Unbalanced spectral Tur\'an problem for color-critical graphs with prescribed large maximum degree

Abstract

Let $F$ be a connected color-critical graph with $\chi(F)=r+1\ge4$, 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 $n$, $\left\lceil\frac{(r-1)n}{r}\right\rceil\le \Delta\le n-\Theta(n^{s_F})$ implies that every $n$-vertex $F$-free graph $G$ with $\Delta(G)=\Delta$ satisfies $\rho(G)\le \rho\bigl(S_{n,\Delta}^{(r)}\bigr)$, 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 Tur\'an problems arising from other extremal parameters.