Cutoff on all Ramanujan graphs

Cutoff on All Ramanujan Graphs

We show that on every Ramanujan graph G, the simple random walk exhibits cutoff: when G has n vertices and degree d, the total-variation distance of the walk from the uniform distribution at time t = d d−2 logd−1 n + s √ logn is asymptotically P(Z > c s) where Z is a standard normal variable and c = c(d) is an explicit constant. Furthermore, for all 1 ≤ p ≤ ∞, d-regular Ramanujan graphs minimiz...

Ramanujan graphs

Second Talk on Ramanujan Graphs

We are especially interested in graphs – or better, in sequences of graphs – whose first eigenvalues λ1 are relatively small. In particular, for any finite graph, define the spectral gap ω(G) = λ0(G) − λ1(G). Define also the isoperimetric constant h(G) to be the infimum #E(V1, V2) min{#V1, #V2} over all partitions of the vertex set into two subsets V1, V2; here E(V1, V2) is the set of edges con...

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