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

Abstract

Let $F$ be a connected color-critical graph with $\chi (F)=r+1\ge 4$, let $S_{n,\Delta }^{(r)}=(n-\Delta )K_1\vee T(\Delta ,r-1)$. We determine the graph of maximum adjacency spectral radius among all $n$-vertex $F$-free graphs with prescribed maximum degree $\Delta$. There is a constant $s_F\in [0,1)$ such that, for all sufficiently large $n$, $\lceil \frac{(r-1)n}{r}\rceil \le \Delta \le n-\Theta (n^{s_F})$ implies that every $n$-vertex $F$-free graph $G$ with $\Delta (G)=\Delta$ satisfies $\rho (G)\le \rho \left(S_{n,\Delta }^{(r)}\right)$, with equality if and only if $G\cong S_{n,\Delta }^{(r)}$. This is the spectral counterpart of the edge theorem of [European J. Combin. 106 (2022), 103576.] and extends the clique result in [arXiv:2608.26634, 2026.]. This result also provides a benchmark for unbalanced spectral Turán problems arising from other extremal parameters.