Haifa University - Dept. of CS Theory Seminar

Speaker : Amnon Ta-Shma, Tel-Aviv University.

Title: "A combinatorial construction of almost-Ramanujan graphs using the zig-zag product ".

Date: Thursday, July 3.

Place: Haifa U. Education Building, room 566.

Time: 12:15 - 13:30


Reingold, Vadhan and Wigderson 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-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 $1-D^{-1/2 + o(1)}$.

Joint work with Avraham Ben-Aroya