The Second Eigenvalue Multiplicity Problem for Bounded Degree Graphs

We study the following problem that arises from the recent solution of equiangular lines with a fixed angle: If a connected graph G has bounded maximum degree Δ, what is the maximum multiplicity of its second eigenvalue as a function of the number of vertices n? We will show that the multiplicity must be sublinear in the number of vertices. We will also discuss constructions of lower bounds and some interesting open directions.