The spectra of the adjacency matrix and Laplacian matrix for some balanced trees

Let T be an unweighted rooted tree of k levels such that in each level the vertices have equal degree. Let dk−j+1 denotes the degree of the vertices in the level j. We find the eigenvalues of the adjacency matrix and of the Laplacian matrix of T. They are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for both matrices are dj-1,2⩽j⩽k-1, and dk, while the diagonal entries are zeros, in the case of the adjacency matrix, and dj, 1 ⩽ j ⩽k, in the case of the Laplacian matrix. Moreover, we give some results concerning to the multiplicity of the above mentioned eigenvalues.