Resumen

ROJO, OSCAR. THE SPECTRUM OF THE LAPLACIAN MATRIX OF A BALANCED 2p-ARY TREE. Proyecciones (Antofagasta) [online]. 2004, vol.23, n.2, pp. 131-149. ISSN 0716-0917.  doi: 10.4067/S0716-09172004000200006.

Let p > 1 be an integer. We consider an unweighted balanced tree B^p_k of k levels with a root vertex of degree 2^p, vertices from the level 2 until the level (k - 1) of degree 2^p +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 (B^p_k) is σ (L (B^p_k)) = U^k_j =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 , scanear fórmula We derive that the multiplicity of each eigenvalue of T_j , as an eigenvalue of L (B^p_k) , is at least 2^{(2p-1)2(k-j-1)p} . Moreover, we show that the multiplicity of the eigenvalue λ = 1 of L (B^p_k) is exactly 2^{(2p-1)2(k-2)p}. Finally, we prove that 3, 7 σ (L (B²_k)) if and only if k is a multiple of 3, that 5 σ (L (B²_k) if and only if k is an even number, and that no others integer eigenvalues exist for L (B²_k).

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