Hasse's theorem on elliptic curves: Difference between revisions

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{{infobox graph
== The strength can be imagined ==
| name = Higman–Sims graph
| image = [[Image:Higman Sims Graph.svg|220px]]
| image_caption = Drawing based on Paul R. Hafner's construction.<ref>{{Cite journal
| last1 = Hafner | first1 = P. R.
| title = [http://www.combinatorics.org/Volume_11/PDF/v11i1r77.pdf On the Graphs of Hoffman&ndash;Singleton and Higman&ndash;Sims]
| journal = [http://www.combinatorics.org the Electronic Journal of Combinatorics]
| volume = 11 | issue = 1 | year = 2004 | pages = R77(1–32)
| postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}
}}.</ref>
| namesake = [[Donald G. Higman]]<br/>[[Charles C. Sims]]
| vertices = 100
| edges = 1100
| radius = 2
| diameter = 2
| girth = 4
| automorphisms = {{formatnum:88704000}} ([[Higman–Sims group|HS]]:2)
| chromatic_number =
| chromatic_index =
| properties = [[Strongly regular graph|Strongly regular]]<br>[[Edge-transitive graph|Edge-transitive]]<br>[[Hamiltonian graph|Hamiltonian]]<br>[[Eulerian graph|Eulerian]]
}}


[[Image:Higman Sims Graph Parts.svg|220px|right|thumb|
, The strength can be imagined, but there is a man behind Qin Yu Lan Shu unfathomable ground.<br>Court<br>stars out of four, Qin Yu and Li child girl, and love the クリスチャンルブタン 偽物 lively Hou fee, plus black Qin Yu to follow.<br>After<br>Green Dragon Palace, clear water House, celestial domain, nine evil Hall, Zi Yan Fel クリスチャンルブタン サイズ is out four or five people, followed クリスチャンルブタン 店舗 by eight to feijian speed flight, and that eight out of the stars of the feijian Court, even fly directly toward the north to go.<br><br>'also stars Court turned north, is really amazing.' Dragon laughed.<br><br>now tied with eight feijian flight, is behind dozens of people, which in addition to dozens of people standing child girl, every other individual's real strength, at least over all virtual hole late.<br><br>Qin Yu Sword flight, クリスチャンルブタン セール and クリスチャンルブタン 値段 even children stand aside Royal Air flight speed was easy to follow the crowd.<br><br>'Li children, your speed ......' Qin Yu stunned.<br><br>Royal Air flight, this is slower than the Sword flight on many, but Li child saver, even Royal Air flight.
The separated parts of Hafner's construction.]]
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In mathematical [[graph theory]], the '''Higman–Sims graph''' is a 22-[[regular graph|regular]] [[undirected graph]] with 100 vertices and 1100 edges. It is the unique [[strongly regular graph]] with 100 vertices and valency 22, where no neighboring pair of vertices share a common neighbor and each non-neighboring pair of vertices share six common neighbors.<ref>{{MathWorld|urlname=Higman-SimsGraph|title=Higman&ndash;Sims Graph}}</ref> It was first constructed by {{harvtxt|Mesner|1956}} and rediscovered in 1968 by Donald G. Higman and Charles C. Sims as a way to define the [[Higman–Sims group]], and that group is a subgroup of [[Index of a subgroup|index]] two in the group of automorphisms of the Higman–Sims graph.<ref>{{Cite journal
== Forber come ==
| last1 = Higman | first1 = Donald G.
| last2 = Sims | first2 = Charles C. | author2-link = Charles Sims (mathematician)
| title = A simple group of order 44,352,000
| journal = [[Mathematische Zeitschrift]]
| volume = 105 | issue = 2 | year = 1968 | pages = 110–113
| doi = 10.1007/BF01110435
| postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.</ref>


