Finite groups with geodetic Cayley graphs

TL;DR

This paper investigates the structure of finite groups with geodetic Cayley graphs, providing theoretical results and computational verification up to size 1024, supporting the conjecture that only odd cycles and complete graphs are geodetic.

Contribution

It offers new theoretical insights and computational evidence supporting the conjecture that finite groups with geodetic Cayley graphs are limited to certain known types.

Findings

The conjecture holds for groups with even-order centers.

Bounds relate group size, generating set, and center for geodetic Cayley graphs.

Verification extends to groups up to size 1024 and specific infinite families.

Abstract

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).