AI 中文总结
本文研究1-平面图的谱Turán型问题，确定了无K₅、无C₅及无2C₅的n顶点1-平面图的谱极值图，建立了相关结构约简定理，将该类问题从团扩展至环及其不交并。
AI 中文摘要
由Nikiforov于2007年提出的谱Turán型问题旨在确定n顶点无H图中谱半径最大的图。本文针对1-平面图研究该问题，1-平面图指可在平面上绘制且每条边最多仅一次交叉的图。近期Xu和Chang证明，所有n顶点无K₅ 1-平面图中谱半径最大的图属于一个小型候选族。首先，本文明确确定了所有n顶点无K₅ 1-平面图中唯一的谱极值图；其次，建立了一个结构约简定理：对于任何含δ(F)≥2的禁用子图F，若F包含于K₂∨P_{n-2}^{2+}但不包含于K₂∨I_{n-2}，则每个谱极值无F 1-平面图都包含一个生成完全二部图K_{2,n-2}，其中P_{n-2}^{2+}是由路径u₁u₂…u_{n-2}添加边u₁u_{n-2}和所有边uᵢu_{i+2}（1≤i≤n-4）得到的图，I_{n-2}表示n-2个顶点的空图。作为应用，确定了所有n顶点无C₅（或无2C₅）1-平面图中谱半径最大的图，这些结果将1-平面图的谱Turán型问题从团扩展到环及其不交并。
英文摘要
The spectral Turán type problem, initiated by Nikiforov in 2007, aims to determine the graphs among $n$-vertex $H$-free graphs having maximum spectral radius. In this paper, we study this problem for $1$-planar graphs, i.e., graphs that admit a drawing in the plane such that each edge is crossed at most once. Recently, Xu and Chang proved that the graphs among all $n$-vertex $K_5$-free $1$-planar graphs having maximum spectral radius lie within a small family of candidates. First, this paper explicitly identifies the unique spectral extremal graph among the $n$-vertex $K_5$-free $1$-planar graphs. Second, it establishes a structural reduction theorem: For any forbidden subgraph $F$ with $δ(F)\ge2$ that is contained in $K_2\vee P_{n-2}^{2+}$ but not in $K_2\vee I_{n-2}$, every spectral extremal $F$-free $1$-planar graph contains a spanning complete bipartite graph $K_{2,n-2}$, where $P^{2+}_{n-2}$ is obtained from a path $u_1u_2\dots u_{n-2}$ by adding edge $u_1u_{n-2}$ and all edges $u_iu_{i+2}$ for $1\le i\le n-4$, and $I_{n-2}$ denotes the empty graph on $n-2$ vertices. As applications, the graph among all $n$-vertex $C_5$-free (resp. $2C_5$-free) $1$-planar graphs having maximum spectral radius is determined. These results extend spectral Turán type problems for $1$-planar graphs from cliques to cycles and their disjoint union.
