From Ramanujan graphs to Ramanujan complexes

Abstract

<jats:p>Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high-dimensional theory has emerged. In this paper, these developments are surveyed. 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. The latter lead to ‘golden gates’ which are of importance in quantum computation.</jats:p> <jats:p>This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.</jats:p>

The Waring’s problem over finite fields through generalized Paley graphs

“…They give the GP-graphs Γ(q ℓ + 1, q m ), with q = p, considered in [20] for q a power of p. Notice that the graphs Γ(3, 2 2t ) with t ≥ 2 and Γ(4, 3 2t ) with t ≥ 1 belong to both families given in (i) and (ii). For instance, Γ (3,16), Γ(3, 64), and Γ(4, 81) are semiprimitive GP-graphs.…”

The Ramanujan conjecture and its applications

“…As shown by Parzanchevski & Sarnak [14], the image of Λ p in PU(2) distributes uniformly, and S p is a Golden Gates Set in quantum computation. For a survey of this topic, the reader is referred to [13].…”

“…The Golden-Gate Sets arise from first representing the Hecke operators of PGL n (F) by global points and then regarding them locally as points at the place at ∞. (See [5,13,14] for more details.) The role of the Ramanujan conjecture is also explained.…”

Axiomatic Construction of Trees from Boundary Arcs

“…Considerable attention has been devoted to the reconstruction of large networks, e.g., expander graphs and Ramanujan graphs [5,8,9], starting from their geometry at large. Possibly infinite graphs with thin or no loops, for instance trees and more general Gromov hyperbolic graphs, real trees and dendrites have also been extensively studied in the literature in relation with their boundaries.…”