发表机构
Kongju National University（Kongju National University）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明n≥5的测地线Leech轮W_n满足n≤40，通过有限傅里叶核等方法完成证明，同时计算机搜索发现W_7到W_13均为测地线Leech轮，否定了Lakshmanan S.等人的相关预期。
AI 中文摘要
设f为有限图G的边到正整数的一个标号，一条路径的权重是其边的标号之和。若所有测地线的权重恰好为1,2,…,t_gp(G)且每个权重仅出现一次，则该标号称为测地线Leech标号，其中t_gp(G)是G的测地线路径数。设W_n为n个顶点的轮图，即一个中心顶点与一个(n-1)阶环相连的图。我们的主要结果是一个上界：若n≥5且W_n是测地线Leech轮，则n≤40。该证明通过有限傅里叶核量化了辐条标号的Sidon型结构，其中仅允许循环相邻对作为缺陷，并利用六变量帕塞瓦尔论证完成了最后三种情形的证明。另一方面，通过计算机搜索找到的W_7到W_13的显式标号，否定了Lakshmanan S.和Manattu提出的一个问题——他们曾找到W_5和W_6的标号，并预期每个n≥7的W_n都是非测地线Leech图。记E为满足W_n是测地线Leech轮的n≥5的集合，我们得到{5,6,…,13}包含于E，而E包含于{5,6,…,40}。
英文摘要
Let f be a labeling of the edges of a finite graph G by positive integers, and let the weight of a path be the sum of the labels of its edges. The labeling is a geodesic Leech labeling if the weights of the geodesics are exactly 1, 2, ..., t_gp(G), each occurring once, where t_gp(G) is the geodesic path number of G. Let W_n be the wheel on n vertices, a hub joined to an (n-1)-cycle. Our main result is an upper bound: if n >= 5 and W_n is geodesic Leech, then n <= 40. The proof quantifies, via a finite Fourier kernel, the Sidon-type structure of the spoke labels, in which only the cyclically adjacent pairs are allowed as defects, and closes the last three cases with a six-variable Parseval argument. In the other direction, explicit labelings of W_7, ..., W_13, found by a computer search, answer in the negative a problem of Lakshmanan S. and Manattu, who had found labelings of W_5 and W_6 and expected every W_n with n >= 7 to be a non-geodesic Leech graph. Writing E for the set of n >= 5 for which W_n is geodesic Leech, we obtain {5, 6, ..., 13} is contained in E, which is contained in {5, 6, ..., 40}.
Comments40 pages, 1 figure. Companion code and data: https://github.com/junyeobe0315/geodesic-leech-wheels (archived at doi:10.5281/zenodo.22254583). Ancillary files: the labelings of W_5,...,W_13 in JSON and the exact-arithmetic verification scripts
