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NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY

Published online by Cambridge University Press:  17 November 2010

KA HIN LEUNG
Affiliation:
Department of Mathematics, National Singapore University, Singapore 119260, Republic of Singapore (email: [email protected])
VINH NGUYEN
Affiliation:
Department of Mathematics, San Jose State University, San Jose, CA 95192-0103, USA (email: [email protected])
WASIN SO*
Affiliation:
Department of Mathematics, San Jose State University, San Jose, CA 95192-0103, USA (email: [email protected])
*
For correspondence; e-mail: [email protected]
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Abstract

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The expansion constant of a simple graph G of order n is defined as where denotes the set of edges in G between the vertex subset S and its complement . An expander family is a sequence {Gi} of d-regular graphs of increasing order such that h(Gi)>ϵ for some fixed ϵ>0. Existence of such families is known in the literature, but explicit construction is nontrivial. A folklore theorem states that there is no expander family of circulant graphs only. In this note, we provide an elementary proof of this fact by first estimating the second largest eigenvalue of a circulant graph, and then employing Cheeger’s inequalities where G is a d-regular graph and λ2(G) denotes the second largest eigenvalue of G. Moreover, the associated equality cases are discussed.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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