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Explicit expanders of every degree and size [article]

Noga Alon
2020 arXiv   pre-print
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]

Shantanu Prasad Burnwal, Kaneenika Sinha, Mathukumalli Vidyasagar
2020 arXiv   pre-print
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]

Alexander Lubotzky
2017 arXiv   pre-print
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

Shantanu Prasad Burnwal, Kaneenika Sinha, Mathukumalli Vidyasagar
2021 The Ramanujan journal  
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]

Alexander Lubotzky, Ori Parzanchevski
2019 arXiv   pre-print
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

Alexander Lubotzky, Ori Parzanchevski
2019 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
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

J Arias de Reyna
1997 Journal of combinatorial theory. Series B (Print)  
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

Yair Glasner
2003 Combinatorica  
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]

Yair Glasner
2003 arXiv   pre-print
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]

Hyungrok Jo, Shingo Sugiyama, Yoshinori Yamasaki
2020 Mathematics for Industry  
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]

Alexander Lubotzky
2013 arXiv   pre-print
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

Alexander Lubotzky
2014 Japanese journal of mathematics  
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

Wen-Ch'ing Winnie Li, Keqin Feng
1992 Journal of Number Theory  
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]

Anamaria Costache, Brooke Feigon, Kristin Lauter, Maike Massierer, Anna Puskás
2018 arXiv   pre-print
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

F. Lazebnik, V. A. Ustimenko, A. J. Woldar
1996 Electronic Journal of Combinatorics  
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 (q1).  ... 
doi:10.37236/1328 fatcat:opqfijgmifcuvixtkzydwcyiiu
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