Graph-theoretic properties of scales: Difference between revisions

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: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-23 17:22:48 UTC</tt>.<br>

: The original revision id was <tt>~~359498985~~</tt>.<br>

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==The dekany==

The standard 2)5 dekany is a [[Combination product sets|combination product set]], Cps([2,3,5,7,11], 2). It consists of ten notes associated to two-element subset of the set of the first five primes, {2,3,5,7,11}, and in one mode is 12/11-5/4-14/11-15/11-3/2-35/22-7/4-20/11-21/11, which we will take as its notes from note 0 to note 9. It has 30 edges, with connectivities 5 ≤ 6 ≤ 6, and the largest element of the Laplace spectrum is 8, so that the complementary graph is also connected. Its radius and diameter are both 2.

The standard 2)5 dekany is a [[Combination product sets|combination product set]], Cps([2,3,5,7,11], 2). It consists of ten notes associated to two-element subset of the set of the first five primes, {2,3,5,7,11}, and in one mode is 12/11-5/4-14/11-15/11-3/2-35/22-7/4-20/11-21/11, which we will take as its notes from note 0 to note 9. It has 30 edges, with connectivities 5 ≤ 6 ≤ 6, and the largest element of the Laplace spectrum is 8, so that the complementary graph is also connected. Its radius and diameter are both 2. The graph is known as the [[Johnson graph]] J(5,2). The 3)5 dekany, which is the inverse of the standard dekany, has the Johnson graph J(5,3) as its graph, which is graph-isomorphic to J(5,2).

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 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.</pre></div>

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.

[[image:dekany.png]]</pre></div>

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<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Graph-theoretic properties of scales</title></head><body><a href="#Graph of a scale">Graph of a scale</a> | <a href="#Connectivity">Connectivity</a> | <a href="#The Characteristic Polynomial">The Characteristic Polynomial</a> | <a href="#The Laplace Spectrum">The Laplace Spectrum</a> | <a href="#The Genus">The Genus</a> | <a href="#The Automorphism Group">The Automorphism Group</a> | <a href="#Examples">Examples</a>

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<br />

<h2 id="toc9"><a name="Examples-The dekany"></a>The dekany</h2>

The standard 2)5 dekany is a <a class="wiki_link" href="/Combination%20product%20sets">combination product set</a>, Cps([2,3,5,7,11], 2). It consists of ten notes associated to two-element subset of the set of the first five primes, {2,3,5,7,11}, and in one mode is 12/11-5/4-14/11-15/11-3/2-35/22-7/4-20/11-21/11, which we will take as its notes from note 0 to note 9. It has 30 edges, with connectivities 5 ≤ 6 ≤ 6, and the largest element of the Laplace spectrum is 8, so that the complementary graph is also connected. Its radius and diameter are both 2.<br />

The standard 2)5 dekany is a <a class="wiki_link" href="/Combination%20product%20sets">combination product set</a>, Cps([2,3,5,7,11], 2). It consists of ten notes associated to two-element subset of the set of the first five primes, {2,3,5,7,11}, and in one mode is 12/11-5/4-14/11-15/11-3/2-35/22-7/4-20/11-21/11, which we will take as its notes from note 0 to note 9. It has 30 edges, with connectivities 5 ≤ 6 ≤ 6, and the largest element of the Laplace spectrum is 8, so that the complementary graph is also connected. Its radius and diameter are both 2. The graph is known as the <a class="wiki_link" href="/Johnson%20graph">Johnson graph</a> J(5,2). The 3)5 dekany, which is the inverse of the standard dekany, has the Johnson graph J(5,3) as its graph, which is graph-isomorphic to J(5,2).<br />

<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 />

<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 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.~~</body></html></pre></div>

<img src="/file/view/dekany.png/359810971/dekany.png" alt="dekany.png" title="dekany.png" /></body></html></pre></div>

