Tangled piecewise-linear embeddings of trivalent graphs

Area of Science:

• Graph theory
• Computational chemistry
• Crystallography

Background:

• Tangled embeddings and graph theory are crucial for understanding complex molecular structures.
• Point-group symmetry is a fundamental concept in chemistry and materials science.
• Exploring graph embeddings under symmetry constraints can reveal novel molecular architectures.

Purpose of the Study:

• To develop a method for generating and exploring tangled piecewise-linear embeddings of trivalent graphs.
• To investigate these embeddings under point-group symmetry constraints.
• To identify potential applications in molecular synthesis.

Main Methods:

• Describing a novel method for generating and exploring graph embeddings.
• Applying point-group symmetry constraints to trivalent graphs.
• Enumerating tangles based on vertex count and symmetry properties.

Main Results:

• Characterizing vertex-transitive tangles as either graphs of vertex-transitive polyhedra or bipartite vertex-transitive nonplanar graphs.
• Identifying specific numbers of tangles for graphs with 6, 8, 10, 12, 16, and 24 vertices.
• Discovering vertex 2-transitive embeddings for graphs with 6 and 8 vertices.
• Describing symmetrical tangles for the Petersen and Desargues graphs.

Conclusions:

• The described method generates diverse and symmetrical graph embeddings.
• The identified tangles represent viable targets for molecular synthesis.
• Extensions to periodic tangles suggest broader applicability in materials science.