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| {{infobox graph
| | They are typically a free website that are pre-designed for enabling businesses of every size in marking the presence on the internet and allows them in showcasing the product services and range through images, contents and various other elements. Good luck on continue learning how to make a wordpress website. Step-4 Testing: It is the foremost important of your Plugin development process. After confirming the account, login with your username and password at Ad - Mob. By using this method one can see whether the theme has the potential to become popular or not and is their any scope of improvement in the theme. <br><br> |
| | name = Herschel graph
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| | image = [[File:Herschel graph LS.svg|220px]]
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| | image_caption = The Herschel graph.
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| | namesake = [[Alexander Stewart Herschel]]
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| | vertices = 11
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| | edges = 18
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| | automorphisms = 12 ([[Dihedral group|D]]<sub>6</sub>)
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| | girth = 4
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| | diameter = 4
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| | radius = 3
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| | chromatic_number = 2
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| | chromatic_index = 4
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| | properties = [[polyhedral graph|Polyhedral]]<br>[[planar graph|Planar]]<br>[[bipartite graph|Bipartite]]<br>[[Perfect graph|Perfect]]
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| }}
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| In [[graph theory]], a branch of [[mathematics]], the '''Herschel graph''' is a [[bipartite graph|bipartite]] [[undirected graph]] with 11 vertices and 18 edges, the smallest [[Hamiltonian graph|non-Hamiltonian]] [[polyhedral graph|polyhedral]] graph. It is named after British astronomer [[Alexander Stewart Herschel]], who wrote an early paper concerning [[William Rowan Hamilton]]'s [[icosian game]]: the Herschel graph describes the smallest [[convex polyhedron]] for which this game has no solution. However, Herschel's paper described solutions for the Icosian game only on the graphs of the [[regular tetrahedron]] and [[regular icosahedron]]; it did not describe the Herschel graph.<ref>{{citation|last=Herschel|first=A. S.|authorlink=Alexander Stewart Herschel|title=Sir Wm. Hamilton's Icosian Game|url=http://books.google.com/books?id=-w8LAAAAYAAJ&pg=PA305|journal=[[The Quarterly Journal of Pure and Applied Mathematics]]|volume=5|page=305|year=1862}}.</ref>
| | These websites can be easily customized and can appear in the top rankings of the major search engines. Infertility can cause a major setback to the couples due to the inability to conceive. Several claim that Wordpress just isn't an preferred tool to utilise when developing a professional site. If you liked this article and you simply would like to get more info relating to [http://shortener.us/wordpress_dropbox_backup_4387132 wordpress backup] generously visit our site. You can up your site's rank with the search engines by simply taking a bit of time with your site. For a Wordpress website, you don't need a powerful web hosting account to host your site. <br><br>Photography is an entire activity in itself, and a thorough discovery of it is beyond the opportunity of this content. Word - Press has ensured the users of this open source blogging platform do not have to troubleshoot on their own, or seek outside help. Those who cannot conceive with donor eggs due to some problems can also opt for surrogacy option using the services of surrogate mother. You or your web designer can customize it as per your specific needs. There are plenty of tables that are attached to this particular database. <br><br>Digg Digg Social Sharing - This plugin that is accountable for the floating social icon located at the left aspect corner of just about every submit. This plugin allows a webmaster to create complex layouts without having to waste so much time with short codes. Thus it is difficult to outrank any one of these because of their different usages. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL. <br><br>Website security has become a major concern among individuals all over the world. An ease of use which pertains to both internet site back-end and front-end users alike. By the time you get the Gallery Word - Press Themes, the first thing that you should know is on how to install it. ) Remote Login: With the process of PSD to Wordpress conversion comes the advantage of flexibility. Get started today so that people searching for your type of business will be directed to you. |
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| ==Properties==
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| The Herschel graph is a [[planar graph]]: it can be drawn in the plane with none of its edges crossing. It is also [[vertex connectivity|3-vertex-connected]]: the removal of any two of its vertices leaves a connected [[Glossary of graph theory#Subgraphs|subgraph]]. Therefore, by [[Steinitz's theorem]], the Herschel graph is a [[polyhedral graph]]: there exists a convex polyhedron (an [[enneahedron]]) having the Herschel graph as its [[skeleton (topology)|skeleton]].<ref name="Coxeter">{{citation|title=[[Regular Polytopes (book)|Regular Polytopes]]|first=H. S. M.|last=Coxeter|authorlink=Harold Scott MacDonald Coxeter|publisher=Dover|year=1973|page=8}}.</ref>
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| The Herschel graph is also a [[bipartite graph]]: its vertices can be separated into two subsets of five and six vertices respectively, such that every edge has an endpoint in each subset (the red and blue subsets in the picture).
