Local martingale: Difference between revisions

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en>Tsirel
→‎Example 1: not "as"; the constant unconditional expectation is necessari but not sufficient for being martingale
 
en>Tsirel
Undid revision 586324790 by Elwwod (talk) No, it was right: otherwise τ<sub>''k''</sub> = ''k''...never infinite
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[[Image:6n-graf.svg|thumb|A graph consisting of 6 vertices and 7 edges]]
:''For other meanings of neighbourhoods in mathematics, see [[Neighbourhood (mathematics)]]. For non-mathematical neighbourhoods, see [[Neighbourhood (disambiguation)]].''
 
In [[graph theory]], an '''adjacent vertex''' of a [[vertex (graph theory)|vertex]] ''v'' in a [[Graph (mathematics)|graph]] is a vertex that is connected to ''v'' by an [[edge (graph theory)|edge]]. The '''neighbourhood''' of a vertex ''v'' in a graph ''G'' is the [[induced subgraph#Subgraphs|induced subgraph]] of ''G'' consisting of all vertices adjacent to ''v'' and all edges connecting two such vertices. For example, the image shows a graph of 6 vertices and 7 edges. Vertex 5 is adjacent to vertices 1, 2, and 4 but it is not adjacent to 3 and&nbsp;6. The neighbourhood of vertex 5 is the graph with three vertices, 1, 2, and&nbsp;4, and one edge connecting vertices 1 and&nbsp;2.
 
The neighbourhood is often denoted ''N''<sub>''G''</sub>(''v'') or (when the graph is unambiguous)&nbsp;''N''(''v''). The same neighbourhood notation may also be used to refer to sets of adjacent vertices rather than the corresponding induced subgraphs. The neighbourhood described above does not include ''v'' itself, and is more specifically the '''open neighbourhood''' of ''v''; it is also possible to define a neighbourhood in which ''v'' itself is included, called the '''closed neighbourhood''' and denoted by ''N''<sub>''G''</sub>[''v'']. When stated without any qualification, a neighbourhood is assumed to be open.
 
Neighbourhoods may be used to represent graphs in computer algorithms, via the [[adjacency list]] and [[adjacency matrix]] representations. Neighbourhoods are also used in the [[clustering coefficient]] of a graph, which is a measure of the average [[Dense graph|density]] of its neighbourhoods. In addition, many important classes of graphs may be defined by properties of their neighbourhoods, or by symmetries that relate neighbourhoods to each other.
 
An [[isolated vertex]] has no adjacent vertices. The [[degree (graph theory)|degree]] of a vertex is equal to the number of adjacent vertices. A special case is a [[loop (graph theory)|loop]] that connects a vertex to itself; if such an edge exists, the vertex belongs to its own neighbourhood.
 
==Local properties in graphs==
[[Image:Octahedron graph.png|thumb|In the [[Octahedron|octahedron graph]], the neighbourhood of any vertex is a 4-[[Cycle graph|cycle]].]]
If all vertices in ''G'' have neighbourhoods that are [[Graph isomorphism|isomorphic]] to the same graph ''H'', ''G'' is said to be ''locally H'', and if all vertices in ''G'' have neighbourhoods that belong to some graph family ''F'', ''G'' is said to be ''locally F'' ({{harvnb|Hell|1978}}, {{harvnb|Sedlacek|1983}}). For instance, in the [[Octahedron|octahedron graph]] shown in the figure, each vertex has a neighbourhood isomorphic to a [[Cycle graph|cycle]] of four vertices, so the octahedron is locally&nbsp;''C''<sub>4</sub>.
 
