On graphs with just three distinct eigenvalues

Let G be a connected non-bipartite graph with exactly three distinct eigenvalues Rho, mu, lambda, where Rho >mu >lambda. 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 mu is non-main and the minimum degree of G is 1 + mu − lambda mu: 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