Document

Spectrum Preserving Short Cycle Removal on Regular Graphs

File

LIPIcs.STACS.2021.55.pdf
• Filesize: 0.78 MB
• 19 pages

Acknowledgements

I am very grateful to Ryan O'Donnell for numerous comments and suggestions, as well as very thorough feedback on an earlier draft of this paper.

Cite As

Pedro Paredes. Spectrum Preserving Short Cycle Removal on Regular Graphs. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 55:1-55:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.55

Abstract

We describe a new method to remove short cycles on regular graphs while maintaining spectral bounds (the nontrivial eigenvalues of the adjacency matrix), as long as the graphs have certain combinatorial properties. These combinatorial properties are related to the number and distance between short cycles and are known to happen with high probability in uniformly random regular graphs. Using this method we can show two results involving high girth spectral expander graphs. First, we show that given d ⩾ 3 and n, there exists an explicit distribution of d-regular Θ(n)-vertex graphs where with high probability its samples have girth Ω(log_{d-1} n) and are ε-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by 2√{d-1} + ε (excluding the single trivial eigenvalue of d). Then, for every constant d ⩾ 3 and ε > 0, we give a deterministic poly(n)-time algorithm that outputs a d-regular graph on Θ(n)-vertices that is ε-near-Ramanujan and has girth Ω(√{log n}), based on the work of [Mohanty et al., 2020].

Subject Classification

ACM Subject Classification
• Mathematics of computing → Spectra of graphs
• Theory of computation → Expander graphs and randomness extractors
Keywords
• Ramanujan Graphs
• High Girth Regular Graphs

Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

References

1. Noga Alon. Eigenvalues and expanders. Combinatorica, 6(2):83-96, 1986.
2. Noga Alon. Explicit expanders of every degree and size. arXiv preprint, 2020. URL: http://arxiv.org/abs/2003.11673.
3. 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.
4. Noga Alon and Shachar Lovett. Almost k-wise vs. k-wise independent permutations, and uniformity for general group actions. Theory of Computing, 9:559-577, 2013.
5. Edward A. Bender and E. Rodney Canfield. The asymptotic number of labeled graphs with given degree sequences. J. Combinatorial Theory Ser. A, 24(3):296-307, 1978. URL: https://doi.org/10.1016/0097-3165(78)90059-6.
6. Yonatan Bilu and Nathan Linial. Lifts, discrepancy and nearly optimal spectral gap. Combinatorica, 26(5):495-519, 2006. URL: https://doi.org/10.1007/s00493-006-0029-7.
7. Béla Bollobás. A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. European J. Combin., 1(4):311-316, 1980. URL: https://doi.org/10.1016/S0195-6698(80)80030-8.
8. Charles Bordenave. A new proof of Friedman’s second eigenvalue theorem and its extension to random lifts. Technical Report 1502.04482v4, arXiv, 2019. To appear in Annales scientifiques de l'École normale supérieure. URL: http://arxiv.org/abs/1502.04482v4.
9. Xavier Dahan. Regular graphs of large girth and arbitrary degree. Combinatorica, 34(4):407-426, 2014. URL: https://doi.org/10.1007/s00493-014-2897-6.
10. Paul Erdős and Horst Sachs. Reguläre graphen gegebener tailenweite mit minimaler knollenzahl. Wiss. Z. Univ. Halle-Willenberg Math. Nat., 12:251-258, 1963.
11. Joel Friedman. Some geometric aspects of graphs and their eigenfunctions. Duke Mathematical Journal, 69(3):487-525, 1993.
12. Joel Friedman. A proof of Alon’s second eigenvalue conjecture and related problems. Memoirs of the American Mathematical Society, 195(910):viii+100, 2008.
13. R. G. Gallager. Low-density parity-check codes. IRE Trans., IT-8:21-28, 1962. URL: https://doi.org/10.1109/tit.1962.1057683.
14. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. American Mathematical Society Bulletin, 43(4):439-561, 2006.
15. Nabil Kahale. Eigenvalues and expansion of regular graphs. Journal of the ACM (JACM), 42(5):1091-1106, 1995.
16. Eyal Kaplan, Moni Naor, and Omer Reingold. Derandomized constructions of k-wise (almost) independent permutations. Algorithmica. An International Journal in Computer Science, 55(1):113-133, 2009.
17. Martin Kassabov. Symmetric groups and expander graphs. Inventiones Mathematicae, 170(2):327-354, 2007.
18. John Lafferty and Dan Rockmore. Codes and iterative decoding on algebraic expander graphs. In the Proceedings of ISITA. Citeseer, 2000.
19. Felix Lazebnik and Vasiliy A. Ustimenko. Explicit construction of graphs with an arbitrary large girth and of large size. Discrete Appl. Math., 60(1-3):275-284, 1995. ARIDAM VI and VII (New Brunswick, NJ, 1991/1992). URL: https://doi.org/10.1016/0166-218X(94)00058-L.
20. Nati Linial and Michael Simkin. A randomized construction of high girth regular graphs. arXiv preprint, 2019. URL: http://arxiv.org/abs/1911.09640.
21. A. Lubotzky, R. Phillips, and P. Sarnak. Ramanujan graphs. Combinatorica, 8(3):261-277, 1988. URL: https://doi.org/10.1007/BF02126799.
22. Mohammad M Mansour and Naresh R Shanbhag. Construction of ldpc codes from ramanujan graphs. In 36th Annu. Conf. on Information Sciences and Systems, 2002.
23. G. A. Margulis. Explicit constructions of graphs without short cycles and low density codes. Combinatorica, 2(1):71-78, 1982. URL: https://doi.org/10.1007/BF02579283.
24. G. A. Margulis. Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Problemy Peredachi Informatsii, 24(1):51-60, 1988.
25. Brendan D. McKay, Nicholas C. Wormald, and Beata Wysocka. Short cycles in random regular graphs. Electron. J. Combin., 11(1):Research Paper 66, 12, 2004. URL: http://www.combinatorics.org/Volume_11/Abstracts/v11i1r66.html.
26. 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.
27. Moshe Morgenstern. Existence and explicit constructions of q+1 regular Ramanujan graphs for every prime power q. J. Combin. Theory Ser. B, 62(1):44-62, 1994. URL: https://doi.org/10.1006/jctb.1994.1054.
28. A. Nilli. On the second eigenvalue of a graph. Discrete Mathematics, 91(2):207-210, 1991.
29. Mark S Pinsker. On the complexity of a concentrator. In 7th International Telegraffic Conference, volume 4, pages 1-318. Citeseer, 1973.
30. Joachim Rosenthal and Pascal O Vontobel. Constructions of ldpc codes using ramanujan graphs and ideas from margulis. In in Proc. of the 38-th Allerton Conference on Communication, Control, and Computing. Citeseer, 2000.
X

Feedback for Dagstuhl Publishing