvalues of the ndimensional Manhattan network Mn = M (N1 , N2 , . . . , Nn ) to the study of the spectra of the following n × n matrix W , corresponding to the adjacency matrix of a weighted conjugate n-cube, defined in the following way: Let
F. Comellas et al. / Electronic Notes in Discrete Mathematics 29 (2007) 267–271
271
i0 . . . in−1 and j0 . . . jn−1 be the binary expressions of the vertices i, j (row and column of W ). Then, the entries of W = (wij ) are, ⎧ ⎪ ⎪ if im = jm for all m = k and ik = jk = 0, λ , ⎪ ⎨ k wij = λ−1 k = λk , if im = jm for all m = k and ik = jk = 1, ⎪ ⎪ ⎪ ⎩0, otherwise, where i, j ∈ {0, 1, . . . , n − 1} and λk is an eigenvector of the cycle CNk . Theorem 2.3 For any given n integers N1 , N2 , . . . , Nn , the different eigenvalues of the n-dimensional Manhattan network Mn = M (N1 , N2 , . . . , Nn ) ∗ coincide with the distinct eigenvalues of the weighted conjugate n-cube Qn : ∗
ev Mn = ev Qn .
References [1] Balbuena, C., D. Ferrero, X. Marcote and I. Pelayo, Algebraic properties of a digraph and its line digraph, J. Intercon. Networks 4 (2003), 377–393. [2] N. Biggs, “Algebraic Graph Theory,” 2nd Ed., Cambridge University Press, Cambridge 1993. [3] Comellas, F., C. Dalf´o and M.A. Fiol, The multidimensional Manhattan networks, Siam J. Discrete Math., submitted. [4] Fiol, M.A., and M. Mitjana, The spectra of some families of digraphs, Linear Algebra Appl. 423(1) (2007), 109–118. [5] Maxemchuk, N.F., Routing in the Manhattan Street Network, IEEE Trans. Commun. 35 (1987), no. 5, 503–512. [6] Morillo, P., M.A. Fiol and J. F` abrega, The diameter of directed graphs associated to plane tessellations, Ars Comb. 20A (1985), 17–27.
A Spectral Study of the Manhattan Networks F. Comellas, C. Dalf´o, M. A. Fiol 1,2 Departament de Matem` atica Aplicada IV Universitat Polit`ecnica de Catalunya Barcelona, Spain
M. Mitjana 1,3 Departament de Matem` atica Aplicada I Universitat Polit`ecnica de Catalunya Barcelona, Spain
Abstract The multidimensional Manhattan networks are a family of digraphs with many appealing properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we fully determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dimensional) case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity. Keywords: Manhattan Network, Spectra, Eigenvalues, Line digraph.
1
Research supported by the Secretaria de Estado de Universidades e Investigaci´ on (Ministerio de Educaci´on y Ciencia), Spain, and the European Regional Development Fund (ERDF) under projects MTM2005-08990-C02-01 and TEC2005-03575. 2 Email: {comellas,dalfo,fiol}@ma4.upc.edu 3 Email: [email protected] 1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.07.045
268
1
F. Comellas et al. / Electronic Notes in Discrete Mathematics 29 (2007) 267–271
The Multidimensional Manhattan Network
The study of a class of directed torus networks known as Manhattan (Street) Networks has received significant attention since they were introduced independently and in different contexts by Morillo et al. [6] and Maxemchuk [5] as an unidirectional regular mesh structure resembling locally the topology of the avenues and streets of Manhattan (or l’Eixample in downtown Barcelona). In this work we address the question of computing the spectrum of the ndimensional Manhattan networks, showing, among other things, that they contain the spectra of the hypercubes. The standard Manhattan (Street) Network is a 2-regular digraph. Every vertex is represented by a pair of integers u = (u1 , u2 ), with 0 ≤ ui ≤ Ni , for some even integers Ni , i = 1, 2 and vertex u has two outgoing arcs: one horizontal (u1 ± 1, u2 ); and the other vertical (u1 , u2 ± 1) (where the sign depends on the parity of the other component and the arithmetic must be understood modulo Ni ). The natural extension of Manhattan networks to higher dimensions was introduced in [3] and its standard definition goes as follows: Definition 1.1 Given n even positive integers N1 , N2 , . . . , Nn , the n-dim Manhattan network Mn = M (N1 , N2 , . . . , Nn ) is a digraph with vertex set V (Mn ) = ZN1 × ZN2 × . . . × ZNn . Thus, each of its vertices is represented by an n-vector u = (u1 , u2 , . . . , un ), with 0 ≤ ui ≤ Ni − 1, i = 1, 2, . . . , n. The arc set A(Mn ) is defined by the following adjacencies:
(u1 , . . . , ui , . . . , un ) → (u1 , . . . , ui + (−1)
j=i
uj
, . . . , un )
(1 ≤ i ≤ n).