For example:
* Any [[complete graph]] ''K''<sub>''n''</sub> is locally ''K''<sub>''n-1''</sub>. The only graphs that are locally complete are disjoint unions of complete graphs.
* A [[Turán graph]] ''T''(''rs'',''r'') is locally ''T''((''r''-1)''s'',''r''-1). More generally any Turán graph is locally Turán.
* Every [[planar graph]] is locally [[Outerplanar graph|outerplanar]]. However, not every locally outerplanar graph is planar.
* A graph is [[triangle-free graph|triangle-free]] if and only if it is locally [[Independent set (graph theory)|independent]].
* Every ''k''-[[Chromatic number|chromatic]] graph is locally (''k''-1)-chromatic. Every locally ''k''-chromatic graph has chromatic number <math>O(\sqrt{kn})</math> {{harv|Wigderson|1983}}.
* If a graph family ''F'' is closed under the operation of taking induced subgraphs, then every graph in ''F'' is also locally ''F''. For instance, every [[chordal graph]] is locally chordal; every [[perfect graph]] is locally perfect; every [[comparability graph]] is locally comparable.
* A graph is locally cyclic if every neighbourhood is a [[Cycle graph|cycle]]. For instance, the [[octahedron]] is the unique locally ''C''<sub>4</sub> graph, the [[icosahedron]] is the unique locally ''C''<sub>5</sub> graph, and the [[Paley graph]] of order 13 is locally ''C''<sub>6</sub>. Locally cyclic graphs other than ''K''<sub>4</sub> are exactly the underlying graphs of [[Triangulation (topology)|Whitney triangulations]], embeddings of graphs on surfaces in such a way that the faces of the embedding are the cliques of the graph ({{harvnb|Hartsfeld|Ringel|1981}}; {{harvnb|Larrión|Neumann-Lara|Pizaña|2002}}; {{harvnb|Malnič|Mohar|1992}}). Locally cyclic graphs can have as many as <math>n^{2-o(1)}</math> edges {{harv|Seress|Szabó|1995}}.
* [[Claw-free graph]]s are the graphs that are locally co-[[triangle-free graph|triangle-free]]; that is, for all vertices, the [[complement graph]] of the neighborhood of the vertex does not contain a triangle. A graph that is locally ''H'' is claw-free if and only if the [[independence number]] of ''H'' is at most two; for instance, the graph of the regular icosahedron is claw-free because it is locally ''C''<sub>5</sub> and ''C''<sub>5</sub> has independence number two.
 
==Neighbourhood of a Set==
For a set ''A'' of vertices, the neighbourhood of ''A'' is the union of the neighbourhoods of the vertices, and so it is the set of all vertices adjacent to at least one member of&nbsp;''A''.
 
A set ''A'' of vertices in a graph is said to be a module if every vertex in ''A'' has the same set of neighbours outside of ''A''. Any graph has a uniquely recursive decomposition into modules, its [[modular decomposition]], which can be constructed from the graph in [[linear time]]; modular decomposition algorithms have applications in other graph algorithms including the recognition of [[comparability graph]]s.
 
==See also==
* [[Moore neighborhood]]
* [[Von Neumann neighborhood]]
* [[Vertex figure]], a related concept in [[polyhedral geometry]]
 
==References==
*{{citation
| last1 = Hartsfeld | first1 = Nora
| last2 = Ringel | first2 = Gerhard | author2-link = Gerhard Ringel
| doi = 10.1007/BF01206358
| journal = [[Combinatorica]]
| pages = 145–155
| title = Clean triangulations
| volume = 11
| year = 1991
| issue = 2}}.
*{{citation
| last = Hell | first = Pavol
| contribution = Graphs with given neighborhoods I
| pages = 219–223
| series = Colloque internationaux C.N.R.S.
| title = Problems Combinatories et theorie des graphes
| volume = 260
| year = 1978}}.
*{{citation
| last1 = Larrión | first1 = F.
| last2 = Neumann-Lara | first2 = V.
| last3 = Pizaña | first3 = M. A.
| doi = 10.1016/S0012-365X(02)00266-2
| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
| pages = 123–135
| title = Whitney triangulations, local girth and iterated clique graphs
| url = http://xamanek.izt.uam.mx/map/papers/cuello10_DM.ps
| volume = 258
| year = 2002}}.
*{{citation
| last1 = Malnič | first1 = Aleksander
| last2 = Mohar | first2 = Bojan | author2-link = Bojan Mohar
| doi = 10.1016/0095-8956(92)90015-P
| issue = 2
| journal = [[Journal of Combinatorial Theory|Journal of Combinatorial Theory, Series B]]
| pages = 147–164
| title = Generating locally cyclic triangulations of surfaces
| volume = 56
| year = 1992}}.
*{{citation
| last = Sedlacek | first = J.
| contribution = On local properties of finite graphs
| doi = 10.1007/BFb0071634
| pages = 242–247
| publisher = Springer-Verlag
| series = Lecture Notes in Mathematics
| title = Graph Theory, Lagów
| volume = 1018
| year = 1983
| chapter = On local properties of finite graphs
| isbn = 978-3-540-12687-4}}.
*{{citation
| last1 = Seress | first1 = Ákos
| last2 = Szabó | first2 = Tibor
| doi = 10.1006/jctb.1995.1020
| journal = [[Journal of Combinatorial Theory|Journal of Combinatorial Theory, Series B]]
| pages = 281–293
| title = Dense graphs with cycle neighborhoods
| url = http://www.inf.ethz.ch/personal/szabo/PS/kornyezetek.ps
| volume = 63
| year = 1995
| issue = 2}}.
*{{citation
| last = Wigderson | first = Avi | author-link = Avi Wigderson
| doi = 10.1145/2157.2158
| issue = 4
| journal = [[Journal of the ACM]]
| pages = 729–735
| title = Improving the performance guarantee for approximate graph coloring
| volume = 30
| year = 1983}}.
 
