Abstract
Reingold et al. introduced the notion zig-zag product on two different graphs, and presented a fully explicit construction of d-regular expanders with the second largest eigenvalue O(d − 1/3). In the same paper, they ask whether or not the similar technique can be used to construct expanders with the second largest eigenvalue O(d − 1/2). Such graphs are called Ramanujan graphs. Recently, zig-zag product has been generalized by Ben-Aroya and Ta-Shma. Using this technique, they present a family of expanders with the second largest eigenvalue d − 1/2 + o(1), what they call almost-Ramanujan graphs. However, their construction relies on local invertible functions and the dependence between the big graph and several small graphs, which makes the construction more complicated.
In this paper, we shall give a generalized theorem of zig-zag product. Specifically, the zig-zag product of one “big” graph and several “small” graphs with the same size will be formalized. By choosing the big graph and several small graphs individually, we shall present a family of fully explicitly almost-Ramanujan graphs with locally invertible function waived.
The research was supported by Shanghai Leading Academic Discipline Project with Number B412 and B114, Research and Development Project of High-Technology of China with Number 2007AA01Z189.
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References
Ajtai, M.: Recursive Construction for 3-Regular Expanders. Combinatorica 14(4), 379–416 (1994)
Alon, N.: Spectral Techniques in Graph Algorithms. In: Proceedings of the 3rd International Conference on Latin American Theoretical Informatics, pp. 206–215 (1998)
Alon, N., Schwartz, O., Shapira, A.: An Elementary Construction of Constant-Degree Expanders. Combinatorics, Probability and Computing 17, 319–328 (2008)
Ben-Aroya, A., Ta-Shma, A.: A Combinatorial Construction of Almost-Ramanujan Graphs using the Zig-Zag Product. In: Proceedings of the 40th Symposium on Theory of Computation, pp. 325–334 (2008)
Friedman, J.: A Proof of Alon’s Second Eigenvalue Conjecture. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing, pp. 720–724 (2003)
Hoory, S., Linial, N., Wigderson, A.: Expander Graphs and their Applications. Bulletin of the American Mathematical Society 43(4), 439–561 (2006)
Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan Graphs. Combinatorica 8(3), 261–277 (1988)
Margulis, G.A.: Explicit Construction of Expanders. Problemy Peredaci Informacii 9(4), 71–80 (1973)
Morgenstern, M.: Existence and Explicit Construction of q + 1 Regular Ramanujan Graphs for Every Prime Power q. Journal of Combinatorial Theory, Series B 62(1), 44–62 (1994)
Reingold, O., Vadhan, S., Wigderson, A.: Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders. Annuals of Mathematics 155(1), 157–187 (2002)
Reingold, O.: Undirected st-Connectivity in Log-Space. In: Proceedings of the 37th Symposium on Theory of Computation, pp. 376–385 (2005)
Schöning, U.: Construction of Expanders and Superconcentrators Using Kolmogorov Complexity. Random Structures and Algorithms 17(1), 64–77 (2000)
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Sun, H., Zhu, H. (2009). On Construction of Almost-Ramanujan Graphs. In: Du, DZ., Hu, X., Pardalos, P.M. (eds) Combinatorial Optimization and Applications. COCOA 2009. Lecture Notes in Computer Science, vol 5573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02026-1_18
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DOI: https://doi.org/10.1007/978-3-642-02026-1_18
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