Abstract

Let L(?k) be the Laplacian matrix of an unweighted balanced binary tree ?k of k levels. In this note, we show that (a) ? = 1 is an eigenvalue of L(?k) with multiplicity ?l=0?(k-2)/4?2k-4l-2 if k is not a multiple of 4 or 1 + ?l=0?(k-2)/4?2k-4l-2 if k is a multiple of 4, where ?(k - 2)/4? is the greatest integer not exceeding (k - 2)/4. (b) ? = 3 is an eigenvalue of L(?k) if and only if k is even. (c) ? = 5 is an eigenvalue of L(?k) if and only if k is a multiple of 4. © 2003 Elsevier Science Inc. All rights reserved.