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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) v.23 n.2 Antofagasta ago. 2004

http://dx.doi.org/10.4067/S0716-09172004000200006 

 

Proyecciones
Vol. 23, No 2, pp. 131-149, August 2004.
Universidad Católica del Norte
Antofagasta - Chile

THE SPECTRUM OF THE LAPLACIAN MATRIX OF A BALANCED 2p-ARY TREE

 

OSCAR ROJO *

Universidad Católica del Norte, Chile

Received : February 2004. Accepted : June 2004

correspondencia a:


Abstract

Let p > 1 be an integer. We consider an unweighted balanced tree Bpk of k levels with a root vertex of degree 2p, vertices from the level 2 until the level (k - 1) of degree 2p +1 and vertices in the level k of degree 1. The case p = 1 it was studied in [8, 9, 10]. We prove that the spectrum of the Laplacian matrix L (Bpk) is σ (L (Bpk)) = Ukj =1σ (T(p) j where, for 1< j < k < 1, T(p)j is the j ×j principal submatrix of the tridiagonal k×k singular matrix T(p)k ,

We derive that the multiplicity of each eigenvalue of Tj , as an eigenvalue of L (Bpk) , is at least 2(2p-1)2(k-j-1)p . Moreover, we show that the multiplicity of the eigenvalue λ = 1 of L (Bpk) is exactly 2(2p-1)2(k-2)p. Finally, we prove that 3, 7 σ (L (B2k)) if and only if k is a multiple of 3, that 5 σ (L (B2k) if and only if k is an even number, and that no others integer eigenvalues exist for L (B2k).

AMS classification: 5C50, 15A48.

Keywords: Tree; balanced tree; binary tree; n-ary tree; Laplacian matrix.


*Work supported by Fondecyt 1040218, Chile. This research was conducted while the author was visitor at Centro de Modelamiento Matemático, Universidad de Chile, Chile.

References

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[6] F. B. Hildebrand, Finite-Difference Equations and Simulations, Prentice-Hall,Inc., Englewood Cliffs, N.J., (1968).        [ Links ]

[7] R. Merris, Laplacian Matrices of Graphs: A Survey, Linear Algebra Appl. a Appl. 197, 198: pp. 143-176, (1994).        [ Links ]

[8] J. J. Molitierno, M. Neumann and B. L. Shader, Tight bounds on the algebraic connectivity of a balanced binary tree, Electronic Journal of Linear Algebra, Vol. 6, pp. 62-71, March (2000).        [ Links ]

[9] O. Rojo, The spectrum of the Laplacian matrix of a balanced binary tree, Linear Algebra Appl. 349, pp. 203-219, (2002).        [ Links ]

[10] O. Rojo and M. Peña, A note on the integer eigenvalues of the Laplacian matrix of a balanced binary tree, Linear Algebra Appl. 362, pp. 293-300 (2003).        [ Links ]

Oscar Rojo
Departamento de Matematicas
Universidad Catolica del Norte
Casilla 1280
Antofagasta
Chile
e-mail : orojo@ucn.cl

 

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