Construction begins with the '''M<sub>22</sub> graph''', whose 77 vertices are the blocks of the S(3,6,22) [[Steiner system]] W<sub>22</sub>. Adjacent vertices are defined to be disjoint blocks. This graph is [[strongly regular graph|strongly regular]]; any vertex has 16 neighbors, any 2 adjacent vertices have no common neighbors, and any 2 non-adjacent vertices have 4 common neighbors. This graph has M<sub>22</sub>:2 as its automorphism group, M<sub>22</sub> being a [[Mathieu group]].
Its power, because it was complicated to a limit, so is the use of some low-grade rock gods lineup to be testing ground. '<br><br>'You mean, I should be FIGHTING test? how that test it?' Qin Yu heart began to think again.<br><br>black feather? Blue House? クリスチャンルブタン セール Wu He? Let them feel FIGHTING.<br><br>not!<br><br>have too much power in battle formation, three seriously injured lost their lives how to do, クリスチャンルブタン メンズ 通販 but other people can not waste time, after all, they are also to cultivate black ground.<br><br>Forber come?<br><br>Forber no soul, the soul of some attacks is essentially useless, クリスチャンルブタン 銀座 can not test FIGHTING power.<br><br>own experience FIGHTING power?<br><br>does not work, in the community of their own, although Jiang Lan can unscathed, if the force Jiang クリスチャンルブタン パンプス Lan sector クリスチャンルブタン アウトレット can use to offset the attack space, then how test FIGHTING power it?<br><br>'Master, it is very simple, and the old master science, casually looking for a robber forces gathering in the square, you arranged at some of the parties
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The Higman–Sims graph is then formed by appending the 22 points of W<sub>22</sub> and a 100th vertex C. The neighbors of C are defined to be those 22 points. A point adjacent to a block is defined to be one that is included.
== Yu Huang to think so ==


A Higman–Sims graph can be partitioned into two copies of the [[Hoffman&ndash;Singleton graph]] in 352 ways.
Ten meters away completely negligible, exudes a sharp Qijin between two parties troops, opposition to each other.<br><br>'Qin Yu, you reach nine cents Emperor of?' Yu Huang little face suddenly changed, and he found himself only a vague feeling Qin Yu's strength, spirit realm and he [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_13.php クリスチャンルブタン 日本] seemed almost.<br><br>'nine cents Emperor, be it.' Qin Yu looked Yu Huang said.<br><br>Yu Huang to think so, Qin Yu would not be denied.<br><br>'how could that be? then reef yellow star battle you, but only reached level Emperor Xian, [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン 店舗] just open the third layer beasts spectrum can now ...... this [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_11.php クリスチャンルブタン ブーツ] for hundreds of years? you ......' Yu Wong and his face slightly pale .<br><br>insufficient millennium, reached nine cents Emperor, this is really horrible thing enough.<br><br>Qin Yu just smiling.<br><br>Qin Yu he did not say any of this cranky Yu Huang, Yu Huang Qin Yu soul realm and fairly. Qin Yu could not clearly see through, this [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン 店舗 東京] has been [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_11.php クリスチャンルブタン 偽物] a series of speculation.<br><br>'Hey, you're the Yu Huang?' Hou Fei evil ruthless
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==Algebraic properties==
== and had to use the horn. Qin Yu relatively similar. ==
The [[automorphism group]] of the Higman–Sims graph is a group of order {{formatnum:88704000}} isomorphic to the [[semidirect product]] of the [[Higman–Sims group]] of order {{formatnum:44352000}} with the [[cyclic group]] of order 2.<ref>{{cite web
  | url = http://www.win.tue.nl/~aeb/drg/graphs/Higman-Sims.html
  | title = Higman&ndash;Sims graph
  | authorlink = Andries E. Brouwer
  | author = Brouwer, Andries E.}}</ref> It has automorphisms that take any edge to any other edge, making the Higman–Sims graph an [[edge-transitive graph]].<ref>Brouwer, A. E. and Haemers, W. H. "The Gewirtz Graph: An Exercise in the Theory of Graph Spectra." Euro. J. Combin. 14, 397&ndash;407, 1993.</ref>


