Let G be a connected non-regular non-bipartite graph whose adjacency matrix has spectrum p,?(k),?(l), where k,l ? N and p > ? > ?. We show that if ? is non-main then ?(G) ? 1 + ? - ??, with equality if and only if G is of one of three types, derived from a strongly regular graph, a symmetric design or a quasi-symmetric design (with appropriate parameters in each case).