[[Category:Graph theory objects]]

Revision as of 20:26, 16 December 2013

A graph consisting of 6 vertices and 7 edges
For other meanings of neighbourhoods in mathematics, see Neighbourhood (mathematics). For non-mathematical neighbourhoods, see Neighbourhood (disambiguation).

In graph theory, an adjacent vertex of a vertex v in a graph is a vertex that is connected to v by an edge. The neighbourhood of a vertex v in a graph G is the induced subgraph of G consisting of all vertices adjacent to v and all edges connecting two such vertices. For example, the image shows a graph of 6 vertices and 7 edges. Vertex 5 is adjacent to vertices 1, 2, and 4 but it is not adjacent to 3 and 6. The neighbourhood of vertex 5 is the graph with three vertices, 1, 2, and 4, and one edge connecting vertices 1 and 2.

The neighbourhood is often denoted NG(v) or (when the graph is unambiguous) N(v). The same neighbourhood notation may also be used to refer to sets of adjacent vertices rather than the corresponding induced subgraphs. The neighbourhood described above does not include v itself, and is more specifically the open neighbourhood of v; it is also possible to define a neighbourhood in which v itself is included, called the closed neighbourhood and denoted by NG[v]. When stated without any qualification, a neighbourhood is assumed to be open.

Neighbourhoods may be used to represent graphs in computer algorithms, via the adjacency list and adjacency matrix representations. Neighbourhoods are also used in the clustering coefficient of a graph, which is a measure of the average density of its neighbourhoods. In addition, many important classes of graphs may be defined by properties of their neighbourhoods, or by symmetries that relate neighbourhoods to each other.

An isolated vertex has no adjacent vertices. The degree of a vertex is equal to the number of adjacent vertices. A special case is a loop that connects a vertex to itself; if such an edge exists, the vertex belongs to its own neighbourhood.

Local properties in graphs

In the octahedron graph, the neighbourhood of any vertex is a 4-cycle.

If all vertices in G have neighbourhoods that are isomorphic to the same graph H, G is said to be locally H, and if all vertices in G have neighbourhoods that belong to some graph family F, G is said to be locally F (Template:Harvnb, Template:Harvnb). For instance, in the octahedron graph shown in the figure, each vertex has a neighbourhood isomorphic to a cycle of four vertices, so the octahedron is locally C4.

For example:

Neighbourhood of a Set

For a set A of vertices, the neighbourhood of A is the union of the neighbourhoods of the vertices, and so it is the set of all vertices adjacent to at least one member of A.

A set A of vertices in a graph is said to be a module if every vertex in A has the same set of neighbours outside of A. Any graph has a uniquely recursive decomposition into modules, its modular decomposition, which can be constructed from the graph in linear time; modular decomposition algorithms have applications in other graph algorithms including the recognition of comparability graphs.

See also

References

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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.

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  • 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.
  • 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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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    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.
  • 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.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    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.
  • 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.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    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.