On Cayley graphs of {\bb Z}^4.

Area of Science:

• Graph Theory
• Computational Geometry
• Discrete Mathematics

Background:

• Enumeration of generating sets for {Z}^4 integral vectors.
• Investigating Cayley graphs embeddable in Euclidean space.
• Challenges in computing graph properties due to computational complexity.

Purpose of the Study:

• To enumerate generating sets of {Z}^4.
• To identify Cayley graphs with straight-edge embeddings in 4D Euclidean space.
• To characterize these graphs using coordination sequences, shortest cycles, and automorphism groups.

Main Methods:

• Enumeration of integral vectors with components -1, 0, 1.
• Fixing graph valency to 10 due to computational constraints.
• Novel strategy for computing automorphism groups using vertex stabilizers of finite balls.

Main Results:

• 58 non-isomorphic graphs were found and characterized.
• Six exceptional, dense graphs were identified, locally isomorphic to a 5D cubic lattice.
• These dense graphs contain Hopf links between quadrangular cycles, arising from interpenetrated lattices.

Conclusions:

• The study successfully enumerated and characterized specific {Z}^4 Cayley graphs.
• A new method for computing automorphism groups was developed and applied.
• Local combinatorial isomorphism does not guarantee local isotopy in these complex graph structures.