High-Girth Near-Ramanujan Graphs with Lossy Vertex Expansion
Kahale proved that linear sized sets in d-regular Ramanujan graphs have vertex expansion at least d/2 and complemented this with construction of near-Ramanujan graphs with vertex expansion no better than d/2. However, the construction of Kahale encounters highly local obstructions to better vertex expansion. In particular, the poorly expanding sets are associated with short cycles in the graph. Thus, it is natural to ask whether the vertex expansion of high-girth Ramanujan graphs breaks past the d/2 bound. Our results are two-fold:
1) For every d = p+1 for prime p ≥ 3 and infinitely many n, we exhibit an n-vertex d-regular graph with girth Ω(log_{d-1} n) and vertex expansion of sublinear sized sets bounded by (d+1)/2 whose nontrivial eigenvalues are bounded in magnitude by 2√{d-1}+O(1/(log_{d-1} n)).
2) In any Ramanujan graph with girth Clog_{d-1} n, all sets of size bounded by n^{0.99C/4} have near-lossless vertex expansion (1-o_d(1))d. The tools in analyzing our construction include the nonbacktracking operator of an infinite graph, the Ihara-Bass formula, a trace moment method inspired by Bordenave’s proof of Friedman’s theorem [Bordenave, 2019], and a method of Kahale [Kahale, 1995] to study dispersion of eigenvalues of perturbed graphs.
expander graphs
Ramanujan graphs
vertex expansion
girth
Mathematics of computing~Spectra of graphs
Theory of computation~Expander graphs and randomness extractors
96:1-96:15
Track A: Algorithms, Complexity and Games
We would like to thank Shirshendu Ganguly and Nikhil Srivastava for their highly valuable insights, intuition, and comments. We would also like to thank Amitay Kamber for helpful comments on the initial version of the preprint.
Theo
McKenzie
Theo McKenzie
Department of Mathematics, University of California, Berkeley, CA, USA
https://math.berkeley.edu/~mckenzie/
This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1752814.
Sidhanth
Mohanty
Sidhanth Mohanty
Department of Computer Science, University of California, Berkeley, CA, USA
http://sidhanthm.com/
Supported by NSF grant CCF-1718695.
10.4230/LIPIcs.ICALP.2021.96
Noga Alon. Eigenvalues and expanders. Combinatorica, 6:83-96, 1986.
Noga Alon. Explicit expanders of every degree and size. Combinatorica, pages 1-17, 2021.
Noga Alon, Shirshendu Ganguly, and Nikhil Srivastava. High-girth near-Ramanujan graphs with localized eigenvectors. arXiv preprint, 2019. URL: http://arxiv.org/abs/1908.03694.
http://arxiv.org/abs/1908.03694
Noga Alon, Shlomo Hoory, and Nathan Linial. The Moore bound for irregular graphs. Graphs and Combinatorics, 18(1):53-57, 2002.
Nalini Anantharaman. Quantum ergodicity on regular graphs. Communications in Mathematical Physics, 353(2):633-690, 2017.
Nalini Anantharaman and Etienne Le Masson. Quantum ergodicity on large regular graphs. Duke Math. J., 164(4):723-765, 2015.
Omer Angel, Joel Friedman, and Shlomo Hoory. The non-backtracking spectrum of the universal cover of a graph. Transactions of the American Mathematical Society, 367(6):4287-4318, 2015.
Charles Bordenave. A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts. In Annales scientifiques de l'Ecole normale supérieure, 2019.
Shimon Brooks, Etienne Le Masson, and Elon Lindenstrauss. Quantum ergodicity and averaging operators on the sphere. International Mathematics Research Notices, 19:6034-6064, 2016.
Michael Capalbo, Omer Reingold, Salil Vadhan, and Avi Wigderson. Randomness conductors and constant-degree lossless expanders. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing, pages 659-668, 2002.
Joel Friedman. A proof of Alon’s second eigenvalue conjecture. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 720-724, 2003.
Shirshendu Ganguly and Nikhil Srivastava. On non-localization of eigenvectors of high girth graphs. International Mathematics Research Notices, 2018.
Dima Grigoriev. Linear lower bound on degrees of positivstellensatz calculus proofs for the parity. Theoretical Computer Science, 259(1-2):613-622, 2001.
Venkatesan Guruswami, James Lee, and Alexander Razborov. Almost euclidean subspaces of 𝓁ⁿ₁ via expander codes. In Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms, pages 353-362, 2008.
Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc., 43(4):439–561, 2018.
Nabil Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM (JACM), 42(5):1091-1106, 1995.
Nati Linial and Michael Simkin. A randomized construction of high girth regular graphs. Random Structures & Algorithms, 58(2):345-369, 2021.
Alexander Lubotzky, Ralph Phillips, and Peter Sarnak. Ramanujan graphs. Combinatorica, 8:261-277, 1988.
Michael Luby, Michael Mitzenmacher, M. Amin Shokrollahi, and Daniel Spielman. Improved low-density parity-check codes using irregular graphs. IEEE Trans. Inform. Theory, 42(2):585-598, 2001.
Grigorii Aleksandrovich Margulis. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problemy peredachi informatsii, 24(1):51-60, 1988.
Sidhanth Mohanty, Ryan O'Donnell, and Pedro Paredes. Explicit near-ramanujan graphs of every degree. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, pages 510-523, 2020.
Moshe Morgenstern. Existence and explicit constructions of q+ 1 regular ramanujan graphs for every prime power q. Journal of Combinatorial Theory, Series B, 62(1):44-62, 1994.
Pedro Paredes. Spectrum preserving short cycle removal on regular graphs. In 38th International Symposium on Theoretical Aspects of Computer Science, STACS 2021, March 16-19, 2021, Saarbrücken, Germany (Virtual Conference), volume 187 of LIPIcs, pages 55:1-55:19, 2021.
Gregory Quenell. Notes on an example of McLaughlin, 1996.
Grant Schoenebeck. Linear level lasserre lower bounds for certain k-csps. In 2008 49th Annual IEEE Symposium on Foundations of Computer Science, pages 593-602. IEEE, 2008.
Michael Sipser and Daniel Spielman. Expander codes. IEEE Trans. Inform. Theory, 42(6, part 1):1710-1722, 1996.
Daniel Spielman. Linear-time encodable and decodable error-correcting codes. IEEE Trans. Inform. Theory, 42(6, part 1):1723-1731, 1996.
Theo McKenzie and Sidhanth Mohanty
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