• Corpus ID: 11435821

Towards Constructing Ramanujan Graphs Using Shift Lifts

@article{Chandrasekaran2015TowardsCR,
title={Towards Constructing Ramanujan Graphs Using Shift Lifts},
author={Karthekeyan Chandrasekaran and Ameya Velingker},
journal={arXiv: Combinatorics},
year={2015}
}
• Published 25 February 2015
• Mathematics
• arXiv: Combinatorics
In a breakthrough work, Marcus-Spielman-Srivastava recently showed that every $d$-regular bipartite Ramanujan graph has a 2-lift that is also $d$-regular bipartite Ramanujan. As a consequence, a straightforward iterative brute-force search algorithm leads to the construction of a $d$-regular bipartite Ramanujan graph on $N$ vertices in time $2^{O(dN)}$. Shift $k$-lifts studied by Agarwal-Kolla-Madan lead to a natural approach for constructing Ramanujan graphs more efficiently. The number of…
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References

SHOWING 1-10 OF 25 REFERENCES
Small Lifts of Expander Graphs are Expanding
• Mathematics
ArXiv
• 2013
It is shown that, for random shift $k$-lifts, if all the nontrivial eigenvalues of the base graph G are at most $\lambda$ in absolute value, then with high probability depending only on the number n of nodes of G (and not on k), the absolute value of every nontrivials eigenvalue of the lift is at most $O(\lambda)$.
Lifts, Discrepancy and Nearly Optimal Spectral Gap*
• Mathematics
Comb.
• 2006
It is shown that every graph of maximal degree d has a 2-lift such that all “new” eigenvalues are in the range, leading to a deterministic polynomial time algorithm for constructing arbitrarily large d-regular graphs, with second eigenvalue O(d/α)+1.
Cryptographic Hash Functions from Expander Graphs
• Mathematics, Computer Science
Journal of Cryptology
• 2007
This work investigates two specific families of optimal expander graphs for provable collision resistant hash function constructions: the families of Ramanujan graphs constructed by Lubotzky-Phillips-Sarnak and Pizer respectively.
Ramanujan graphs and Hecke operators
We associate to the Hecke operator Tp , p a prime, acting on a space of theta series an explicit p + 1 regular Ramanujan graph G having large girth. Such graphs have high "magnification" and thus
Ramanujan Graphs
In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication
Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees
• Mathematics
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.
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.
Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes
A new, self-contained construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al. (STOC ‘03) and improving upon it when the error parameter is small.
Signatures, lifts, and eigenvalues of graphs
• Mathematics
Discrete and Continuous Models in the Theory of Networks
• 2020
It is proved that the existence of an infinite tower of $3$-cyclic lifts, each of which is again Ramanujan, is possible.
The PCP theorem by gap amplification
• Irit Dinur
• Mathematics, Computer Science
STOC '06
• 2006
A new combinatorial amplification transformation that doubles the unsat-value of a constraint-system, with only a linear blowup in the size of the system, is described.