Explicit Near-Ramanujan Graphs of Every Degree

Ramanujan graphs

Explicit Constructions of Ramanujan Complexes

In [LSV] we defined and proved the existence of Ramanujan complexes, see also [B1], [CSZ] and [Li]. The goal of this paper is to present an explicit construction of such complexes. Our work is based on the lattice constructed by Cartwright and Steger [CS]. This remarkable discrete subgroup Γ of PGLd(F ), when F is a local field of positive characteristic, acts simply transitively on the vertice...

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

Near-Optimal Induced Universal Graphs for Bounded Degree Graphs

A graph U is an induced universal graph for a family F of graphs if every graph in F is a vertex-induced subgraph of U . For the family of all undirected graphs on n vertices Alstrup, Kaplan, Thorup, and Zwick [STOC 2015] give an induced universal graph with O ( 2n/2 ) vertices, matching a lower bound by Moon [Proc. of Glasgow Math. Association 1965]. Let k = dD/2e. Improving asymptotically on ...