Hudde's rules: Difference between revisions

Latest revision as of 19:27, 7 July 2013

In the mathematical field of graph theory, the Folkman graph, named after Jon Folkman, is a bipartite 4-regular graph with 20 vertices and 40 edges.^[1]

The Folkman graph is Hamiltonian and has chromatic number 2, chromatic index 4, radius 3, diameter 4 and girth 4. It is also a 4-vertex-connected and 4-edge-connected perfect graph.

Algebraic properties

The automorphism group of the Folkman graph acts transitively on its edges but not on its vertices. It is the smallest undirected graph that is edge-transitive and regular, but not vertex-transitive.^[2] Such graphs are called semi-symmetric graphs and were first studied by Folkman in 1967 who discovered the graph on 20 vertices that now is named after him.^[3]

As a semi-symmetric graph, the Folkman graph is bipartite, and its automorphism group acts transitively on each of the two vertex sets of the bipartition. In the diagram below indicating the chromatic number of the graph, the green vertices can not be mapped to red ones by any automorphism, but any red vertex can be mapped on any other red vertex and any green vertex can be mapped on any other green vertex.

The characteristic polynomial of the Folkman graph is $(x - 4) x^{10} (x + 4) (x^{2} - 6)^{4}$ .

Gallery

The chromatic index of the Folkman graph is 4.
The chromatic number of the Folkman graph is 2.
The Folkman graph is Hamiltonian.

References

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↑ Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 186-187, 1990
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[1] 

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[2] Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 186-187, 1990

[3] Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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[2]

[3]

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+{{infobox graph
+ | name = Folkman graph
+ | image = [[Image:Folkman graph alt.svg|220px]]
+ | image_caption = The Folkman graph
+ | namesake = J. Folkman
+ | vertices = 20
+ | edges = 40
+ | girth = 4
+ | diameter = 4
+ | radius = 3
+ | chromatic_number = 2
+ | chromatic_index = 4
+ | properties = [[Hamiltonian graph|Hamiltonian]]<br>[[regular graph|Regular]]<br>[[Bipartite graph|Bipartite]]<br>[[semi-symmetric graph|Semi-symmetric]]<br>[[Eulerian graph|Eulerian]]<br>[[perfect graph|Perfect]]
+}}
+In the [[mathematics|mathematical]] field of [[graph theory]], the '''Folkman graph''', named after [[Jon Folkman]], is a [[Bipartite graph|bipartite]] 4-[[regular graph|regular]] graph with 20 [[vertex (graph theory)|vertices]] and 40 edges.<ref>{{MathWorld|title=Folkman graph|urlname=FolkmanGraph}}</ref>
+The Folkman graph is [[Hamiltonian graph|Hamiltonian]] and has [[chromatic number]] 2, [[chromatic index]] 4, radius 3, diameter 4 and [[girth (graph theory)|girth]]&nbsp;4. It is also a 4-[[k-vertex-connected graph|vertex-connected]] and 4-[[k-edge-connected graph|edge-connected]] [[perfect graph]].
+==Algebraic properties==
+The [[automorphism group]] of the Folkman graph acts transitively on its edges but not on its vertices.  It is the smallest undirected graph that is [[edge-transitive graph|edge-transitive]] and regular, but not [[vertex-transitive graph|vertex-transitive]].<ref>Skiena, S. Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 186-187, 1990</ref> Such graphs are called [[semi-symmetric graph]]s and were first studied by Folkman in 1967 who discovered the  graph on 20 vertices that now is named after him.<ref>{{citation
+ | first = J. | last = Folkman | authorlink=Jon Folkman
+ | title = Regular line-symmetric graphs
+ | journal = Journal of Combinatorial Theory
+ | volume =3 | year = 1967
+ | issue = 3 | pages = 215–232
+ | doi = 10.1016/S0021-9800(67)80069-3}}</ref>
+As a semi-symmetric graph, the Folkman graph is [[bipartite graph|bipartite]], and its automorphism group acts transitively on each of the two vertex sets of the bipartition. In the diagram below indicating the chromatic number of the graph, the green vertices can not be mapped to red ones by any automorphism, but any red vertex can be mapped on any other red vertex and any green vertex can be mapped on any other green vertex.
+The [[characteristic polynomial]] of the Folkman graph is <math>(x-4) x^{10} (x+4) (x^2-6)^4</math>.
+==Gallery==
+<gallery>
+Image:Folkman graph 4color edge.svg|The [[chromatic index]] of the Folkman graph is&nbsp;4.
+Image:Folkman graph.svg|The [[chromatic number]] of the Folkman graph is&nbsp;2.
+File:Folkman Lombardi.svg|The Folkman graph is [[Hamiltonian graph|Hamiltonian]].
+</gallery>
+== References ==
+{{reflist}}
+[[Category:Individual graphs]]
+[[Category:Regular graphs]]

Hudde's rules: Difference between revisions

Latest revision as of 19:27, 7 July 2013

Algebraic properties

Gallery

References

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