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

Abstract

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 sufficiently large $n$, $\left\lceil\frac{(r-1)n}{r}\right\rceil\le Δ\le n-Θ(n^{s_F})$ implies that every $n$-vertex $F$-free graph $G$ with $Δ(G)=Δ$ satisfies $ρ(G)\le ρ\bigl(S_{n,Δ}^{(r)}\bigr)$, with equality if and only if $G\cong S_{n,Δ}^{(r)}$. 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.