Consider a random $d$-regular graph on $n$ vertices. What can be said about its nontrivial (i.e. orthogonal to the constant) eigenfunctions? For example, I'm interested whether there are "nodal zones", i.e. if the graph can be divided into groups such that the eigenfunction has the same sign on every vertex of the same groups. Are there any results of this kind?

Another question - suppose I divide the graph arbitrarily into $k$ groups of vertices ($k$ may depend on $n$); is it possible to say something about how large the sum of eigenfunction's values on one group typically is?

essentiallyat most two nodal zones for each eigenfunction: arxiv.org/abs/0807.3675 – alex o. Sep 13 '11 at 19:16