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In the [[mathematics|mathematical]] fields of [[graph theory]] and [[combinatorics]], a '''matching polynomial''' (sometimes called an '''acyclic polynomial''') is a [[generating function]] of the numbers of [[Matching (graph theory)|matching]]s of various sizes in a graph.
==Definition==
Several different types of matching polynomials have been defined. Let ''G'' be a graph with ''n'' vertices and let ''m<sub>k</sub>'' be the number of ''k''-edge matchings.


One matching polynomial of ''G'' is
:<math>m_G(x) := \sum_{k\geq0} m_k x^k.</math>


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Another definition gives the matching polynomial as
:<math>M_G(x) := \sum_{k\geq0} (-1)^k m_k x^{n-2k}.</math>
 
A third definition is the polynomial
:<math>\mu_G(x,y) := \sum_{k\geq0} m_k x^k y^{n-2k}.</math>
 
Each type has its uses, and all are equivalent by simple transformations.  For instance,
:<math>M_G(x) = x^n m_G(-x^{-2})</math>
and
:<math>\mu_G(x,y) = y^n m_G(x/y^2).</math>
 
==Connections to other polynomials==
The first type of matching polynomial is a direct generalization of the [[rook polynomial]].
 
The second type of matching polynomial has remarkable connections with [[orthogonal polynomials]]. For instance, if ''G''&nbsp;=&nbsp;''K''<sub>''m'',''n''</sub>, the [[complete bipartite graph]], then the second type of matching polynomial is related to the generalized [[Laguerre polynomial]] ''L''<sub>''n''</sub><sup>''&alpha;''</sup>(''x'') by the identity:
 
: <math>M_{K_{m,n}}(x) = n! L_n^{(m-n)}(x^2). \, </math>
 
If ''G'' is the [[complete graph]] ''K''<sub>''n''</sub>, then ''M''<sub>''G''</sub>(''x'') is an Hermite polynomial:
: <math>M_{K_n}(x) = H_n(x), \, </math>
where ''H''<sub>''n''</sub>(''x'') is the "probabilist's Hermite polynomial" (1) in the definition of [[Hermite polynomial]]s. These facts were observed by {{harvtxt|Godsil|1981}}.
 
If ''G'' is a [[Tree (graph theory)|forest]], then its matching polynomial is equal to its [[characteristic polynomial]].
 
If ''G'' is a [[Path (graph theory)|path]] or a [[Cycle (graph theory)|cycle]], then ''M''<sub>''G''</sub>(''x'') is a [[Chebyshev polynomial]]. In this case
μ<sub>''G''</sub>(1,''x'') is a [[Fibonacci polynomial]] or [[Lucas polynomial]] respectively.
 
==Complementation==
The matching polynomial of a graph ''G'' with ''n'' vertices is related to that of its complement by a pair of (equivalent) formulas. One of them is a simple combinatorial identity due to {{harvtxt|Zaslavsky|1981}}.  The other is an integral identity due to {{harvtxt|Godsil|1981}}.
 
There is a similar relation for a subgraph ''G'' of ''K''<sub>''m'',''n''</sub> and its complement in ''K''<sub>''m'',''n''</sub>.  This relation, due to Riordan (1958), was known in the context of non-attacking rook placements and rook polynomials.
 
==Applications in chemical informatics==
The [[Hosoya index]] of a graph ''G'', its number of matchings, is used in [[chemoinformatics]] as a structural descriptor of a molecular graph. It may be evaluated as ''m''<sub>''G''</sub>(1) {{harv|Gutman|1991}}.
 
The third type of matching polynomial was introduced by {{harvtxt|Farrell|1980}} as a version of the "acyclic polynomial" used in [[chemistry]].
 
==Computational complexity==
On arbitrary graphs, or even [[planar graph]]s, computing the matching polynomial is [[Sharp-P-complete|#P-complete]] {{harv|Jerrum|1987}}. However, it can be computed more efficiently when additional structure about the graph is known. In particular,
computing the matching polynomial on ''n''-vertex graphs of [[treewidth]] ''k'' is [[fixed-parameter tractability|fixed-parameter tractable]]: there exists an algorithm whose running time, for any fixed constant ''k'', is a [[polynomial time|polynomial]] in ''n'' with an exponent that does not depend on ''k'' {{harv|Courcelle|Makowsky|Rotics|2001}}.
The matching polynomial of a graph with ''n'' vertices and [[clique-width]] ''k'' may be computed in time ''n''<sup>O(''k'')</sup> {{harv|Makowsky|Rotics|Averbouch|Godlin|2006}}
 
