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Spatial graph embeddings and coupler curves - source code and results

Legerský, Jan; Bartzos, Evangelos


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    <subfield code="u">Research Institute for Symbolic Computation, \\ Johannes Kepler University Linz, Austria</subfield>
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    <subfield code="a">Legerský, Jan</subfield>
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    <subfield code="a">&lt;p&gt;This is supplementary material for the paper&amp;nbsp;&lt;em&gt;On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs&lt;/em&gt; accepted to ISSAC 2018, see&amp;nbsp;&lt;a href="https://arxiv.org/abs/1802.05860"&gt;preprint&lt;/a&gt;&amp;nbsp;and&amp;nbsp;&lt;a href="http://jan.legersky.cz/project/spatialgraphembeddings/"&gt;project website&lt;/a&gt;&amp;nbsp;for details. It includes source code and results.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Construction of Geiringer graphs&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Sage and Matlab code for construction of Geiringer graphs using Henneberg steps.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Computation of mixed volume and complex solutions&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Functions that&amp;nbsp;compute the mixed volume and the (complex) embeddings of a given graph using different algebraic formulations.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Coupler curve visualization and sampling&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Implementation of&amp;nbsp;our&amp;nbsp;method for obtaining edge lengths of a Geiringer&amp;nbsp;graph with many real spatial embeddings.&lt;br&gt;
Visualization of coupler curves of the 7-vertex Geiringer graph with the maximal number of&amp;nbsp;real embeddings (G48).&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Maple parametric_search&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Maple worksheet illustrating a method to improve number of real embeddings of G48 using RootFinding[Parametric] subpackage.&lt;/p&gt;

&lt;p&gt;&lt;strong&gt;Results&lt;/strong&gt;&lt;/p&gt;

&lt;p&gt;Full list of Geiringer graphs up to 8 vertices.&lt;br&gt;
Edge lengths of graphs obtained by our method that maximizes number of real embeddings.&amp;nbsp;&lt;br&gt;
Maple script verifying the results.&lt;/p&gt;</subfield>
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