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 -