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Symmetric vertex- and edge-transitive 3-periodic nets
Michael O'Keeffe1, Michael M J Treacy2
1School of Molecular Sciences, Arizona State University, Tempe, Arizona 85287, USA.
Summary
This study identifies new vertex- and edge-transitive graphs with symmetric embeddings. These symmetric tessellate graphs are surprisingly common in crystal structures, unlike their untessellable counterparts.
Area of Science:
- Graph theory
- Crystallography
- Symmetry in materials science
Background:
- Vertex- and edge-transitive graphs capable of tessellation (tessellate graphs) have been studied for two decades.
- Symmetric embeddings in graphs are crucial for understanding structural properties.
Purpose of the Study:
- To identify novel families of vertex- and edge-transitive graphs exhibiting symmetric embeddings.
- To investigate the relationship between point symmetry order and coordination number in these graphs.
- To compare the prevalence of symmetric tessellate graphs versus symmetric decussate graphs in crystal structures.
Main Methods:
- Graph theory analysis to identify graph families.
- Examination of graph embeddings for symmetry properties.
- Computational analysis of crystal structure databases.
Main Results:
- Discovery of additional families of vertex- and edge-transitive graphs with symmetric embeddings.
- Demonstration that the point symmetry order at a vertex matches its coordination number in these graphs.
- Observation of over 150,000 occurrences of symmetric tessellate graphs in crystal structures.
- Identification of only one instance of a symmetric decussate (untessellable) graph.
Conclusions:
- Symmetric tessellate graphs represent a significant structural motif in crystalline materials.
- The rarity of symmetric decussate graphs suggests specific constraints on their formation or stability.
- This research expands the understanding of symmetric graph structures and their implications in materials science.
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