Glasgow Mathematical Journal

Research Article

COMB GRAPHS AND SPECTRAL DECIMATION

JONATHAN JORDANa1

a1 Department of Probability and Statistics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, UK e-mail: jonathan.jordan@shef.ac.uk

Abstract

We investigate the spectral properties of matrices associated with comb graphs. We show that the adjacency matrices and adjacency matrix Laplacians of the sequences of graphs show a spectral similarity relationship in the sense of work by L. Malozemov and A. Teplyaev (Self-similarity, operators and dynamics, Math. Phys. Anal. Geometry 6 (2003), 201–218), and hence these sequences of graphs show a spectral decimation property similar to that of the Laplacians of the Sierpiński gasket graph and other fractal graphs.

(Received December 03 2007)

(Accepted August 01 2008)

2000 AMS Subject Classification

  • Primary 47A10;
  • Secondary 05C99;
  • 28A80