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NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY

  • KA HIN LEUNG (a1), VINH NGUYEN (a2) and WASIN SO (a3)

Abstract

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.

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Copyright

Corresponding author

For correspondence; e-mail: so@math.sjsu.edu

References

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[1]Alon, N. and Roichman, Y., ‘Random Cayley graphs and expanders’, Random Structures Algorithms 5 (1994), 271284.
[2]Blum, M., Karp, R. M., Vornberger, O., Papadimitriou, C. H. and Yannakakis, M., ‘The complexity of testing whether a graph is a superconcentrator’, Inform. Process. Lett. 13 (1981), 164167.
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[5]Hoory, S., Linial, N. and Wigderson, A., ‘Expander graphs and their applications’, Bull. Amer. Math. Soc. 43 (2006), 439561.
[6]Margulis, G. A., ‘Explicit constructions of concentrators’, Probl. Inf. Transm. 9 (1973), 325332.
[7]Pinsker, M. S., ‘On the complexity of a concentrator’, in: Proceedings, 7th International Teletraffic Conference, Stockholm, 1973, pp. 318/1–318/4.
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NONEXISTENCE OF A CIRCULANT EXPANDER FAMILY

  • KA HIN LEUNG (a1), VINH NGUYEN (a2) and WASIN SO (a3)

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