Therefore, Mn is a n-regular digraph on N = ni=1 Ni vertices. In particular, when Ni = 2, 1 ≤ i ≤ n, the n-dimensional Manhattan network is isomorphic to the symmetric digraph Q∗n , with Qn being the hypercube of dimension n or n-cube. In [3], the authors provided some structural results concerning the symmetries of the Manhattan networks which lead to the following useful alternative definition. Definition 1.2 The vertex set of Mn = M (N1 , N2 , . . . , Nn ) is, as before, ZN1 × · · · × ZNn and the arcs are now: (u1 , . . . , ui , . . . , un ) → (−u1 , . . . , −ui−1 , ui + 1, −ui+1 , . . . , −un )(1 ≤ i ≤ n). Recall that, given a digraph G = (V, A) with n vertices and m arcs, its line digraph LG = (VL , AL ) has vertices representing the arcs of G, and its
269
F. Comellas et al. / Electronic Notes in Discrete Mathematics 29 (2007) 267–271
(0,3)
(1,3)
(2,3)
(3,3)
(4,3)
(5,3)
(0,3)
(5,1)
(4,3)
(3,1)
(2,3)
(1,1)
(0,2)
(1,2)
(2,2)
(3,2)
(4,2)
(5,2)
(0,2)
(1,2)
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(5,2)
(0,1)
(1,1)
(2,1)
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(4,1)
(3,3)
(2,1)
(1,3)
(0,0)
(1,0)
(2,0)
(3,0)
(4,0)
(5,0)
(0,0)
(1,0)
(2,0)
(3,0)
(4,0)
(5,0)
Fig. 1. The vertices in a Manhattan network M (6, 4) with their standard labels and with the new labels.(Directed dashed lines stand for the identification of parallel sides of the rectangle representing the torus surface).
adjacencies are naturally induced by the arc adjacencies in G. The spectrum of the line digraph LG has the same non-zero eigenvalues as G. In fact the characterisc polynomials satisfy xn pLG = xm pG , (see [1]). In our case, for any N1 , N2 , the 2-dimensional Manhattan network M2 is a line digraph, whereas for n ≥ 3 it is not. In fact, a direct consequence of this and the results in [4], is that the spectrum of M2 = (N1 , N2 ) has the eigenvalue 0 with (geometric) multiplicity at least N1 N2 /2.
2
The spectrum
We first recall that the eigenvalues of the directed CN are the N roots of the 2π unity, λk = ei N k , 0 ≤ k ≤ N − 1. Moreover, the spectrum of the n-cube n Qn is sp Qn = {(n − 2k)(k ) : k = 0, 1, . . . , n}, where the superscripts denote multiplicities; see for instance [2]. The following proposition shows that the spectrum of a n-dimensional Manhattan network contains the spectrum of the n-cube Qn . Proposition 2.1 The spectrum of the n-dimensional Manhattan network Mn = M (N1 , . . . , Nn ) contains all the eigenvalues (including multiplicities) of the n-cube Qn : sp Qn ⊆ sp Mn , with equality when Ni = 2, 1 ≤ i ≤ n. Furthermore, for every subset Ik ⊂ {1, 2, . . . , n} of cardinality k, 0 ≤ k ≤ n, the vector v with components vu = i∈Ik (−1)ui is an eigenvector with the eigenvalue n − 2k. The spectrum of the 2-dimensional Manhattan network M2 can be computed in terms of the eigenvalues λj of the directed cycles CNj . Proposition 2.2 The spectrum of the 2-dimensional Manhattan network
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F. Comellas et al. / Electronic Notes in Discrete Mathematics 29 (2007) 267–271
M2 = M (N1 , N2 ) is (the superscript denotes multiplicity) N1 N2 sp M2 = 0 2 , ± 2 cos 4kN11π + 2 cos 4kN22π 0 ≤ ki <
Ni ,i 2
= 1, 2 .
