发表机构
College of Science, National University of Defense Technology（国防科技大学理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究指定最大度下无F图的不平衡Turán问题和谱Turán问题，针对不同类型的F给出极值图结论，建立边界与谱半径界的关联，丰富了Turán理论的相关结果。
AI 中文摘要
经典Turán型问题用于确定n顶点无F图的最大边数和谱半径，且无度约束。我们研究n顶点无F图G在指定最大度Δ(G)=Δ下的对应问题。设χ(F)=r+1≥3，且⌈(r-1)n/r⌉≤Δ≤n-1，最大度条件会导出完全r部图S_{n,Δ}^{(r)}=(n−Δ)K₁∨T(Δ,r−1)，其大小为n−Δ的部分通常小于其他部分，这是本文所考虑的不平衡Turán问题的来源。设ex_F(n,Δ)和spex_F(n,Δ)分别表示该类图中的最大边数和邻接谱半径。对于F=K_{r+1}，我们证明S_{n,Δ}^{(r)}是这两个参数的唯一极值图。对于一般图F，设a(F)为满足χ(F−I)≤r的独立集I的最小大小。若a(F)=1，我们证明相对于S_{n,Δ}^{(r)}的边稳定性和谱稳定性；若a(F)>1，极值值具有通常的Erdős–Stone–Simonovits渐近性，且边极值图和谱极值图与T(n,r)的距离为o(n²)。最后，对于有限禁止族，我们证明阶为O(n^{1+s})的分解族边界会产生误差项为O(n^s)的谱半径界，其中0≤s<1，这可用于从其他不平衡Turán问题的分解族界中获取谱半径估计。
英文摘要
Classical Turán theory shows that, without additional constraints, the extremal structure of $F$-free graphs is governed by the chromatic number of $F$ and is asymptotically the balanced Turán graph. We investigate how prescribing a large maximum degree changes this picture and leads to an unbalanced Turán problem. Let $F$ be a graph with $χ(F)=r+1\ge3$, and let $\lceil(r-1)n/r\rceil\leΔ\le n-1$. We study the maximum number of edges and adjacency spectral radius of an $n$-vertex $F$-free graph with maximum degree exactly $Δ$. For $F=K_{r+1}$, the unbalanced $r$-partite graph $S_{n,Δ}^{(r)}=(n-Δ)K_1\vee T(Δ,r-1)$ is the unique maximizer of both quantities. Let $a(F)$ be the minimum size of an independent set $I$ such that $χ(F-I)\le r$. If $a(F)=1$, we prove edge and spectral stability with respect to $S_{n,Δ}^{(r)}$. If $a(F)>1$, the extremal values are $t(n,r)+o(n^2)$ and $ρ(T(n,r))+o(n)$, respectively, and every extremal graph differs from $T(n,r)$ in $o(n^2)$ edges, uniformly all $\lceil(r-1)n/r\rceil\leΔ\le n-1$. We further establish a general spectral transfer principle: for finite forbidden families, a decomposition-family bound of order $O(n^{1+s})$ yields a spectral bound with error $O(n^s)$ for $0\le s<1$.
