Many large eigenvalues in sparse graphs

It is shown that, in every family G of graphs that is closed under taking induced subgraphs and whose members have bounded average degree, the following properties are roughly equivalent for every G ∈ G : (a) G has many large eigenvalues; (b) G has many large negative eigenvalues; (c) G has many vertices of large degree. By a rough equivalence we mean that, in the quantitative version of the result, specifying the values of “how many” eigenvalues we want and “how large” they are, each implication may change these values by a constant factor.