  Let F(G) be a fixed linear combination of the k extremal eigenvalues of a
graph G and of its complement. The problem of finding max{F(G):v(G)=n}
generalizes a number of problems raised previously in the literature. We show
that the limit max{F(G):v(G)=n}/n exists when n tends to infinity. We also
answer a question of Gernert about the sum of the two maximal eigenvalues of a
graph.