Proof (Sketch) Let A be the adjacency matrix of the 2-dimensional Manhattan network M2 = M (N1 , N2 ). Let λ1 and λ2 be eigenvalues of the cycles CN1 and CN2 , respectively. If we consider the entries of an eigenvector as charges in each vertex, the following equalities hold: ui+1 = λ1 ui , 0 ≤ i ≤ N1 − 1, vj+1 = λ2 vj , 0 ≤ j ≤ N2 − 1. We define a λ-eigenvector w in M2 , whose components are of the form: w(i,j) = α ui vj + β u−i v−j + γ ui v−j + δ u−i vj , or some constants α, β, γ, δ to be determined. Taking into account that vertex (i, j) in M2 is adjacent to vertices (i + 1, −j) and (−i, j + 1), and (Aw)(i,j) =
w(i ,j ) = λ w(i,j) ,
(i ,j )←(i,j)
it turns out that
⎛
⎞⎛ ⎞ ⎛ ⎞ 0 0 λ1 λ2 α α ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 0 λ12 λ11 ⎟ ⎜β ⎟ ⎜β ⎟ ⎜ ⎟⎜ ⎟ = λ⎜ ⎟. ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜λ1 λ1 0 0 ⎟ ⎜ γ ⎟ ⎜γ ⎟ 2 ⎝ ⎠⎝ ⎠ ⎝ ⎠ 1 λ2 λ1 0 0 δ δ
Then λ is an eigenvalue of the above of its character matrix and the zeros −2 2 + λ + λ . 2 istic polynomial are λ = 0 and λ = ± λ21 + λ−2 1 2 2 ˜ n , is obtained from the hypercube Qn by adding The folded (n + 1)-cube, Q ˜ n is a (n+1)-regular an edge between each pair of antipodal vertices. Hence, Q n graph on 2 vertices, with diameter D = (n + 1)/2 and distinct eigenvalues ˜ n = {n + 1 > n − 3 > n − 7 > · · · }. ev Q The graph that we call the conjugate n-cube, Qn , has the same vertex set as the hipercube, Zn2 , and two vertices are adjacent whenever the corresponding strings differ in all components but one. Thus, Qn is an n-regular graph which, coincides either with the hypercube Qn , or the two folded hypercube, ˜ n−1 depending on the dimension. ˜ n−1 ∪ Q Q The following result reduces the computation of the eigenvalues of the ndimensional Manhattan network Mn = M (N1 , N2 , . . . , Nn ) to the study of the spectra of the following n × n matrix W , corresponding to the adjacency matrix of a weighted conjugate n-cube, defined in the following way: Let
F. Comellas et al. / Electronic Notes in Discrete Mathematics 29 (2007) 267–271
271
i0 . . . in−1 and j0 . . . jn−1 be the binary expressions of the vertices i, j (row and column of W ). Then, the entries of W = (wij ) are, ⎧ ⎪ ⎪ if im = jm for all m = k and ik = jk = 0, λ , ⎪ ⎨ k wij = λ−1 k = λk , if im = jm for all m = k and ik = jk = 1, ⎪ ⎪ ⎪ ⎩0, otherwise, where i, j ∈ {0, 1, . . . , n − 1} and λk is an eigenvector of the cycle CNk . Theorem 2.3 For any given n integers N1 , N2 , . . . , Nn , the different eigenvalues of the n-dimensional Manhattan network Mn = M (N1 , N2 , . . . , Nn ) ∗ coincide with the distinct eigenvalues of the weighted conjugate n-cube Qn : ∗
ev Mn = ev Qn .
References [1] Balbuena, C., D. Ferrero, X. Marcote and I. Pelayo, Algebraic properties of a digraph and its line digraph, J. Intercon. Networks 4 (2003), 377–393. [2] N. Biggs, “Algebraic Graph Theory,” 2nd Ed., Cambridge University Press, Cambridge 1993. [3] Comellas, F., C. Dalf´o and M.A. Fiol, The multidimensional Manhattan networks, Siam J. Discrete Math., submitted. [4] Fiol, M.A., and M. Mitjana, The spectra of some families of digraphs, Linear Algebra Appl. 423(1) (2007), 109–118. [5] Maxemchuk, N.F., Routing in the Manhattan Street Network, IEEE Trans. Commun. 35 (1987), no. 5, 503–512. [6] Morillo, P., M.A. Fiol and J. F` abrega, The diameter of directed graphs associated to plane tessellations, Ars Comb. 20A (1985), 17–27.