Quantum dissipation: Difference between revisions
Jump to navigation
Jump to search
en>Ybband correction of the formula for the bath spectral function |
en>Crown Prince Disambiguated: path integral → path integral formulation |
||
| Line 1: | Line 1: | ||
{{infobox graph | |||
| name = Wagner graph | |||
| image = [[File:Wagner graph ham.svg|220px]] | |||
| image_caption = The Wagner graph | |||
| namesake = [[Klaus Wagner]] | |||
| vertices = 8 | |||
| edges = 12 | |||
| automorphisms = 16 ([[dihedral group|D]]<sub>8</sub>) | |||
| girth = 4 | |||
| diameter = 2 | |||
| radius = 2 | |||
| chromatic_number = 3 | |||
| chromatic_index = 3 | |||
| notation = ''M''<sub>''8''</sub> | |||
| properties = [[Cubic graph|Cubic]]<br>[[Hamiltonian graph|Hamiltonian]]<br>[[Triangle-free graph|Triangle-free]]<br>[[Vertex-transitive graph|Vertex-transitive]]<br>[[Toroidal graph|Toroidal]]<br>[[Apex graph|Apex]] | |||
}} | |||
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Wagner graph''' is a 3-[[regular graph]] with 8 vertices and 12 edges.<ref>{{cite book | |||
| author = [[John Adrian Bondy|Bondy, J. A.]]; [[U. S. R. Murty|Murty, U. S. R.]] | |||
| title = Graph Theory | |||
| year = 2007 | |||
| publisher = Springer | |||
| isbn = 978-1-84628-969-9 | |||
| pages = 275–276}}</ref> It is the 8-vertex [[Möbius ladder]] graph. | |||
==Properties== | |||
As a Möbius ladder, the Wagner graph is [[Planar graph|nonplanar]] but has [[Crossing number (graph theory)|crossing number]] one, making it an [[apex graph]]. It can be embedded without crossings on a [[torus]] or [[projective plane]], so it is also a [[toroidal graph]]. It has girth 4, diameter 2, radius 2, [[chromatic number]] 3, [[chromatic index]] 3 and is both 3-[[k-vertex-connected graph|vertex-connected]] and 3-[[k-edge-connected graph|edge-connected]]. | |||
The Wagner graph has 392 [[spanning tree]]s; it and the [[complete graph]] ''K''<sub>3,3</sub> have the most spanning trees among all cubic graphs with the same number of vertices.<ref>{{cite arxiv | |||
| author = Jakobson, Dmitry; Rivin, Igor | |||
| title = On some extremal problems in graph theory | |||
| year = 1999 | |||
| eprint = math.CO/9907050 | |||
| postscript = .}}</ref> | |||
The Wagner graph is a [[vertex-transitive graph]] but is not [[edge-transitive graph|edge-transitive]]. Its full automorphism group is isomorphic to the [[dihedral group]] D<sub>8</sub> of order 16, the group of symmetries of an [[octagon]], including both rotations and reflections. | |||
The [[characteristic polynomial]] of the Wagner graph is <math>(x-3)(x-1)^2(x+1)(x^2+2x-1)^2</math>. It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum. | |||
The Wagner graph is [[triangle-free graph|triangle-free]] and has [[independence number]] three, providing one half of the proof that the [[Ramsey number]] ''R''(3,4) (the least number ''n'' such that any ''n''-vertex graph contains either a triangle or a four-vertex independent set) is 9.<ref>{{Cite book|first=Alexander|last=Soifer|year=2008|title=The Mathematical Coloring Book|publisher=Springer-Verlag|isbn=978-0-387-74640-1|page=245|postscript=.}}.</ref> | |||
==Graph minors== | |||
Möbius ladders play an important role in the theory of [[graph minor]]s. The earliest result of this type is a 1937 theorem of [[Klaus Wagner]] (part of a cluster of results known as [[Wagner's theorem]]) that graphs with no ''K''<sub>5</sub> minor can be formed by using [[clique-sum]] operations to combine planar graphs and the Möbius ladder ''M''<sub>8</sub>.<ref>{{cite journal | |||
| author = Wagner, K. | authorlink = Klaus Wagner | |||
| title = Über eine Eigenschaft der ebenen Komplexe | |||
| journal = Mathematische Annalen | |||
| volume = 114 | |||
| issue = 1 | |||
| year = 1937 | |||
| pages = 570–590 | |||
| doi = 10.1007/BF01594196 | |||
| postscript = .}}</ref> For this reason ''M''<sub>8</sub> is called the Wagner graph. | |||
The Wagner graph is also one of four minimal [[forbidden minor]]s for the graphs of [[treewidth]] at most three (the other three being the [[complete graph]] ''K''<sub>5</sub>, the graph of the [[regular octahedron]], and the graph of the [[pentagonal prism]]) and one of four minimal forbidden minors for the graphs of [[branchwidth]] at most three (the other three being ''K''<sub>5</sub>, the graph of the octahedron, and the [[hypercube graph|cube graph]].<ref>{{Cite journal | |||
