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.