A combinatorial construction of almost-Ramanujan graphs using the zig-zag product

Avraham Ben-Aroya, Amnon Ta-Shma

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Reingold, Vadhan and Wigderson [21] introduced the graph zig-zag product. This product combines a large graph and a small graph into one graph, such that the resulting graph inherits its size from the large graph, its degree from the small graph and its spectral gap from both. Using this product they gave the first fully-explicit combinatorial construction of expander graphs. They showed how to construct D-regular graphs having spectral gap 1 - 0(D-1/3). In the same paper, they posed the open problem of whether a similar graph product could be used to achieve the almost-optimal spectral gap 1 - 0(D -1/2). In this paper we propose a generalization of the zig-zag product that combines a large graph and several small graphs. The new product gives a better relation between the degree and the spectral gap of the resulting graph. We use the new product to give a fully-explicit combinatorial construction of D-regular graphs having spectral gap 1 - D-1/2+0(1).

Original languageEnglish
Title of host publicationSTOC'08
Subtitle of host publicationProceedings of the 2008 ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery (ACM)
Pages325-334
Number of pages10
ISBN (Print)9781605580470
DOIs
StatePublished - 2008
Event40th Annual ACM Symposium on Theory of Computing, STOC 2008 - Victoria, BC, Canada
Duration: 17 May 200820 May 2008

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference40th Annual ACM Symposium on Theory of Computing, STOC 2008
Country/TerritoryCanada
CityVictoria, BC
Period17/05/0820/05/08

Keywords

  • Expander graphs
  • Zig-zag product

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