THE SPECTRUM OF THE LAPLACIAN MATRIX OF A BALANCED 2p-ARY TREE
Let p > 1 be an integer. We consider an unweighted balanced tree Bp k 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 (Bp k) is σ (L (Bp k)) = Uk 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 Tj , as an eigenvalue of L (Bp 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 (Bp k) is exactly 2(2p-1)2(k-2)p. Finally, we prove that 3, 7 <IMG SRC="/fbpe/img/proy/v23n2/e.jpg" WIDTH=18 HEIGHT=16>σ (L (B²k)) if and only if k is a multiple of 3, that 5 <IMG SRC="/fbpe/img/proy/v23n2/e.jpg" WIDTH=18 HEIGHT=16>σ (L (B2k) if and only if k is an even number, and that no others integer eigenvalues exist for L (B²k).