| last = Bodlaender | first = Hans L. | authorlink = Hans L. Bodlaender | |||
| title = A partial ''k''-arboretum of graphs with bounded treewidth | |||
| journal = Theoretical Computer Science | |||
| volume = 209 | issue = 1–2 | pages = 1–45 | year = 1998 | doi = 10.1016/S0304-3975(97)00228-4 | |||
| postscript = <!--None-->}}.</ref><ref>{{Cite journal | |||
| last1 = Bodlaender | first1 = Hans L. | author1-link = Hans L. Bodlaender | |||
| last2 = Thilikos | first2 = Dimitrios M. | |||
| doi = 10.1006/jagm.1999.1011 | |||
| issue = 2 | |||
| journal = Journal of Algorithms | |||
| pages = 167–194 | |||
| title = Graphs with branchwidth at most three | |||
| volume = 32 | |||
| year = 1999 | |||
| postscript = <!--None-->}}.</ref> | |||
==Construction== | |||
The Wagner graph is a [[cubic graph|cubic]] [[Hamiltonian graph]] and can be defined by the [[LCF notation]] [4]<sup>8</sup>. | |||
It can be drawn as a [[ladder graph]] with 4 rungs made cyclic on a topological [[Möbius strip]]. | |||
==Gallery== | |||
<gallery> | |||
Image:Wagner graph 3COL.svg|The [[chromatic number]] of the Wagner graph is 3. | |||
Image:Wagner graph 3color edge.svg|The [[chromatic index]] of the Wagner graph is 3. | |||
Image:Möbius_ladder_on_Möbius_strip.svg|The Wagner graph drawn on the Möbius strip. | |||
</gallery> | |||
== References == | |||
{{reflist}} | |||
[[Category:Individual graphs]] | |||
[[Category:Regular graphs]] | |||
Revision as of 17:17, 19 November 2013
Template:Infobox graph In the mathematical field of graph theory, the Wagner graph is a 3-regular graph with 8 vertices and 12 edges.[1] It is the 8-vertex Möbius ladder graph.
Properties
As a Möbius ladder, the Wagner graph is nonplanar but has crossing number one, making it an apex graph. It can be embedded without crossings on a torus or projective plane, so it is also a toroidal graph. It has girth 4, diameter 2, radius 2, chromatic number 3, chromatic index 3 and is both 3-vertex-connected and 3-edge-connected.
The Wagner graph has 392 spanning trees; it and the complete graph K3,3 have the most spanning trees among all cubic graphs with the same number of vertices.[2]
The Wagner graph is a vertex-transitive graph but is not edge-transitive. Its full automorphism group is isomorphic to the dihedral group D8 of order 16, the group of symmetries of an octagon, including both rotations and reflections.
The characteristic polynomial of the Wagner graph is . It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.
The Wagner graph is triangle-free and has independence number three, providing one half of the proof that the Ramsey number R(3,4) (the least number n such that any n-vertex graph contains either a triangle or a four-vertex independent set) is 9.[3]
Graph minors
Möbius ladders play an important role in the theory of graph minors. The earliest result of this type is a 1937 theorem of Klaus Wagner (part of a cluster of results known as Wagner's theorem) that graphs with no K5 minor can be formed by using clique-sum operations to combine planar graphs and the Möbius ladder M8.[4] For this reason M8 is called the Wagner graph.
The Wagner graph is also one of four minimal forbidden minors for the graphs of treewidth at most three (the other three being the complete graph K5, the graph of the regular octahedron, and the graph of the pentagonal prism) and one of four minimal forbidden minors for the graphs of branchwidth at most three (the other three being K5, the graph of the octahedron, and the cube graph.[5][6]
Construction
The Wagner graph is a cubic Hamiltonian graph and can be defined by the LCF notation [4]8.
It can be drawn as a ladder graph with 4 rungs made cyclic on a topological Möbius strip.
Gallery
-
The chromatic number of the Wagner graph is 3. -
The chromatic index of the Wagner graph is 3. -
The Wagner graph drawn on the Möbius strip.
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Template:Cite arxiv
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534. - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang. - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang.