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 Ḡ be the complement of G. Suppose F (G) is a fixed linear combination of μi (G), μn-i+1 (G), μi (Ḡ), and μn-i+1 (Ḡ), 1 ≤ i ≤ k. It is shown that the limit limn→∞ 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)=nμ1 (G) + μ2 (G) ≤ 2/√3 n. This inequality answers in the negative a question of Gernert.