Eigenvalues of graphs with vertices of large degree at distance three apart

We investigate the family of graphs with many large eigenvalues. It is not hard to see that every graph with many vertices of large degree that are pairwise at distance at least four from each other, has many large eigenvalues. We show that this does not hold if the vertices of large degree are at mutual distance three from each other. We explore this class of graphs further and provide some bounds on their eigenvalues.