On graphs with just three distinct eigenvalues

Let G   be a connected non-bipartite graph with exactly three distinct eigenvalues ρ,μ,λρ,μ,λ, where ρ>μ>λρ>μ>λ. In the case that G has just one non-main eigenvalue, we find necessary and sufficient spectral conditions on a vertex-deleted subgraph of G for G to be the cone over a strongly regular graph. Secondly, we determine the structure of G when just μ is non-main and the minimum degree of G   is 1+μ−λμ1+μ−λμ: such a graph is a cone over a strongly regular graph, or a graph derived from a symmetric 2-design, or a graph of one further type.