AI 中文总结
本文研究了不含C_{k,l}的平面图和外平面图的谱极值问题，确定了最大谱半径及其极值图结构。
AI 中文摘要
令spex_{P}(n,F)和spex_{OP}(n,F)分别表示所有n个顶点F-free的平面图和外平面图中的最大谱半径。定义C_{k,l}为由C_k∪C_l构成的图，其中两个环共享一个顶点，其中l≥k≥3。20世纪90年代，Cvetković和Rowlinson猜测K_1+P_{n-1}在n个顶点的外平面图中最大化谱半径，而Boots和Royle（独立地，Cao和Vince）猜测K_2+P_{n-2}在平面图中如此。Tait和Tobin[J. Combin. Theory Ser. B, 2017]确定了基本结构是确认这两个猜想的关键，对于足够大的n。最近，Yin和Li[Discrete Mathematics, 2026]基于这一关键思想，Characterized了平面图和外平面图中spex_{P}(n,B_{t,l})和spex_{OP}(n,B_{t,l})的极值图。在本文中，我们关注不含C_{k,l}的平面图和外平面图，并确定spex_{P}(n,C_{k,l})和spex_{OP}(n,C_{k,l})及其唯一的极值图，对于所有l≥k≥3和大的n。
英文摘要
Let $\emph{spex}_{\mathcal{P}}(n,F)$ and $\emph{spex}_{\mathcal{OP}}(n,F)$ be the maximum spectral radius among all $n$-vertex $F$-free planar graphs and outerplanar graphs, respectively. Define $C_{k,l}$ as a graph obtained from $C_k \cup C_l$ such that the two cycles share a common vertex, where $l \ge k \ge 3$. In the 1990s, Cvetković and Rowlinson conjectured $K_1 + P_{n-1}$ maximizes spectral radius in outerplanar graphs on $n$ vertices, while Boots and Royle (independently, Cao and Vince) conjectured $K_2 + P_{n-2} $ does so in planar graphs. Tait and Tobin [J. Combin. Theory Ser. B, 2017] determined the fundamental structure as the key to confirming these two conjectures for sufficiently large $n$. Recently, Yin and Li [Discrete Mathematics, 2026] characterized the extremal graphs for $\emph{spex}_{\mathcal{P}}(n,B_{t,l})$ and $\emph{spex}_{\mathcal{OP}}(n,B_{t,l})$ in planar and outerplanar graphs on the basis of this key idea, where $B_{t,l}$ denotes the graph obtained by $t$ edge-disjoint $l$-cycles sharing a common vertex. In this paper, we focus on planar and outerplanar graphs without $C_{k,l}$, and determine $\emph{spex}_{\mathcal{P}}(n,C_{k,l})$ and $\emph{spex}_{\mathcal{OP}}(n,C_{k,l})$ along with their unique extremal graphs for all $l \geq k \geq 3$ and large $n$.
