Filters








1,713 Hits in 3.7 sec

From Ramanujan Graphs to Ramanujan Complexes [article]

Alexander Lubotzky, Ori Parzanchevski
2019 arXiv   pre-print
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.  ...  Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science.  ...  This illustrates the connection between the notion of Ramanujan graph and the Ramanujan conjecture.  ... 
arXiv:1904.03533v1 fatcat:2xohc75slbckddgjawpa5jg3iu

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.  ...  This illustrates the connection between the notion of Ramanujan graph and the Ramanujan conjecture.  ...  So, Ramanujan graphs are expanders. Expander graphs are of great importance in combinatorics and computer science (cf. [HLW] and the references therein) and also in pure mathematics (cf. [L2] ).  ... 
arXiv:1711.06558v1 fatcat:fo76ykueazba7d2g3umd7im2ji

From Ramanujan graphs to Ramanujan complexes

Alexander Lubotzky, Ori Parzanchevski
2019 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
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.  ...  Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science.  ...  This illustrates the connection between the notion of Ramanujan graph and the Ramanujan conjecture.  ... 
doi:10.1098/rsta.2018.0445 pmid:31813373 pmcid:PMC6939227 fatcat:xzsc3nqhqvbx7dxf7rl7uvadmi

Algebraic Connectivity Ratio of Ramanujan Graphs

Reza Olfati-Saber
2007 American Control Conference (ACC)  
Explicit construction algorithms exist for Ramanujan graphs that create regular graphs with especial degree and scale that depend on a pair of prime numbers.  ...  In this paper, we explore spectral properties of a class of regular Cayley graphs known as Ramanujan graphs and prove that the ratio of their algebraic connectivity to that of regular lattices grows exponentially  ...  In other words, if this conjecture holds, quasi Ramanujan graphs have the same algebraic connectivity ratio as Ramanujan graphs.  ... 
doi:10.1109/acc.2007.4282254 dblp:conf/acc/Olfati-Saber07a fatcat:jb5lpjou35ge3oc3ee7hsg6hpu

The Ramanujan conjecture and its applications

Wen-Ching Winnie Li
2019 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
In this paper, we review the Ramanujan conjecture in classical and modern settings and explain its various applications in computer science, including the explicit constructions of the spectrally extremal  ...  The connection between Ramanujan graphs/complexes and their zeta functions satisfying the Riemann hypothesis is also discussed.  ...  These include the explicit constructions of infinite families of Ramanujan graphs and Ramanujan complexes.  ... 
doi:10.1098/rsta.2018.0441 pmid:31813366 pmcid:PMC6939229 fatcat:54oknv6cozeo3glo3xhtnjqdxm

Ramanujan in Computing Technology [article]

V. N. Krishnachandran
2021 arXiv   pre-print
Many were completely novel; his original and highly unconventional results, such as the Ramanujan prime, the Ramanujan theta function, partition formulae and mock theta functions, have opened entire new  ...  Ramanujan independently compiled nearly 3,900 results in the form of identities and equations.  ...  Ramanujan graphs has applications in the construction and study of expander graphs.  ... 
arXiv:2103.09654v1 fatcat:t2hkkyaivzhexdkxurrmuhfqk4

Ramanujan Graphs and Digraphs [article]

Ori Parzanchevski
2020 arXiv   pre-print
Other topics explored are the connection to Cayley graphs and digraphs, the spectral radius of universal covers, Alon's conjecture for random digraphs, and explicit constructions of almost-normal Ramanujan  ...  Ramanujan graphs have fascinating properties and history.  ...  (Ramanujan graph) While the generalized Ramanujan conjecture appears in the first constructions of such graphs [LPS88, Mar88] , the reason that lead Lubotzky, Phillips and Sarnak to coin the term Ramanujan  ... 
arXiv:1804.08028v4 fatcat:vmithvuihvff5nkrl5o3tjvjhe

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  ...  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.  ...  I am especially grateful to Ori Parzanchevski, whose help and advice in preparing these notes have improved them significantly.  ... 
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

