Elementary number theory, group theory, and Ramanujan graphs

By Giuliana P. Davidoff, Peter Sarnak, Alain Valette. 2003

372 citations recorded by the source corpus.

Subfields: Geometry and Topology, Physical and Theoretical Chemistry, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1d2f1602325f26693de04540d51eae371d54ce24

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Ben Steinberg
2. Lihua Xie
3. Yuval Peres
4. Igor Rivin
5. Sanming Zhou
6. Rostislav I. Grigorchuk
7. Eyal Lubetzky
8. Xiaogang Liu

In Physical and Theoretical Chemistry

1. Igor Rivin

In Statistical and Nonlinear Physics

1. Ben Steinberg
2. Pavel Panteleev
3. Lihua Xie
4. Yuval Peres
5. Igor Rivin
6. Ji-feng Zhang
7. Gleb Kalachev
8. Rostislav I. Grigorchuk

Citations in the corpus

• Some topics in the dynamics of group actions on rooted trees (2011, background): “Finding infinite sequences of m-regular finite Ramanujan graphs is a difficult and enthralling problem [127, 50].”
• Representation Theory of Finite Groups: An Introductory Approach (2011, background): “For more on the relationship between graph theory and representation theory, as well as the related subject of random walks on graphs, see [1, 3, 4].”
• Integral Cayley graphs over dihedral groups (2016, background): “le 1. For this purpose, we need to introduce the famous Remanujan sum. Let s ≥ 0,n ≥ 1 be two integers, the Remanujan sum is defined by c(s,n) = P (k,n)=1 e 2skπ n i, which is known as an integer (see [8] for reference): c(s,n) = ϕ(n) ϕ( n (s,n) ) µ( n (s,n) ) (6) 10 where ϕ(·) and µ(·) are the Euler”
• Distributed Consensus With Limited Communication Data Rate (2011, background): “Ramanujan graphs are expanders which achieve the largest possible algebraic connectivity ([39]–[42]).”
• Asymptotically good Quantum and locally testable classical LDPC codes (2021, background): “There are a number of explicit constructions of such graphs [37, 38] that use Cayley graphs of some non-commutative groups (see [39] for a good survey).”
• Cutoff on all Ramanujan graphs (2015, background): “Introduction A family of d-regular graphs Gn with d ≥ 3 fixed is called an expander, following the works of Alon and Milman [4,6] from the 1980’s, if all nontrivial eigenvalues of the adjacency matrices are uniformly bounded away from d. Lubotzky, Phillips, and Sarnak [25] defined a connected…”
• Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies (2011, background): “Thus, in contrast to ordinary curves, there are a finite number of isomorphism classes of curves in any given isogeny class; this number is in fact the genus g of the modular curve X 0 ( p ), which is roughly p +112 .”