== References ==
{{No footnotes|date=April 2011}}
*{{citation
| last1 = Courcelle | first1 = B.
| last2 = Makowsky | first2 = J. A.
| last3 = Rotics | first3 = U.
| doi = 10.1016/S0166-218X(00)00221-3
| issue = 1-2
| journal = Discrete Applied Mathematics
| pages = 23–52
| title = On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic
| url = http://www.labri.fr/perso/courcell/CoursMaster/CMR-Dam.pdf
| volume = 108
| year = 2001}}.
*{{citation
| last = Farrell | first = E. J.
| journal = Ars Combinatoria
| pages = 221–228
| title = The matching polynomial and its relation to the acyclic polynomial of a graph
| volume = 9
| year = 1980}}.
*{{citation
| last = Godsil | first = C.D. | author-link = Chris Godsil
| doi = 10.1007/BF02579331
| issue = 3
| journal = Combinatorica
| pages = 257–262
| title = Hermite polynomials and a duality relation for matchings polynomials
| volume = 1
| year = 1981}}.
*{{citation
| last = Gutman | first = Ivan
| editor1-last = Bonchev | editor1-first = D.
| editor2-last = Rouvray | editor2-first = D. H.
| contribution = Polynomials in graph theory
| isbn = 978-0-85626-454-2
| pages = 133–176
| publisher = Taylor & Francis
| series = Mathematical Chemistry
| title = Chemical Graph Theory: Introduction and Fundamentals
| volume = 1
| year = 1991}}.
*{{citation
| last = Jerrum | first = Mark
| doi = 10.1007/BF01010403
| issue = 1
| journal = Journal of Statistical Physics
| pages = 121–134
| title = Two-dimensional monomer-dimer systems are computationally intractable
| volume = 48
| year = 1987}}.
*{{citation
| last1 = Makowsky | first1 = J. A.
| last2 = Rotics | first2 = Udi
| last3 = Averbouch | first3 = Ilya
| last4 = Godlin | first4 = Benny
| contribution = Computing graph polynomials on graphs of bounded clique-width
| doi = 10.1007/11917496_18
| pages = 191–204
| publisher = Springer-Verlag
| series = Lecture Notes in Computer Science
| title = Proc. 32nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG '06)
| url = http://www.cs.technion.ac.il/~admlogic/TR/2006/WG06_makowsky.pdf
| volume = 4271
| year = 2006}}.
*{{citation
| last = Riordan | first = John | author-link = John Riordan (mathematician)
| location = New York
| publisher = Wiley
| title = An Introduction to Combinatorial Analysis
| year = 1958}}.
*{{citation
| last = Zaslavsky | first = Thomas
| journal = European Journal of Combinatorics
| pages = 91–103
| title = Complementary matching vectors and the uniform matching extension property
| volume = 2
| year = 1981}}.
 
[[Category:Algebraic graph theory]]
[[Category:Matching]]

Latest revision as of 06:05, 18 July 2013

In the mathematical fields of graph theory and combinatorics, a matching polynomial (sometimes called an acyclic polynomial) is a generating function of the numbers of matchings of various sizes in a graph.

Definition

Several different types of matching polynomials have been defined. Let G be a graph with n vertices and let mk be the number of k-edge matchings.

One matching polynomial of G is

Another definition gives the matching polynomial as

A third definition is the polynomial

Each type has its uses, and all are equivalent by simple transformations. For instance,

and

Connections to other polynomials

The first type of matching polynomial is a direct generalization of the rook polynomial.

The second type of matching polynomial has remarkable connections with orthogonal polynomials. For instance, if G = Km,n, the complete bipartite graph, then the second type of matching polynomial is related to the generalized Laguerre polynomial Lnα(x) by the identity:

If G is the complete graph Kn, then MG(x) is an Hermite polynomial:

where Hn(x) is the "probabilist's Hermite polynomial" (1) in the definition of Hermite polynomials. These facts were observed by Template:Harvtxt.

If G is a forest, then its matching polynomial is equal to its characteristic polynomial.

If G is a path or a cycle, then MG(x) is a Chebyshev polynomial. In this case μG(1,x) is a Fibonacci polynomial or Lucas polynomial respectively.

Complementation

The matching polynomial of a graph G with n vertices is related to that of its complement by a pair of (equivalent) formulas. One of them is a simple combinatorial identity due to Template:Harvtxt. The other is an integral identity due to Template:Harvtxt.

There is a similar relation for a subgraph G of Km,n and its complement in Km,n. This relation, due to Riordan (1958), was known in the context of non-attacking rook placements and rook polynomials.

Applications in chemical informatics

The Hosoya index of a graph G, its number of matchings, is used in chemoinformatics as a structural descriptor of a molecular graph. It may be evaluated as mG(1) Template:Harv.

The third type of matching polynomial was introduced by Template:Harvtxt as a version of the "acyclic polynomial" used in chemistry.

Computational complexity

On arbitrary graphs, or even planar graphs, computing the matching polynomial is #P-complete Template:Harv. However, it can be computed more efficiently when additional structure about the graph is known. In particular, computing the matching polynomial on n-vertex graphs of treewidth k is fixed-parameter tractable: there exists an algorithm whose running time, for any fixed constant k, is a polynomial in n with an exponent that does not depend on k Template:Harv. The matching polynomial of a graph with n vertices and clique-width k may be computed in time nO(k) Template:Harv

References

Template:No footnotes

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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.

    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.
  • 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.