Generalized Nordhaus--Gaddum Inequalities for Eigenvalues

For a graph $G$, let $ λ_1(G)\ge λ_2(G)\ge \cdots \ge λ_n(G)$ denote the adjacency eigenvalues of $G$. We prove that for all looped graphs $G$ on $n$ vertices, \[λ_1(G) + λ_2(\overline{G}) \le \frac87 n. \] Our method also gives a short new proof of the Nordhaus-Gaddum result for the spectral radius