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The distribution of eigenvalues of arithmetic inænite graphs

The distribution of eigenvalues of arithmetic inænite graphs Abstract. Let q be a principal congruence subgroups of PGL2Y Fq t, then it is known that qnX is an in®nite Ramanujan diagram, where X is the q 1-regular tree. We obtain the limit distributions of eigenvalues of the adjacency operator for qnX . The spectral density of X and the Sato-Tate measure appear as the limit distributions. 2000 Mathematics Subject Classi®cation: 11F72, 11F03; 05C25, 58J50. 1 Introduction A Ramanujan graph is de®ned as a k-regular ®nite graph whose nontrivial eigenp values of the adjacency operator have absolute values bounded by 2 k À 1. Such eigenvalue bounds force graphs to have high magni®cations and small diameters, hence these graphs give good communication networks. They also have important applications in computer science. However, it is not easy to determine whether a given graph is Ramanujan, especially when the size of a graph is large. So it is a dif®cult task to give explicit constructions of an in®nite family of Ramanujan graphs whose sizes increase. The ®rst construction of an in®nite family of Ramanujan graphs were given by Lubotzky, Phillips and Sarnak [LPS] and Margulis [Ma]. It is based on the arithmetic of quaternion algebras and the Ramanujan conjecture http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

The distribution of eigenvalues of arithmetic inænite graphs

Nagoshi, Hirofumi
Forum Mathematicum , Volume 14 (6) – Jul 29, 2002
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References (17)

Publisher
de Gruyter
Copyright
Copyright © 2002 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2002.035
Publisher site
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Abstract

Abstract. Let q be a principal congruence subgroups of PGL2Y Fq t, then it is known that qnX is an in®nite Ramanujan diagram, where X is the q 1-regular tree. We obtain the limit distributions of eigenvalues of the adjacency operator for qnX . The spectral density of X and the Sato-Tate measure appear as the limit distributions. 2000 Mathematics Subject Classi®cation: 11F72, 11F03; 05C25, 58J50. 1 Introduction A Ramanujan graph is de®ned as a k-regular ®nite graph whose nontrivial eigenp values of the adjacency operator have absolute values bounded by 2 k À 1. Such eigenvalue bounds force graphs to have high magni®cations and small diameters, hence these graphs give good communication networks. They also have important applications in computer science. However, it is not easy to determine whether a given graph is Ramanujan, especially when the size of a graph is large. So it is a dif®cult task to give explicit constructions of an in®nite family of Ramanujan graphs whose sizes increase. The ®rst construction of an in®nite family of Ramanujan graphs were given by Lubotzky, Phillips and Sarnak [LPS] and Margulis [Ma]. It is based on the arithmetic of quaternion algebras and the Ramanujan conjecture

Journal

Forum Mathematicum – de Gruyter

Published: Jul 29, 2002

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