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Explicit expanders of every degree and size
[article]
2020
arXiv
pre-print
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For every ϵ >0, d>d_0(ϵ) and n>n_0(d,ϵ) we present a strongly explicit construction of an (m,d,λ)-graph with λ < (2+ϵ) √(d) and m=n+o(n). ...
For any d=p+2 with p ≡ 1 4 prime and all sufficiently large n, we describe a strongly explicit construction of an (n,d, λ)-graph (on exactly n vertices) with λ≤√(2(d-1)) + √(d-2) +o(1) (< (1+√(2)) √(d- ...
Acknowledgment I thank László Babai, Oded Goldreich, Ryan O'Donnell, Ori Parzanchevski and Peter Sarnak for helpful discussions. ...
arXiv:2003.11673v1
fatcat:ep7kjk5ufnd73m4tyxtd2azjhm
New and Explicit Constructions of Unbalanced Ramanujan Bipartite Graphs
[article]
2020
arXiv
pre-print
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2019 pre-print arXiv:1910.03937v1 fatcat:ckzxsupjqfec7cvempo6jzadg4
Secondly, we revisit some of the known methods for constructing Ramanujan graphs and discuss the computational work required in actually implementing the various construction methods. ...
The objectives of this article are three-fold. Firstly, we present for the first time explicit constructions of an infinite family of unbalanced Ramanujan bigraphs. ...
Alex Lubotzky of Hebrew University, Jerusalem, Israel for useful hints on the construction of Ramanujan graphs of high degree, and Prof. ...
arXiv:1910.03937v2
fatcat:dfxkenurzjbl7czlky7vewxkt4
Ramanujan Graphs
[article]
2017
arXiv
pre-print
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This is an item on Ramanujan Graphs for a planned encyclopedia on Ramanujan. The notion of Ramanujan graphs is explained, as well as the reason to name these graphs after Ramanujan. ...
First examples of infinite families of such graphs were given by explicit construction in [LPS] and [M] for k = q + 1, q prime. ...
In [MSV] , it is shown, by a non constructive method, that for every k ≥ 3 there exist infinitely many k-regular bi-partite Ramanujan graphs. Why are Ramanujan graphs named after Ramanujan? ...
arXiv:1711.06558v1
fatcat:fo76ykueazba7d2g3umd7im2ji
New and explicit constructions of unbalanced Ramanujan bipartite graphs
2021
The Ramanujan journal
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Secondly, we revisit some of the known methods for constructing Ramanujan graphs and discuss the computational work required in actually implementing the various construction methods. ...
AbstractThe objectives of this article are threefold. Firstly, we present for the first time explicit constructions of an infinite family of unbalanced Ramanujan bigraphs. ...
The authors also thank the referee for constructive comments and for suggesting the references [10, 20] . ...
doi:10.1007/s11139-021-00384-0
fatcat:rhajaj6rv5cvtglgsmppazyiqu
From Ramanujan Graphs to Ramanujan Complexes
[article]
2019
arXiv
pre-print
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Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. ...
After explaining their connection to the Ramanujan conjecture we will present some old and new results with an emphasis on random walks on these discrete objects and on the Euclidean spheres. ...
It is known that for every 3 ≤ k ∈ N, there exist infinitely many k-regular Ramanujan graphs (explicit constructions for every k = p e + 1, p prime [32] , and non explicit for every k [30] ). ...
arXiv:1904.03533v1
fatcat:2xohc75slbckddgjawpa5jg3iu
From Ramanujan graphs to Ramanujan complexes
2019
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. ...
This article is part of a discussion meeting issue 'Srinivasa Ramanujan: in celebration of the centenary of his election as FRS'. ...
It is known that for every 3 ≤ k ∈ N, there exist infinitely many k-regular Ramanujan graphs (explicit constructions for every k = p e + 1, p prime [14] , and non explicit for every k [8] ). ...
doi:10.1098/rsta.2018.0445
pmid:31813373
pmcid:PMC6939227
fatcat:xzsc3nqhqvbx7dxf7rl7uvadmi
Finite Fields and Ramanujan Graphs
1997
Journal of combinatorial theory. Series B (Print)
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We give a new method to construct Ramanujan graphs and apply it to define two new families of Ramanujan graphs. ...
These new graphs have a very easy description and we prove that they are Ramanujan graphs in an elementary way. 1997 Academic Press article no. TB961740 259 ...
The well-known relations between the size of +(G) and the expansion properties of a graph are a cause for the great interest in the explicit constructions of families of graphs [G n, k ] , where k is ...
doi:10.1006/jctb.1996.1740
fatcat:azlubvmgozedxj2u5tnepd4qxy
Ramanujan Graphs with Small Girth
2003
Combinatorica
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We construct an infinite family of (q +1)−regular Ramanujan graphs X n of girth 1. ...
