Hostname: page-component-cb9f654ff-kl2l2 Total loading time: 0 Render date: 2025-08-21T05:23:11.968Z Has data issue: false hasContentIssue false
  • English
  • Français

COMB GRAPHS AND SPECTRAL DECIMATION

Published online by Cambridge University Press:  01 January 2009

JONATHAN JORDAN
JONATHAN JORDAN*
Affiliation:
Department of Probability and Statistics, University of Sheffield, Hounsfield Road, Sheffield S3 7RH, UK e-mail: jonathan.jordan@shef.ac.uk
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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. Geometry6 (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.

Keywords

Primary 47A10Secondary 05C9928A80

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 51 , Issue 1 , January 2009 , pp. 71 - 81
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Accardi, L., Ben Ghorbal, A. and Obata, N., Monotone independence, comb graphs and Bose-Einstein condensation, Infin. Dimen. Anal. Quantum Probab. Relat. Top 7 (2004), 419–435.CrossRefGoogle Scholar
2.Bajorin, N., Chen, T., Dagan, A., Emmons, C., Hussein, M., Khalil, M., Mody, P., Steinhurst, B. and Teplyaev, A., Vibration nodes of 3n-gaskets and other fractals, J. Phys. A 41 (015101), 2008.CrossRefGoogle Scholar
3.Bartholdi, L. and Grigorchuk, R. L., On the spectrum of Kecke type operators related to some fractal groups, Proc. Steklov Inst. Math. 231 (2000), 1–41.Google Scholar
4.Chung, F., Lu, L. and Vu, V., Spectra of random graphs with given expected degrees, Proc. Nat. Acad. Sci. 100 (2003), 6313–6318.CrossRefGoogle ScholarPubMed
5.Chung, F. R. K., Spectral graph theory (AMS, Providence, Rhode Island, 1997). Number 92 in CBMS Regional Conference Series.Google Scholar
6.Fukushima, M. and Shima, T., On a spectral analysis for the Sierpiński gasket, Potential Anal. 1 (1992), 1–35.CrossRefGoogle Scholar
7.Kigami, J., Analysis on fractals (Cambridge University Press, Cambridge, 2001).CrossRefGoogle Scholar
8.Malozemov, L. and Teplyaev, A., Self-similarity, operators and dynamics, Math. Phys. Anal. Geometry 6 (2003), 201–218.CrossRefGoogle Scholar
9.Rammal, R. and Toulouse, G., Random walks on fractal structures and percolation clusters, J. Physique Letters 44 (1983), L13–L22.CrossRefGoogle Scholar
10.Schenker, J. H. and Aizenman, M., The creation of spectral gaps by graph decoration, Lett. Math. Phys. 53 (2000), 253–262.CrossRefGoogle Scholar
11.Shima, T., On eigenvalue problems for the random walks on the Sierpiński pre-gaskets, Japan. J. Indust. Appl. Math. 8 (1991), 127–141.CrossRefGoogle Scholar
12.Strichartz, R. S., Differential equations on fractals: A tutorial (Princeton University Press, Princeton, New Jersey, 2006).CrossRefGoogle Scholar