Ramanujan Complexes and bounded degree topological expanders [article]

Tali Kaufman, David Kazhdan, Alexander Lubotzky
2014 arXiv   pre-print
More precisely, our main result says that the 2-skeletons of the 3-dimensional Ramanujan complexes are topological expanders.  ...  Assuming a conjecture of Serre on the congruence subgroup property, infinitely many of them are also coboundary expanders.  ...  We thank also the ERC, ISF, BSF and NSF for their support.  ... 
arXiv:1408.6351v1 fatcat:qf5aeit4hneexcptjdxuemh4fu

The Cayley Graphs Associated With Some Quasi-Perfect Lee Codes Are Ramanujan Graphs

Khodakhast Bibak, Bruce M. Kapron, Venkatesh Srinivasan
2016 IEEE Transactions on Information Theory  
They also conjectured that 𝒢_p is a Ramanujan graph for every prime p such that p≡ 3 4. In this paper, we solve this conjecture.  ...  Our proof techniques may motivate more work in the interactions between spectral graph theory, character theory, and coding theory, and may provide new ideas towards the famous Golomb–Welch conjecture  ...  ACKNOWLEDGEMENTS During the preparation of this work the first author was supported by a Fellowship from the University of Victoria (UVic Fellowship).  ... 
doi:10.1109/tit.2016.2595778 fatcat:p2ysclxnlfabzcf2j5j763yrva

Page 4040 of Mathematical Reviews Vol. , Issue 96g [page]

1996 Mathematical Reviews  
These are the deepest tools used here (leading among other things to the solution of the Petersson-Ramanujan conjecture).  ...  The Appendix, written by J. Rogawski, explains the Jacquet-Langlands theory and indicates Deligne’s proof of the Petersson-Ramanujan conjecture. It would merit its own review.  ... 

Bounded cutoff window for the non-backtracking random walk on Ramanujan Graphs [article]

Evita Nestoridi, Peter Sarnak
2021 arXiv   pre-print
We prove that the non-backtracking random walk on Ramanujan graphs with large girth exhibits the fastest possible cutoff with a bounded window.  ...  Acknowledgements We would like to thank Eyal Lubetzky for his comments and insights concerning cutoff for the NBRW.  ...  Our Conjecture 1.8 implies that the NBRW on these Ramanujan graphs exhibit cutoff with an explicit and tight bounded window, namely t x (ε) ≤ log p n + 2 log p ε −1 , for every starting point x.  ... 
arXiv:2103.15176v2 fatcat:2d4rjvo3cbfsll6u4mzu5ycpc4

Euler Product Asymptotics for Dirichlet L-Functions [article]

Ikuya Kaneko
2021 arXiv   pre-print
Understanding the behaviour of Euler products on the critical line is called the Deep Riemann Hypothesis (DRH). This work manifests the relation between GRH and DRH.  ...  Via the work of Ramanujan, we establish the asymptotic behaviour of partial Euler products for Dirichlet L-functions under the Generalised Riemann Hypothesis (GRH).  ...  Given an elliptic curve /Q with the conductor, Kuo-Murty [5] established the equivalence between the Birch and Swinnerton-Dyer conjecture and the bound ˜ = ( ).  ... 
arXiv:1902.04203v2 fatcat:nlmgxgtcvndwllgnbcgyez4pfy

Expander graphs and gaps between primes

Sebastian M Cioabă, M Ram Murty
2008 Forum mathematicum  
The explicit construction of infinite families of d-regular graphs which are Ramanujan is known only in the case d À 1 is a prime power.  ...  The main result is that by perturbing known Ramanujan graphs and using results about gaps between consecutive primes, we are able to construct infinite families of "almost" Ramanujan graphs for almost  ...  Our goal is to begin with the infinite families of Ramanujan graphs described above and perturb them in an explicit way to obtain what we call almost Ramanujan graphs.  ... 
doi:10.1515/forum.2008.035 fatcat:mxlt3uy5tvcg7o4rv3ozeyfyrq
« Previous Showing results 1 — 15 out of 1,713 results