We also give covering maps X n+1 → X n such that the minimal common covering of all the X n 's is the universal covering tree. ...
The regularity restriction is not really necessary, our method will work also for variants of the LPS construction which yield (q + 1)−regular Ramanujan graphs for every prime power q. ...
doi:10.1007/s00493-003-0029-9
fatcat:hsceplsr75fpzkvussemgwkbkm
Ramanujan Graphs with Small Girth
[article]
2003
arXiv
pre-print
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We construct an infinite family of (q+1)-regular Ramanujan graphs X_n of girth 1. ...
We also give covering maps X_n+1 --> X_n such that the minimal common covering of all the graphs is the universal covering tree. ...
The regularity restriction is not really necessary, our method will work also for variants of the LPS construction which yield (q + 1)−regular Ramanujan graphs for every prime power q. ...
arXiv:math/0306196v1
fatcat:tbk576xxefhzbbqvvof3cc6nxe
Ramanujan Graphs for Post-Quantum Cryptography
[chapter]
2020
Mathematics for Industry
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We propose a generalized version of explicit constructions of Ramanujan graphs, which are seen as an optimal structure of expander graphs in a spectral sense, from the previous works of Lubotzky, Phillips ...
We also describe the relationship between the security of Cayley hash functions and word problems for group theory. ...
The authors would like to thank Meghan Delaney for pointing out grammatical errors. ...
doi:10.1007/978-981-15-5191-8_17
fatcat:bnonrygdnvaynmkgsbrvkhxmka
Ramanujan Complexes and High Dimensional Expanders
[article]
2013
arXiv
pre-print
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Expander graphs in general, and Ramanujan graphs in particular, have been of great interest in the last three decades with many applications in computer science, combinatorics and even pure mathematics ...
In these notes we describe various efforts made in recent years to generalize these notions from graphs to higher dimensional simplicial complexes. ...
The author is indebted to Konstantin Golubev, Gil Kalai, Tali Kaufman, Roy Meshulam and Uli Wagner for many discussions regarding the material of these notes. ...
arXiv:1301.1028v2
fatcat:w3bfzrdewnfxplesvtewfkt52e
Ramanujan complexes and high dimensional expanders
2014
Japanese journal of mathematics
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Expander graphs in general, and Ramanujan graphs in particular, have been of great interest in the last three decades with many applications in computer science, combinatorics and even pure mathematics ...
In these notes we describe various efforts made in recent years to generalize these notions from graphs to higher dimensional simplicial complexes. * This paper is based on notes prepared for the Takagi ...
The author is indebted to Konstantin Golubev, Gil Kalai, Tali Kaufman, Roy Meshulam and Uli Wagner for many discussions regarding the material of these notes. ...
doi:10.1007/s11537-014-1265-z
fatcat:xj7bwp3o4ncidmbjqycpfbc26y
Character sums and abelian Ramanujan graphs
1992
Journal of Number Theory
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Hence for each prime p 2 5 and each k satisfying p+l dkdpfl +2p"*, there exists a k-regular Ramanujan graph with p'k vertices. ...
In Section 4 we also construct, using elliptic curves, k-regular Ramanujan graphs with p*k vertices for any prime p 2 5 and k satisfying p + 1 6 k 6 p + 1 + 2~"~. ...
doi:10.1016/0022-314x(92)90120-e
fatcat:567wn4sgprcmpesmrlpkus74wq
Ramanujan graphs in cryptography
[article]
2018
arXiv
pre-print
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In this paper we study the security of a proposal for Post-Quantum Cryptography from both a number theoretic and cryptographic perspective. ...
Charles-Goren-Lauter in 2006 [CGL06] proposed two hash functions based on the hardness of finding paths in Ramanujan graphs. ...
The Corollary describes the set of primes p for which G(p, 5) is a six-regular Ramanujan graph. ...
arXiv:1806.05709v2
fatcat:rt6uqp5ap5dk3c6wutbi2sjhqu
Upper Bounds on the Order of Cages
1996
Electronic Journal of Combinatorics
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In this paper we establish general upper bounds on $ {v(k,g)}$ which are roughly the 3/2 power of the lower bounds, and we provide explicit constructions for such $ {(k,g)}$-graphs. ...
Let $ { k\ge 2}$ and $ {g\ge3}$ be integers. A $ {(k,g)}$-graph is a $ {k}$-regular graph with girth (length of a smallest cycle) exactly $ {g}$. ...
In [10] , Lazebnik and Ustimenko constructed the family of graphs D(n, q), n ≥ 2, q a prime power, for which γ ≥ log q (q − 1). ...
doi:10.37236/1328
fatcat:opqfijgmifcuvixtkzydwcyiiu
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