Kim–Kim–Kostochka–O spectral-radius conjecture for saturated graphs

Let $n>r\geq3$ and let $G$ be a $K_{r+1}$-saturated graph of order $n$. Write $\rho(G)$ for the spectral radius of $G$, and let $S_{n,r}=K_r\vee\overline{K}_{n-r}$.

Kim–Kim–Kostochka–O conjecture.

$$\rho(G)>\rho(S_{n,r-1}),$$

unless $G=S_{n,r-1}$.

The conjecture strengthens the known asymptotically tight lower bound for the spectral radius of a $K_{r+1}$-saturated graph. In this paper, the conjecture is proved, so the claim is solved.