Finite groups with geodetic Cayley graphs

2024-06-01

arXiv.org

AI Summary

This paper addresses the geodeticity problem for Cayley graphs of finite groups: determining when there exists a unique shortest path between every pair of vertices. It focuses on verifying the conjecture that only odd cycles and complete graphs are geodetic Cayley graphs. Methodologically, the authors combine group-theoretic analysis, graph-theoretic geodeticity criteria, GAP-assisted enumeration, combinatorial construction, and inductive reasoning over infinite families. Their key contributions include: (i) proving that the conjecture holds whenever the center of the group has even order—significantly narrowing the search space for potential counterexamples; (ii) establishing tight bounds linking group order, generating set size, and center order to enable efficient computational verification; and (iii) exhaustively verifying the conjecture for all groups of order ≤ 1024, confirming it for dihedral groups and several families of nilpotent groups, while definitively ruling out counterexamples among 2-groups.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Causality

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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).

Problem

• Cayley graphs
• geodesics
• finite groups

Innovation

• Cayley Graphs
• Geodesic Optimization
• Automated Verification