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 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-23 14:31:26 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-08-23 17:22:48 UTC</tt>.<br>
-: The original revision id was <tt>359498985</tt>.<br>
+: The original revision id was <tt>359531781</tt>.<br>
 : The revision comment was: <tt></tt><br>
 The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
@@ Line 70: / Line 70: @@
 ==The dekany==
-The standard 2)5 dekany is a [[Combination product sets|combination product set]], Cps([2,3,5,7,11], 2). It consists of ten notes associated to two-element subset of the set of the first five primes, {2,3,5,7,11}, and in one mode is 12/11-5/4-14/11-15/11-3/2-35/22-7/4-20/11-21/11, which we will take as its notes from note 0 to note 9. It has 30 edges, with connectivities 5 ≤ 6 ≤ 6, and the largest element of the Laplace spectrum is 8, so that the complementary graph is also connected. Its radius and diameter are both 2.
+The standard 2)5 dekany is a [[Combination product sets|combination product set]], Cps([2,3,5,7,11], 2). It consists of ten notes associated to two-element subset of the set of the first five primes, {2,3,5,7,11}, and in one mode is 12/11-5/4-14/11-15/11-3/2-35/22-7/4-20/11-21/11, which we will take as its notes from note 0 to note 9. It has 30 edges, with connectivities 5 ≤ 6 ≤ 6, and the largest element of the Laplace spectrum is 8, so that the complementary graph is also connected. Its radius and diameter are both 2. The graph is known as the [[Johnson graph]] J(5,2). The 3)5 dekany, which is the inverse of the standard dekany, has the Johnson graph J(5,3) as its graph, which is graph-isomorphic to J(5,2).
-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 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.</pre></div>
+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.
+[[image:dekany.png]]</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Graph-theoretic properties of scales&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextTocRule:20:&amp;lt;img id=&amp;quot;wikitext@@toc@@flat&amp;quot; class=&amp;quot;WikiMedia WikiMediaTocFlat&amp;quot; title=&amp;quot;Table of Contents&amp;quot; src=&amp;quot;/site/embedthumbnail/toc/flat?w=100&amp;amp;h=16&amp;quot;/&amp;gt; --&gt;&lt;!-- ws:end:WikiTextTocRule:20 --&gt;&lt;!-- ws:start:WikiTextTocRule:21: --&gt;&lt;a href="#Graph of a scale"&gt;Graph of a scale&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:21 --&gt;&lt;!-- ws:start:WikiTextTocRule:22: --&gt; | &lt;a href="#Connectivity"&gt;Connectivity&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:22 --&gt;&lt;!-- ws:start:WikiTextTocRule:23: --&gt; | &lt;a href="#The Characteristic Polynomial"&gt;The Characteristic Polynomial&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:23 --&gt;&lt;!-- ws:start:WikiTextTocRule:24: --&gt; | &lt;a href="#The Laplace Spectrum"&gt;The Laplace Spectrum&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:24 --&gt;&lt;!-- ws:start:WikiTextTocRule:25: --&gt; | &lt;a href="#The Genus"&gt;The Genus&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:25 --&gt;&lt;!-- ws:start:WikiTextTocRule:26: --&gt; | &lt;a href="#The Automorphism Group"&gt;The Automorphism Group&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:26 --&gt;&lt;!-- ws:start:WikiTextTocRule:27: --&gt; | &lt;a href="#Examples"&gt;Examples&lt;/a&gt;&lt;!-- ws:end:WikiTextTocRule:27 --&gt;&lt;!-- ws:start:WikiTextTocRule:28: --&gt;&lt;!-- ws:end:WikiTextTocRule:28 --&gt;&lt;!-- ws:start:WikiTextTocRule:29: --&gt;&lt;!-- ws:end:WikiTextTocRule:29 --&gt;&lt;!-- ws:start:WikiTextTocRule:30: --&gt;&lt;!-- ws:end:WikiTextTocRule:30 --&gt;&lt;!-- ws:start:WikiTextTocRule:31: --&gt;
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 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:18:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc9"&gt;&lt;a name="Examples-The dekany"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:18 --&gt;The dekany&lt;/h2&gt;
-The standard 2)5 dekany is a &lt;a class="wiki_link" href="/Combination%20product%20sets"&gt;combination product set&lt;/a&gt;, Cps([2,3,5,7,11], 2). It consists of ten notes associated to two-element subset of the set of the first five primes, {2,3,5,7,11}, and in one mode is 12/11-5/4-14/11-15/11-3/2-35/22-7/4-20/11-21/11, which we will take as its notes from note 0 to note 9. It has 30 edges, with connectivities 5 ≤ 6 ≤ 6, and the largest element of the Laplace spectrum is 8, so that the complementary graph is also connected. Its radius and diameter are both 2.&lt;br /&gt;
+The standard 2)5 dekany is a &lt;a class="wiki_link" href="/Combination%20product%20sets"&gt;combination product set&lt;/a&gt;, Cps([2,3,5,7,11], 2). It consists of ten notes associated to two-element subset of the set of the first five primes, {2,3,5,7,11}, and in one mode is 12/11-5/4-14/11-15/11-3/2-35/22-7/4-20/11-21/11, which we will take as its notes from note 0 to note 9. It has 30 edges, with connectivities 5 ≤ 6 ≤ 6, and the largest element of the Laplace spectrum is 8, so that the complementary graph is also connected. Its radius and diameter are both 2. The graph is known as the &lt;a class="wiki_link" href="/Johnson%20graph"&gt;Johnson graph&lt;/a&gt; J(5,2). The 3)5 dekany, which is the inverse of the standard dekany, has the Johnson graph J(5,3) as its graph, which is graph-isomorphic to J(5,2).&lt;br /&gt;
+&lt;br /&gt;
+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.&lt;br /&gt;
 &lt;br /&gt;
-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 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.&lt;/body&gt;&lt;/html&gt;</pre></div>
+&lt;!-- ws:start:WikiTextLocalImageRule:35:&amp;lt;img src=&amp;quot;/file/view/dekany.png/359810971/dekany.png&amp;quot; alt=&amp;quot;&amp;quot; title=&amp;quot;&amp;quot; /&amp;gt; --&gt;&lt;img src="/file/view/dekany.png/359810971/dekany.png" alt="dekany.png" title="dekany.png" /&gt;&lt;!-- ws:end:WikiTextLocalImageRule:35 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>