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| As with any bipartite graph, the Herschel graph is a [[perfect graph]] : the [[chromatic number]] of every [[induced subgraph]] equals the size of the largest [[Glossary of graph theory#Cliques|clique]] of that subgraph. It has also [[chromatic index]] 4, girth 4, radius 3 and diameter 4.
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| ==Hamiltonicity==
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| Because it is a bipartite graph that has an odd number of vertices, the Herschel graph does not contain a [[Hamiltonian cycle]] (a cycle of edges that passes through each vertex exactly once). For, in any bipartite graph, any cycle must alternate between the vertices on either side of the bipartition, and therefore must contain equal numbers of both types of vertex and must have an even length. Thus, a cycle passing once through each of the eleven vertices cannot exist in the Herschel graph. It is the smallest non-Hamiltonian polyhedral graph, whether the size of the graph is measured in terms of its number of vertices, edges, or faces;<ref>{{citation|title=Hamiltonian circuits on 3-polytopes|first1=David|last1=Barnette|first2=Ernest|last2=Jucovič|journal=Journal of Combinatorial Theory|volume=9|issue=1|year=1970|pages=54–59|doi=10.1016/S0021-9800(70)80054-0}}.</ref> there exist other polyhedral graphs with 11 vertices and no Hamiltonian cycles (notably the [[Goldner–Harary graph]]<ref>{{mathworld|title=Goldner-Harary Graph|urlname=Goldner-HararyGraph}}.</ref>) but none with fewer edges.<ref name="Coxeter"/>
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| All but three of the vertices of the Herschel graph have degree three; [[Tait's conjecture]]<ref>{{citation|first=P. G.|last=Tait|authorlink=P. G. Tait|title=Listing's ''Topologie''|url=http://books.google.com/books?id=8VMwAAAAIAAJ&pg=PA30|journal=[[Philosophical Magazine]] (5th ser.)|volume=17|year=1884|pages=30–46}}. Reprinted in ''Scientific Papers'', Vol. II, pp. 85–98.</ref> states that a polyhedral graph in which [[cubic graph|every vertex has degree three]] must be Hamiltonian, but this was disproved when [[W. T. Tutte]] provided a counterexample, the much larger [[Tutte graph]].<ref>{{citation|first=W. T.|last=Tutte|authorlink=W. T. Tutte|title=On Hamiltonian circuits|year=1946|url=http://jlms.oxfordjournals.org/cgi/reprint/s1-21/2/98.pdf | doi = 10.1112/jlms/s1-21.2.98|journal=Journal of the London Mathematical Society|volume=21|issue=2|pages=98–101}}.</ref> A refinement of Tait's conjecture, [[Barnette's conjecture]] that every bipartite 3-regular polyhedral graph is Hamiltonian, remains open.<ref>{{citation|url=http://garden.irmacs.sfu.ca/?q=op/barnettes_conjecture |title=Barnette's conjecture|publisher=the Open Problem Garden|first=Robert |last=Samal
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| |date=11 June 2007| accessdate=24 Feb 2011}}</ref>
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| The Herschel graph also provides an example of a polyhedral graph for which the [[medial graph]] cannot be decomposed into two edge-disjoint Hamiltonian cycles. The medial graph of the Herschel graph is a 4-[[regular graph]] with 18 vertices, one for each edge of the Herschel graph; two vertices are adjacent in the medial graph whenever the corresponding edges of the Herschel graph are consecutive on one of its faces.<ref>{{citation|title= Edge-disjoint Hamilton cycles in 4-regular planar graphs|first1=J. A.|last1=Bondy|first2=R.|last2=Häggkvist|journal=Aequationes Mathematicae|volume=22|issue=1|year=1981|pages=42–45|doi=10.1007/BF02190157}}.</ref>
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| ==Algebraic properties==
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| The Herschel graph is not a [[vertex-transitive graph]] and its full automorphism group is isomorphic to the [[dihedral group]] of order 12, the group of symmetries of a [[regular hexagon]], including both rotations and reflections. Every permutation of its three degree-four vertices can be realized by an automorphism of the graph, and there also exists a nontrivial automorphism that exchanges the remaining vertices while leaving the degree-four vertices unchanged.
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| The [[characteristic polynomial]] of the Herschel graph is <math>-x^3 (x^2-11) (x^2-3) (x^2-2)^2</math>.
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| ==References==
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| {{reflist}}
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| ==External links==
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| *{{mathworld|title=Herschel Graph|urlname=HerschelGraph}}
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| [[Category:Individual graphs]]
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| [[Category:Planar graphs]]
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| [[Category:Hamiltonian paths and cycles]]
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