Given a planar graph family , let and be the maximum size and maximum spectral radius over all ‐vertex ‐free planar graphs, respectively. Let be the disjoint union of copies of ‐cycles, and be the family of vertex‐disjoint cycles without length restriction. Tait and Tobin determined that is the extremal spectral graph among all planar graphs with sufficiently large order , which implies the extremal graphs of both and for are . In this paper, we first determine and and characterize the unique extremal graph for , and sufficiently large . Second, we obtain the exact values of and , which solve a conjecture of Li for .

中文翻译：

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无顶点不相交循环的平面图的极值谱结果

给定一个平面图族 ，令 和 分别为所有无顶点平面图的最大尺寸和最大谱半径。令 为 循环副本的不相交并，为无长度限制的顶点不相交循环族。Tait 和 Tobin 确定 是所有具有足够大阶数的平面图中的极值谱图，这意味着 和 的极值图是 。在本文中，我们首先确定 和 并表征 和 足够大的唯一极值图。其次，我们获得 和 的精确值，从而解决了 Li 对 的猜想。