发表机构
College of Mathematics and Systems Science, Xinjiang University（新疆大学数学与系统科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究不含友谊图 $F_t$ 的 1-平面图的谱极值问题，通过建立极值图的结构定理并利用 $K_{3,6}$ 的绘图性质，确定了谱半径最大值 $\textit{spex}_{\mathcal{P}_1}(n,F_t)$ 及其唯一极值图。
AI 中文摘要
设 $\textit{spex}_{\mathcal{P}_1}(n,F)$ 为所有 $n$ 个顶点且不含 $F$ 作为子图的 $1$-平面图中谱半径的最大值。定义 $F_t$ 为由 $t$ 个三角形恰好共享一个公共顶点构成的友谊图。Tait 和 Tobin (2017) 利用谱极值图的基本结构确定了在阶数足够大时具有最大谱半径的唯一平面图。随后，Zhang、Wang 和 Wang (2024) 刻画了 $1$-平面图类中相应的极值图。本文研究不含 $F_t$ 的 $1$-平面图，并对所有 $t\geq1$ 及足够大的 $n$，建立其谱极值图的结构定理。更精确地，每个极值图都是连通的且包含一个 $K_{2,n-2}$ 副本；当 $t\geq2$ 时，两个特殊顶点相邻，且剩余顶点诱导的子图是二分图。基于该结构结果以及 $K_{3,6}$ 的绘图性质，我们确定了 $\textit{spex}_{\mathcal{P}_1}(n,F_t)$ 并刻画了其唯一极值图。
英文摘要
Let $\textit{spex}_{\mathcal{P}_1}(n,F)$ be the maximum spectral radius among all $n$-vertex $F$-free $1$-planar graphs. Define $F_t$ as the friendship graph formed by $t$ triangles sharing exactly one common vertex. Tait and Tobin (2017)~\cite{Tait2017} used the fundamental structure of spectral extremal graphs to determine the unique planar graph with maximum spectral radius for sufficiently large order. Subsequently, Zhang, Wang and Wang (2024)~\cite{Zhang2024} characterized the corresponding extremal graph in the class of $1$-planar graphs. In this paper, we focus on $F_t$-free $1$-planar graphs and establish a structural theorem for their spectral extremal graphs for all $t\geq1$ and sufficiently large $n$. More precisely, every extremal graph is connected and contains a copy of $K_{2,n-2}$, and for $t\geq2$ the two distinguished vertices are adjacent and the subgraph induced by the remaining vertices is a bipartite graph. Based on this structure result together with the drawing properties of $K_{3,6}$, we determine $\textit{spex}_{\mathcal{P}_1}(n,F_t)$ and characterize its unique extremal graph.
