Stevanović's Nordhaus–Gaddum extremal conjecture for graph spectral radius

Let $G$ be a simple undirected graph of order $n$, with adjacency matrix $A(G)$ and spectral radius $\rho(G)=\rho(A(G))$. For a graph $G$, let $\overline{G}$ denote its complement. For vertex-disjoint graphs $G$ and $H$, let $G\vee H$ be their join; write $K_q$ for the complete graph of order $q$ and $N_q$ for the null graph of order $q$. A graph $K_q\vee N_{n-q}$ is called a complete split graph.

Stevanović's conjecture. The maximum value of

$$\rho(G)+\rho(\overline{G})$$

among graphs $G$ of order $n$ is attained by the complete split graph $K_{\lfloor\frac{n}{3}\rfloor}\vee N_{\lceil\frac{2n}{3}\rceil}$ and its complement. If $n\equiv 2 \pmod{3}$, the maximum is also attained by $K_{\lceil\frac{n}{3}\rceil}\vee N_{\lfloor\frac{2n}{3}\rfloor}$ and its complement.

This is a Nordhaus–Gaddum type extremal problem for the spectral radius. The paper's abstract states that the authors determine the extremal graph, thereby resolving the conjecture.