Linear combinations of graph eigenvalues

Abstract

Let µ1 (G) ≥ ...≥ µn (G) be the eigenvalues of the adjacency matrix of a graph G of order n, and G be the complement of G. Suppose F (G) is a fixed linear combination of µi (G), µn−i+1 (G) ,µ i G , and µn−i+1 G , 1 ≤ i ≤ k. It is shown that the limit lim n→∞ 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 Kr-free or r-partite graphs. It is also shown that 29 + √ 329 42 n − 25 ≤ max v(G)=n µ1 (G )+ µ2 (G) ≤ 2 √ 3 n. This inequality answers in the negative a question of Gernert.