The characteristic polynomial of the Higman–Sims graph is (''x''&nbsp;&minus;&nbsp;22)(''x''&nbsp;&minus;&nbsp;2)<sup>77</sup>(''x''&nbsp;+&nbsp;8)<sup>22</sup>. Therefore the Higman–Sims graph is an [[integral graph]]: its [[Spectral graph theory|spectrum]] consists entirely of integers. It is also the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
Once [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_10.php クリスチャンルブタン 銀座] upon a time people did not create. 'Qin Yu heart sigh.<br><br>received this Excalibur 'Happening', Qin Yu know \u003cHappening jianjue\u003e simply refining [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン 店舗] Zhebing Excalibur year by people left in the above.<br><br>'such a long sword, I can not accustomed to using, or short some good.' Qin Yu mind a move sharply reduced Zhebing sword lengths, becoming only three inches long from three feet three inches in length, into a handle dagger, and had to [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_10.php クリスチャンルブタン 銀座] use the horn. Qin Yu relatively similar.<br><br>this Excalibur 'Happening' revenue body, Qin Yu also took the robe armor.<br><br>'jersey name 'black Ningxue. was beneath contempt artifact level. Sword physical defense is not strong enough, when I had this level of defense artifact treasures. cooperate' Happening 'sharp attack and I was fully achieved fairy magic demon world [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_10.php クリスチャンルブタン 直営店] the pinnacle. 'Emperor cents against the central voice [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン 店舗] sounded in QINFA mind.<br><br>Qin Yu smiled, but also identify the main blood up.<br><br>when it absorbed a drop of blood after Qin Yu, this artifact armor actually emit little white brilliance, it is like snowflakes attack in Jersey
 
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==Inside the Leech lattice==
<ul>
[[Image:Higman-Sims-19.svg|220px|right|thumb|
 
A projection of the Higman–Sims graph inside the Leech lattice.]]
  <li>[http://lutongning.com/plus/feedback.php?aid=1 http://lutongning.com/plus/feedback.php?aid=1]</li>
The Higman–Sims graph naturally occurs inside the [[Leech lattice]]: if ''X'', ''Y'' and ''Z'' are three points in the Leech lattice such that the distances ''XY'', ''XZ'' and ''YZ'' are <math>2, \sqrt{6}, \sqrt{6}</math> respectively, then there are exactly 100 Leech lattice points ''T'' such that all the distances ''XT'', ''YT'' and ''ZT'' are equal to 2, and if we connect two such points ''T'' and ''T''&prime; when the distance between them is <math> \sqrt{6} </math>, the resulting graph is isomorphic to the Higman–Sims graph. Furthermore, the set of all automorphisms of the Leech lattice (that is, Euclidean congruences fixing it) which fix each of ''X'', ''Y'' and ''Z'' is the Higman–Sims group (if we allow exchanging ''X'' and ''Y'', the order 2 extension of all graph automorphisms is obtained). This shows that the Higman–Sims group occurs inside the [[Conway group]]s Co<sub>2</sub> (with its order 2 extension) and Co<sub>3</sub>, and consequently also Co<sub>1</sub>.<ref>{{Cite book | last1=Conway | first1=John H. | authorlink1=John Horton Conway | last2=Sloane | first2=Neil J. A. | authorlink2=Neil Sloane | title=Sphere Packings, Lattices and Groups | edition=3rd | series=Grundlehren der mathematischen Wissenschaften | isbn=1-4419-3134-1 | publisher=[[Springer-Verlag]] }} chapter 10 (John H. Conway, &ldquo;Three Lectures on Exceptional Groups&rdquo;), §3.5 (&ldquo;The Higman–Sims and McLaughlin groups&rdquo;), p. 292&ndash;293.</ref>
 
 
  <li>[http://www.aotaier.com/plus/view-1811-1.html http://www.aotaier.com/plus/view-1811-1.html]</li>
== References ==
 
{{reflist}}
  <li>[http://www.jitaweb.com/plus/feedback.php?aid=158 http://www.jitaweb.com/plus/feedback.php?aid=158]</li>
 
 
*{{Citation | last1=Mesner | first1=Dale Marsh | title=An investigation of certain combinatorial properites of partially balanced incomplete block experimental designs and association schemes, with a detailed study of designs of Latin square and related types | publisher=Department of Statistics, Michigan State University | series=Doctoral Thesis | id={{MR|2612633}} | year=1956}}
</ul>
 
{{DEFAULTSORT:Higman-Sims graph}}
[[Category:Group theory]]
[[Category:Individual graphs]]
[[Category:Regular graphs]]

Revision as of 02:34, 5 March 2014

The strength can be imagined

, The strength can be imagined, but there is a man behind Qin Yu Lan Shu unfathomable ground.
Court
stars out of four, Qin Yu and Li child girl, and love the クリスチャンルブタン 偽物 lively Hou fee, plus black Qin Yu to follow.
After
Green Dragon Palace, clear water House, celestial domain, nine evil Hall, Zi Yan Fel クリスチャンルブタン サイズ is out four or five people, followed クリスチャンルブタン 店舗 by eight to feijian speed flight, and that eight out of the stars of the feijian Court, even fly directly toward the north to go.

