TY - JOUR

T1 - A combinatorial construction of almost-ramanujan graphs using the zig-zag product

AU - Ben-Aroya, Avraham

AU - Ta-Shma, Amnon

PY - 2011

Y1 - 2011

N2 - Reingold, Vadhan, and Wigderson [Ann. of Math. (2), 155 (2002), pp. 157-187] introduced the graph zig-zag product. This product combines a large and a small graph into one, 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 a fully explicit combinatorial construction of D-regular graphs having spectral gap 1 - O(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 - O(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 +o(1).

AB - Reingold, Vadhan, and Wigderson [Ann. of Math. (2), 155 (2002), pp. 157-187] introduced the graph zig-zag product. This product combines a large and a small graph into one, 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 a fully explicit combinatorial construction of D-regular graphs having spectral gap 1 - O(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 - O(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 +o(1).

KW - Combinatorial construction

KW - Expander graphs

KW - Zig-zag product

UR - http://www.scopus.com/inward/record.url?scp=79957443340&partnerID=8YFLogxK

U2 - 10.1137/080732651

DO - 10.1137/080732651

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AN - SCOPUS:79957443340

SN - 0097-5397

VL - 40

SP - 267

EP - 290

JO - SIAM Journal on Computing

JF - SIAM Journal on Computing

IS - 2

ER -