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Generating Strictly Controlled Stimuli for Figure Recognition Experiments
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Published on: March 18, 2019

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.

Acta Crystallographica. Section A, Foundations and Advances
|July 7, 2026
PubMed
Summary
This summary is machine-generated.

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.

Keywords:
decussategraphsnetssymmetric netstessellate

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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.