Unbalanced spectral Turán problem for color-critical graphs with prescribed large maximum degree

Let $F$ be a connected color-critical graph with $χ(F)=r+1\ge4$, let $S_{n,Δ}^{(r)}=(n-Δ)K_1\vee T(Δ,r-1)$. We determine the graph of maximum adjacency spectral radius among all $n$-vertex $F$-free graphs with prescribed maximum degree $Δ$. There is a constant $s_F\in[0,1)$ such that, for all suffic