Combinatorica
Expanders obtained from affine transformations
Combinatorica - Theory of Computing
On the second eigenvalue of a graph
Discrete Mathematics
Universal traversal sequences for expander graphs
Information Processing Letters
Existence and explicit constructions of q+1 regular Ramanujan graphs for every prime power q
Journal of Combinatorial Theory Series B
Eigenvalues and expansion of regular graphs
Journal of the ACM (JACM)
Randomness conductors and constant-degree lossless expanders
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Semi-Direct Product in Groups and Zig-Zag Product in Graphs: Connections and Applications
FOCS '01 Proceedings of the 42nd IEEE symposium on Foundations of Computer Science
Combinatorica
Undirected ST-connectivity in log-space
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Lifts, Discrepancy and Nearly Optimal Spectral Gap
Combinatorica
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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 $1-O(D^{-\frac{1}{3}})$. 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 $1-O(D^{-\frac{1}{2}})$. 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 $1-D^{-\frac{1}{2}+o(1)}$.