Ramanujan coverings of graphs

Ramanujan graphs

Ramanujan Graphs

In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication network theory (see for example [2]). Second, from a more aesthetic viewpoint, they fuse diverse branches of pure mathematics, namely, number theory, representation theory and algebraic geometry. The...

Properties of Ramanujan Graphs

Spectra of Lifted Ramanujan Graphs

A random n-lift of a base graph G is its cover graph H on the vertices [n]×V (G), where for each edge uv in G there is an independent uniform bijection π, and H has all edges of the form (i, u), (π(i), v). A main motivation for studying lifts is understanding Ramanujan graphs, and namely whether typical covers of such a graph are also Ramanujan. Let G be a graph with largest eigenvalue λ1 and l...

Coverings of K-graphs

k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of Cuntz-Krieger type. Here we develop the theory of covering spaces for k-graphs, obtaining a satisfactory version of the usual topological classification in terms of subgroups of a fundamental group. We then use this classification to describe the C∗algebras o...