Spectral conjecture for (r,k)-critical graphs

Let $r\ge 2$ and $k\ge 0$, and let $G$ be a connected graph of order $n\ge 2(2r+k+2)(r+k+2)$ with minimum degree $\delta(G)\ge r+k$. Let $F_n^{r,k}$ be the extremal graph appearing in the spectral threshold, and let $\lambda(G)$ denote the spectral radius of $G$. An $(r,k)$-critical graph is a graph such that, after deleting any $k$ vertices, the remaining graph has an $r$-factor. Spectral conjecture for $(r,k)$-critical graphs. If

$$\lambda(G)\ge\lambda(F_n^{r,k}),$$

then $G$ is an $(r,k)$-critical graph, unless $G\cong F_n^{r,k}$. This conjecture seeks a spectral-radius characterization of $(r,k)$-critical graphs under the stated order and minimum-degree assumptions. The supplied text poses it for future research and gives no resolution.