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 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 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 significantly cuts down the number of generating sets which must be searched).

翻译：一个连通无向图被称为\emph{测地线}的，如果对于每一对顶点，都存在唯一的最短路径连接它们。据推测，对于有限群而言，仅有的测地线凯莱图是奇环和完全图。在本文中，我们提出了一系列理论结果，这些结果支持了一项计算机搜索，该搜索验证了此猜想对于所有阶数不超过1024的群均成立。该猜想也在包括二面体群和一些幂零群族在内的若干无限群族中得到了验证。使得计算机搜索能够达到目前范围的两个关键结果是：如果一个群的中心具有偶数阶，则该猜想成立（这从我们的计算机搜索中排除了所有$2$-群）；如果一个凯莱图是测地线的，那么群的阶数、生成集和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）。