scholarly journals The Ramanujan conjecture and its applications

2019 ◽  
Vol 378 (2163) ◽  
pp. 20180441
Author(s):  
Wen-Ching Winnie Li
Keyword(s):  
Quantum Computing ◽  
Computer Science ◽  
Riemann Hypothesis ◽  
Zeta Functions ◽  
Golden Gate ◽  
Ramanujan Graphs ◽  
Combinatorial Objects ◽  
Explicit Constructions ◽  
Ramanujan Conjecture

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 combinatorial objects, called Ramanujan graphs and Ramanujan complexes, points uniformly distributed on spheres, and Golden-Gate Sets in quantum computing. The connection between Ramanujan graphs/complexes and their zeta functions satisfying the Riemann hypothesis is also discussed. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.

scholarly journals New and explicit constructions of unbalanced Ramanujan bipartite graphs

2021 ◽  
Author(s):  
Shantanu Prasad Burnwal ◽  
Kaneenika Sinha ◽  
Mathukumalli Vidyasagar
Keyword(s):  
Infinite Family ◽  
Bipartite Graphs ◽  
Affirmative Answer ◽  
Ramanujan Graphs ◽  
The Third ◽  
Construction Methods ◽  
Explicit Constructions ◽  
Ramanujan Graph ◽  
Computational Work ◽  
First Time

AbstractThe objectives of this article are threefold. Firstly, we present for the first time explicit constructions of an infinite family of unbalanced Ramanujan bigraphs. 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 third goal of this article is to address the following question: can we construct a bipartite Ramanujan graph with specified degrees, but with the restriction that the edge set of this graph must be distinct from a given set of “prohibited” edges? We provide an affirmative answer in many cases, as long as the set of prohibited edges is not too large.

scholarly journals On an anti-Ramsey property of Ramanujan graphs

1995 ◽  
Vol 6 (4) ◽  
pp. 417-431 ◽  
Author(s):  
P. E. Haxell ◽  
Y. Kohayakawa
Keyword(s):  
Ramsey Property ◽  
Ramanujan Graphs

scholarly journals Gap sets for the spectra of cubic graphs

2021 ◽  
Vol 1 (1) ◽  
pp. 1-38
Author(s):  
Alicia Kollár ◽  
Peter Sarnak
Keyword(s):  
Line Graphs ◽  
Cubic Graphs ◽  
Content Type ◽  
Ramanujan Graphs ◽  
Adjacency Matrices

We study gaps in the spectra of the adjacency matrices of large finite cubic graphs. It is known that the gap intervals ( 2 2 , 3 ) (2 \sqrt {2},3) and [ − 3 , − 2 ) [-3,-2) achieved in cubic Ramanujan graphs and line graphs are maximal. We give constraints on spectra in [ − 3 , 3 ] [-3,3] which are maximally gapped and construct examples which achieve these bounds. These graphs yield new instances of maximally gapped intervals. We also show that every point in [ − 3 , 3 ) [-3,3) can be gapped by planar cubic graphs. Our results show that the study of spectra of cubic, and even planar cubic, graphs is subtle and very rich.

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