Explicit near-Ramanujan graphs of every degree

@article{Mohanty2020ExplicitNG,
title={Explicit near-Ramanujan graphs of every degree},
author={Sidhanth Mohanty and R. O'Donnell and Pedro Paredes},
journal={Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing},
year={2020}
}
• Published 2020
• Computer Science, Mathematics
• Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
For every constant d ≥ 3 and є > 0, we give a deterministic poly(n)-time algorithm that outputs a d-regular graph on Θ(n) vertices that is є-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by 2√d−1 + є (excluding the single trivial eigenvalue of d).

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References

SHOWING 1-10 OF 78 REFERENCES
Ramanujan graphs
• Mathematics, Computer Science
• Comb.
• 1988
The girth ofX is asymptotically ≧4/3 logk−1 ¦X¦ which gives larger girth than was previously known by explicit or non-explicit constructions. Expand
Lifts, Discrepancy and Nearly Optimal Spectral Gaps
• Mathematics
• 2003
Let G be a graph on n vertices. A 2-lift of G is a graph H on 2n vertices, with a covering map � : H → G. It is not hard to see that all eigenvalues of G are also eigenvalues of H. In addition, H hasExpand
Cubic Ramanujan graphs
• P. Chiu
• Mathematics, Computer Science
• Comb.
• 1992
A fimily of cubic Ramanujan graph is explicitly constructed. They are realized as Cayley graphs of a certain free group acting on the 3-regular tree; this group is obtained from a definite quaternionExpand
The Moore Bound for Irregular Graphs
• Mathematics, Computer Science
• Graphs Comb.
• 2002
The Moore bound is extended here to cover irregular graphs as well, yielding an affirmative answer to an old open problem. Expand
Interlacing Families IV: Bipartite Ramanujan Graphs of All Sizes
• Computer Science, Mathematics
• 2015 IEEE 56th Annual Symposium on Foundations of Computer Science
• 2015
It is proved that there exist bipartite Ramanujan graphs of every degree and every number of vertices using the framework of finite free convolutions introduced recently by the authors. Expand
Ramanujan Graphs in Polynomial Time
• Michael B. Cohen
• Mathematics, Computer Science
• 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)
• 2016
A polynomial time algorithm is provided to compute certain expected characteristic polynomials related to this construction of bipartite Ramanujan (multi) graphs of all degrees and all sizes. Expand
Expander graphs and gaps between primes
• Mathematics
• 2008
Abstract The explicit construction of infinite families of d-regular graphs which are Ramanujan is known only in the case d – 1 is a prime power. In this paper, we consider the case when d – 1 is notExpand
Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees
• Mathematics, Computer Science
• 2013 IEEE 54th Annual Symposium on Foundations of Computer Science
• 2013
The existence of infinite families of (c, d)-biregular bipartite graphs with all non-trivial eigenvalues bounded by √c-1+√d-1, for all c, d ≥ q 3 is proved. Expand
Explicit Expanders of Every Degree and Size
• N. Alon
• Computer Science, Mathematics
• ArXiv
• 2020
It is shown that there is a deterministic poly(n) time algorithm that outputs an ( n, d , λ)-graph (on exactly n vertices) with λ ≤ 2 d − 1 + ϵ . Expand
Existence and Explicit Constructions of q + 1 Regular Ramanujan Graphs for Every Prime Power q
For any prime power q, explicit constructions for many infinite linear families of q + 1 regular Ramanujan graphs are given as Cayley graphs of PGL2 or PSL2 over finite fields, with respect to very simple generators. Expand