versión impresa ISSN 0716-0917
Proyecciones (Antofagasta) v.23 n.2 Antofagasta ago. 2004
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
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.
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