ELA, Volume 15, pp. 329-336, December 2006, abstract.
Linear Combinations of Graph Eigenvalues
Vladimir Nikiforov
Let mu_1(G) >= ... >= mu_n(G) be the eigenvalues of the adjacency
matrix of a graph G of order n, and \overline{G} be the complement of G.
Suppose F(G) is a fixed linear combination of mu_i(G), mu_{n-i+1}(G),
mu_i(\overline{G}), and mu_{n-i+1}(\overline{G}), 1 <= i <= k.
It is shown that the limit as n approaches infinity of
1/n max{F(G) : v(G)=n}
always exists. Moreover, the statement remains true if the maximum
is taken over some restricted families like "K_r-free" or
"r-partite" graphs. It is also shown that
(29+sqrt{329})/42 n-25 <=
max_{v(G)=n} mu_1(G) + mu_2(G) <= 2/sqrt{3}} n.
This inequality answers in the negative a question of Gernert.