Generalized Nordhaus--Gaddum Inequalities for Eigenvalues

For a graph $G$, let $\lambda_1(G)\ge \lambda_2(G)\ge \cdots \ge \lambda_n(G)$ denote the adjacency eigenvalues of $G$. We investigate the asymptotic maximum of $$\lambda_i(G)+\lambda_j(\overline G)$$ for fixed $i$ and $j$. We prove general bounds on $\lambda_i(G) + \lambda_{j}(\overline{G})$ for all pairs $(i, j)$ and also give general bounds on the related problem of minimizing $\lambda_{n-i+1}(G) + \lambda_{n-j+1}(\overline{G})$ for fixed $i$ and $j$. We prove that for all looped graphs $G$ on $n$ vertices, $$\lambda_1(G) + \lambda_2(\overline{G}) \le \frac87 n.$$ Our method also gives a new short proof of the Nordhaus-Gaddum result for the spectral radius proved by Terpai that $\lambda_1(G) + \lambda_1(\overline{G}) \le \frac43n - 1$. We also show the close relation of these Nordhaus-Gaddum type problems to recent work on the maximum spectral gaps of graphs by Brooks, Linz and Lu.