A COMBINATORIAL CONSTRUCTION OF ALMOST-RAMANUJAN GRAPHS USING THE ZIG-ZAG PRODUCT

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).