A connected undirected graph is called \emph{geodetic} if for every pair of vertices there is a unique shortest path connecting them. It has been conjectured that for finite groups, the only geodetic Cayley graphs which occur are odd cycles and complete graphs. In this article we present a series of theoretical results which contribute to a computer search verifying this conjecture for all groups of size up to 1024. The conjecture is also verified theoretically for several infinite families of groups including dihedral and some families of nilpotent groups. Two key results which enable the computer search to reach as far as it does are: if the center of a group has even order, then the conjecture holds (this eliminates all 2-groups from our computer search); if a Cayley graph is geodetic then there are bounds relating the size of the group, generating set and center (which cuts down the number of generating sets which must be searched significantly).

翻译：若一个连通无向图中任意两顶点间均存在唯一最短路径，则称该图是测地的。长期以来存在一个猜想：对于有限群，仅可能出现的测地凯莱图是奇环和完全图。本文提出了一系列理论结果，这些结果为通过计算机搜索验证该猜想对所有阶数不超过1024的群均成立提供了支撑。该猜想还在理论上被证明适用于若干无限群族，包括二面体群及某些幂零群族。使计算机搜索能达到当前范围的两个关键结论是：若群的中心具有偶数阶，则猜想成立（这使所有2-群可从计算机搜索中排除）；若凯莱图是测地的，则群的阶数、生成集与中心的大小之间存在约束关系（这显著减少了必须搜索的生成集数量）。