2,147 research outputs found
Cutoff on Graphs and the Sarnak-Xue Density of Eigenvalues
It was recently shown by Lubetzky and Peres (2016) and by Sardari (2018) that
Ramanujan graphs, i.e., graphs with the optimal spectrum, exhibit cutoff of the
simple random walk in optimal time and have optimal almost-diameter. We prove
that this spectral condition can be replaced by a weaker condition, the
Sarnak-Xue density of eigenvalues property, to deduce similar results.
We show that a family of Schreier graphs of the
-action on the projective line satisfies the
Sarnak-Xue density condition, and hence exhibit the desired properties. To the
best of our knowledge, this is the first known example of optimal cutoff and
almost-diameter on an explicit family of graphs that are neither random nor
Ramanujan
The diameter of random Cayley digraphs of given degree
We consider random Cayley digraphs of order with uniformly distributed
generating set of size . Specifically, we are interested in the asymptotics
of the probability such a Cayley digraph has diameter two as and
. We find a sharp phase transition from 0 to 1 at around . In particular, if is asymptotically linear in , the
probability converges exponentially fast to 1.Comment: 11 page
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