Graph-theoretic properties of scales: Difference between revisions
Wikispaces>genewardsmith **Imported revision 359531781 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 359532899 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-23 17: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-23 17:29:52 UTC</tt>.<br> | ||
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The automorphism group is S5, the symmetric group of order 120 on a set of five points, which in this case are the five prime numbers 2 to 11. Any permutation acts faithfully on the notes of the dekany, inducing the transitive permutation representation called 10T13 of S5 on ten points. The dekany has five maximal 4-cliques (tetrads) and ten maximal 3-cliques (triads), and S5 acts faithfully on these also. The graph of triads is isomorphic to the graph of the scale, and the graph of tetrads is the complete graph on five vertices K5; both have automorphism group S5. | The automorphism group is S5, the symmetric group of order 120 on a set of five points, which in this case are the five prime numbers 2 to 11. Any permutation acts faithfully on the notes of the dekany, inducing the transitive permutation representation called 10T13 of S5 on ten points. The dekany has five maximal 4-cliques (tetrads) and ten maximal 3-cliques (triads), and S5 acts faithfully on these also. The graph of triads is isomorphic to the graph of the scale, and the graph of tetrads is the complete graph on five vertices K5; both have automorphism group S5. | ||
Though it has only ten notes, an attempt to compute the genus of the dekany using SAGE caused it to wander off into the weeds and never return, or at least not when it was allowed to run overnight. An inquiry of someone who has published on the Johnson graphs revealed he had no idea what the genus of J(5,2) was, and it may very well not be known. However, the inequalities above show the genus must be at least 1. | |||
[[image:dekany.png]]</pre></div> | [[image:dekany.png]]</pre></div> | ||
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The automorphism group is S5, the symmetric group of order 120 on a set of five points, which in this case are the five prime numbers 2 to 11. Any permutation acts faithfully on the notes of the dekany, inducing the transitive permutation representation called 10T13 of S5 on ten points. The dekany has five maximal 4-cliques (tetrads) and ten maximal 3-cliques (triads), and S5 acts faithfully on these also. The graph of triads is isomorphic to the graph of the scale, and the graph of tetrads is the complete graph on five vertices K5; both have automorphism group S5.<br /> | The automorphism group is S5, the symmetric group of order 120 on a set of five points, which in this case are the five prime numbers 2 to 11. Any permutation acts faithfully on the notes of the dekany, inducing the transitive permutation representation called 10T13 of S5 on ten points. The dekany has five maximal 4-cliques (tetrads) and ten maximal 3-cliques (triads), and S5 acts faithfully on these also. The graph of triads is isomorphic to the graph of the scale, and the graph of tetrads is the complete graph on five vertices K5; both have automorphism group S5.<br /> | ||
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Though it has only ten notes, an attempt to compute the genus of the dekany using SAGE caused it to wander off into the weeds and never return, or at least not when it was allowed to run overnight. An inquiry of someone who has published on the Johnson graphs revealed he had no idea what the genus of J(5,2) was, and it may very well not be known. However, the inequalities above show the genus must be at least 1.<br /> | |||
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<!-- ws:start:WikiTextLocalImageRule:35:&lt;img src=&quot;/file/view/dekany.png/359810971/dekany.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/dekany.png/359810971/dekany.png" alt="dekany.png" title="dekany.png" /><!-- ws:end:WikiTextLocalImageRule:35 --></body></html></pre></div> | <!-- ws:start:WikiTextLocalImageRule:35:&lt;img src=&quot;/file/view/dekany.png/359810971/dekany.png&quot; alt=&quot;&quot; title=&quot;&quot; /&gt; --><img src="/file/view/dekany.png/359810971/dekany.png" alt="dekany.png" title="dekany.png" /><!-- ws:end:WikiTextLocalImageRule:35 --></body></html></pre></div> | ||