发表机构
Okegawa West Junior High School; Faculty of Science and Technology, Oita University（尾高西初中; 大分大学理工学部）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究轮图中的根生成森林，给出基于Fibonacci和Lucas数的计数公式，并解决根配置的极值问题。
AI 中文摘要
本文研究了轮图 $W_{N+1}$ 中的根生成森林。对于由轮缘顶点组成的根集 $R$，我们给出了一个用 Fibonacci 数和 Lucas 数表示的显式公式，用于计算每个连通分量恰好包含 $R$ 中一个顶点的根生成森林的数量。当中心顶点也包含在根集中时，我们得到了一个仅涉及 Fibonacci 数的简单乘积公式。此外，利用根生成森林与顶点识别之间的对应关系，我们推导了通过将若干顶点识别为一个顶点而得到的轮图商图的生成树数量的显式公式。然后，我们将所得公式用路径图和圈图中的匹配数进行组合重写。另外，对于固定数量的 $r$ 个轮缘根顶点，我们将所有大小为 $r$ 的轮缘根集上的根生成森林数量之和表示为生成函数的系数，并解决了根的排列的极值问题。
英文摘要
In this paper, we study rooted spanning forests in the wheel graph $W_{N+1}$. For a root set $R$ consisting of rim vertices, we give an explicit formula, in terms of Fibonacci and Lucas numbers, for the number of rooted spanning forests in which each connected component contains exactly one vertex of $R$. When the central vertex is also included in the root set, we obtain a simple product formula involving only Fibonacci numbers. Furthermore, by using the correspondence between rooted spanning forests and vertex identification, we derive explicit formulas for the number of spanning trees of quotient graphs of wheel graphs obtained by identifying several vertices into one vertex. We then rewrite the obtained formulas combinatorially in terms of the numbers of matchings in path graphs and cycle graphs. In addition, for a fixed number $r$ of rim root vertices, we express the sum of the numbers of rooted spanning forests over all rim root sets of size $r$ as coefficients of generating functions, and we solve the extremal problem for the arrangement of roots.
Comments20 pages
