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## Filters

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Explicit expanders of every degree and size
[article]

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. ...

##
###
New and Explicit Constructions of Unbalanced Ramanujan Bipartite Graphs
[article]

2020
*
arXiv
*
pre-print

Secondly, we revisit some

arXiv:1910.03937v2
fatcat:dfxkenurzjbl7czlky7vewxkt4
*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. ...##
###
Ramanujan Graphs
[article]

2017
*
arXiv
*
pre-print

This is an item on

arXiv:1711.06558v1
fatcat:fo76ykueazba7d2g3umd7im2ji
*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*? ...##
###
New and explicit constructions of unbalanced Ramanujan bipartite graphs

2021
*
The Ramanujan journal
*

Secondly, we revisit some

doi:10.1007/s11139-021-00384-0
fatcat:rhajaj6rv5cvtglgsmppazyiqu
*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] . ...##
###
From Ramanujan Graphs to Ramanujan Complexes
[article]

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] ). ...

##
###
From Ramanujan graphs to Ramanujan complexes

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] ). ...

##
###
Finite Fields and Ramanujan Graphs

1997
*
Journal of combinatorial theory. Series B (Print)
*

We give a new method to

doi:10.1006/jctb.1996.1740
fatcat:azlubvmgozedxj2u5tnepd4qxy
*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 ...##
###
Ramanujan Graphs with Small Girth

2003
*
Combinatorica
*

We

doi:10.1007/s00493-003-0029-9
fatcat:hsceplsr75fpzkvussemgwkbkm
*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*. ...##
###
Ramanujan Graphs with Small Girth
[article]

2003
*
arXiv
*
pre-print

We

arXiv:math/0306196v1
fatcat:tbk576xxefhzbbqvvof3cc6nxe
*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*. ...##
###
Ramanujan Graphs for Post-Quantum Cryptography
[chapter]

2020
*
Mathematics for Industry
*

We propose a generalized version

doi:10.1007/978-981-15-5191-8_17
fatcat:bnonrygdnvaynmkgsbrvkhxmka
*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. ...##
###
Ramanujan Complexes and High Dimensional Expanders
[article]

2013
*
arXiv
*
pre-print

Expander

arXiv:1301.1028v2
fatcat:w3bfzrdewnfxplesvtewfkt52e
*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. ...##
###
Ramanujan complexes and high dimensional expanders

2014
*
Japanese journal of mathematics
*

Expander

doi:10.1007/s11537-014-1265-z
fatcat:xj7bwp3o4ncidmbjqycpfbc26y
*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. ...##
###
Character sums and abelian Ramanujan graphs

1992
*
Journal of Number Theory
*

Hence

doi:10.1016/0022-314x(92)90120-e
fatcat:567wn4sgprcmpesmrlpkus74wq
*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~"~. ...##
###
Ramanujan graphs in cryptography
[article]

2018
*
arXiv
*
pre-print

In this paper we study the security

arXiv:1806.05709v2
fatcat:rt6uqp5ap5dk3c6wutbi2sjhqu
*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*. ...##
###
Upper Bounds on the Order of Cages

1996
*
Electronic Journal of Combinatorics
*

In this paper we establish general upper bounds on $ {v(k,g)}$ which are roughly the 3/2

doi:10.37236/1328
fatcat:opqfijgmifcuvixtkzydwcyiiu
*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*). ...
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