The spectrum of the Laplacian matrix of a balanced binary tree Original Research Article

Let image be the Laplacian matrix of an unweighted balanced binary tree image of k levels. We prove that spectrum of image isimagewhere, for 1less-than-or-equals, slantjless-than-or-equals, slantk−1, Tj is the j×j principal submatrix of the tridiagonal k×k matrix Sk,imageWe derive that the multiplicity of each eigenvalue of Tj,1less-than-or-equals, slantjless-than-or-equals, slantk−1, as an eigenvalue of image is at least 2k−j−1. Finally, for each Tj, using some results in [Electron. J. Linear Algebra 6 (2000) 62], we obtain lower and upper bounds for its smallest eigenvalue and an upper bound for its largest eigenvalue. In particular, we give upper bounds for the second largest eigenvalue and for the largest eigenvalue of image.