'also stars Court turned north, is really amazing.' Dragon laughed.

now tied with eight feijian flight, is behind dozens of people, which in addition to dozens of people standing child girl, every other individual's real strength, at least over all virtual hole late.

Qin Yu Sword flight, クリスチャンルブタン セール and クリスチャンルブタン 値段 even children stand aside Royal Air flight speed was easy to follow the crowd.

'Li children, your speed ......' Qin Yu stunned.

Royal Air flight, this is slower than the Sword flight on many, but Li child saver, even Royal Air flight. 相关的主题文章：

  • ?aid=3
  • ?title=User:Awbiwgwyvl#like_ants_in_his_hands

Forber come

Its power, because it was complicated to a limit, so is the use of some low-grade rock gods lineup to be testing ground. '

'You mean, I should be FIGHTING test? how that test it?' Qin Yu heart began to think again.

black feather? Blue House? クリスチャンルブタン セール Wu He? Let them feel FIGHTING.

not!

have too much power in battle formation, three seriously injured lost their lives how to do, クリスチャンルブタン メンズ 通販 but other people can not waste time, after all, they are also to cultivate black ground.

Forber come?

Forber no soul, the soul of some attacks is essentially useless, クリスチャンルブタン 銀座 can not test FIGHTING power.

own experience FIGHTING power?

does not work, in the community of their own, although Jiang Lan can unscathed, if the force Jiang クリスチャンルブタン パンプス Lan sector クリスチャンルブタン アウトレット can use to offset the attack space, then how test FIGHTING power it?

'Master, it is very simple, and the old master science, casually looking for a robber forces gathering in the square, you arranged at some of the parties 相关的主题文章：

  • ?mod=space&uid=122850
  • ?mod=space&uid=55276
  • ?mod=space&uid=869039

Yu Huang to think so

Ten meters away completely negligible, exudes a sharp Qijin between two parties troops, opposition to each other.

'Qin Yu, you reach nine cents Emperor of?' Yu Huang little face suddenly changed, and he found himself only a vague feeling Qin Yu's strength, spirit realm and he クリスチャンルブタン 日本 seemed almost.

'nine cents Emperor, be it.' Qin Yu looked Yu Huang said.

Yu Huang to think so, Qin Yu would not be denied.

'how could that be? then reef yellow star battle you, but only reached level Emperor Xian, クリスチャンルブタン 店舗 just open the third layer beasts spectrum can now ...... this クリスチャンルブタン ブーツ for hundreds of years? you ......' Yu Wong and his face slightly pale .

insufficient millennium, reached nine cents Emperor, this is really horrible thing enough.

Qin Yu just smiling.

Qin Yu he did not say any of this cranky Yu Huang, Yu Huang Qin Yu soul realm and fairly. Qin Yu could not clearly see through, this クリスチャンルブタン 店舗 東京 has been クリスチャンルブタン 偽物 a series of speculation.

'Hey, you're the Yu Huang?' Hou Fei evil ruthless 相关的主题文章：

and had to use the horn. Qin Yu relatively similar.

Once クリスチャンルブタン 銀座 upon a time people did not create. 'Qin Yu heart sigh.

received this Excalibur 'Happening', Qin Yu know \u003cHappening jianjue\u003e simply refining クリスチャンルブタン 店舗 Zhebing Excalibur year by people left in the above.

'such a long sword, I can not accustomed to using, or short some good.' Qin Yu mind a move sharply reduced Zhebing sword lengths, becoming only three inches long from three feet three inches in length, into a handle dagger, and had to クリスチャンルブタン 銀座 use the horn. Qin Yu relatively similar.

this Excalibur 'Happening' revenue body, Qin Yu also took the robe armor.

'jersey name 'black Ningxue. was beneath contempt artifact level. Sword physical defense is not strong enough, when I had this level of defense artifact treasures. cooperate' Happening 'sharp attack and I was fully achieved fairy magic demon world クリスチャンルブタン 直営店 the pinnacle. 'Emperor cents against the central voice クリスチャンルブタン 店舗 sounded in QINFA mind.

Qin Yu smiled, but also identify the main blood up.

when it absorbed a drop of blood after Qin Yu, this artifact armor actually emit little white brilliance, it is like snowflakes attack in Jersey 相关的主题文章：

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