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NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY

Part of: Graph theory

Published online by Cambridge University Press:  17 November 2010

KA HIN LEUNG ,
VINH NGUYEN  and
WASIN SO
KA HIN LEUNG
Affiliation:
Department of Mathematics, National Singapore University, Singapore 119260, Republic of Singapore (email: matlkh@nus.edu.sg)
VINH NGUYEN
Affiliation:
Department of Mathematics, San Jose State University, San Jose, CA 95192-0103, USA (email: hacbao@gmail.com)
WASIN SO*
Affiliation:
Department of Mathematics, San Jose State University, San Jose, CA 95192-0103, USA (email: so@math.sjsu.edu)
*
✉For correspondence; e-mail: so@math.sjsu.edu
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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.

Keywords

circulant graphCheeger’s inequalityexpander family

MSC classification

Secondary: 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 1 , February 2011 , pp. 87 - 95
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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