From Ramanujan Graphs to Ramanujan Complexes

Abstract

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.

To appear in the proceedings of the Ramanujan Centenary Celebration at the Royal Society

References in corpus (2)

• Super-Golden-Gates for PU(2)
• Random walks on Ramanujan complexes and digraphs

Cited by in corpus (3)

• Quantum-like states on complex synchronized networks
• The largest Laplacian eigenvalue and the balancedness of simplicial complexes
• The largest normalized Laplacian eigenvalue and incidence balancedness of simplicial complexes