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Use of the integrals in the graph theory
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I hope to know some good references about the use of integrals to study the graph theory:
For example, it seems that
$$
int^infty_-infty dx exp(-x^2/2+lambda x^3/3!)
$$
whose coefficients in the $V$ powers of $lambda$, the coefficient of $lambda^V$ counts the possible trivalent graphs of $V$ vertices.For example, it seems that
$$
int dM exp(textTr(-M^2/2+lambda M^3/3!))
$$
where $M$ is a rank-$N$ Hermitian matrix.
The coefficient of $N^2-2glambda^V$ counts the possible trivalent graphs of $V$ vertices on a genus-$g$ Riemann surfaces.
Are there some similar or more general statements of such integrals in the graph theory? Any References are welcome.
Some intuitive way to obtain the above formulas?
functional-analysis graph-theory manifolds triangulation
asked Aug 6 at 19:05
wonderich
1,68821226
 |Â
show 4 more comments
up vote
2
down vote
favorite
I hope to know some good references about the use of integrals to study the graph theory:
For example, it seems that
$$
int^infty_-infty dx exp(-x^2/2+lambda x^3/3!)
$$
whose coefficients in the $V$ powers of $lambda$, the coefficient of $lambda^V$ counts the possible trivalent graphs of $V$ vertices.For example, it seems that
$$
int dM exp(textTr(-M^2/2+lambda M^3/3!))
$$
where $M$ is a rank-$N$ Hermitian matrix.
The coefficient of $N^2-2glambda^V$ counts the possible trivalent graphs of $V$ vertices on a genus-$g$ Riemann surfaces.
Are there some similar or more general statements of such integrals in the graph theory? Any References are welcome.
Some intuitive way to obtain the above formulas?
functional-analysis graph-theory manifolds triangulation
asked Aug 6 at 19:05
wonderich
1,68821226
2
The first integral does not integrate at $+infty$.
– ncmathsadist
Aug 6 at 19:08
Thanks, if you know about that, you may write the answer - I saw this integral from a seminar talk.
– wonderich
Aug 6 at 19:09
I double checked, and it was how the speaker wrote down the expression - I wrote it the same way as he did.
– wonderich
Aug 6 at 19:10
1
wellesley.edu/sites/default/files/assets/kayll-colloq.pdf
– saulspatz
Aug 6 at 19:15
Can you explain how the first integral implies that there is one trivalent graph on four vertices?
– Mike Earnest
Aug 6 at 19:33
 |Â
show 4 more comments
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I hope to know some good references about the use of integrals to study the graph theory:
For example, it seems that
$$
int^infty_-infty dx exp(-x^2/2+lambda x^3/3!)
$$
whose coefficients in the $V$ powers of $lambda$, the coefficient of $lambda^V$ counts the possible trivalent graphs of $V$ vertices.For example, it seems that
$$
int dM exp(textTr(-M^2/2+lambda M^3/3!))
$$
where $M$ is a rank-$N$ Hermitian matrix.
The coefficient of $N^2-2glambda^V$ counts the possible trivalent graphs of $V$ vertices on a genus-$g$ Riemann surfaces.
Are there some similar or more general statements of such integrals in the graph theory? Any References are welcome.
Some intuitive way to obtain the above formulas?
functional-analysis graph-theory manifolds triangulation
asked Aug 6 at 19:05
wonderich
1,68821226
I hope to know some good references about the use of integrals to study the graph theory:
For example, it seems that
$$
int^infty_-infty dx exp(-x^2/2+lambda x^3/3!)
$$
whose coefficients in the $V$ powers of $lambda$, the coefficient of $lambda^V$ counts the possible trivalent graphs of $V$ vertices.For example, it seems that
$$
int dM exp(textTr(-M^2/2+lambda M^3/3!))
$$
where $M$ is a rank-$N$ Hermitian matrix.
The coefficient of $N^2-2glambda^V$ counts the possible trivalent graphs of $V$ vertices on a genus-$g$ Riemann surfaces.
Are there some similar or more general statements of such integrals in the graph theory? Any References are welcome.
Some intuitive way to obtain the above formulas?
functional-analysis graph-theory manifolds triangulation
asked Aug 6 at 19:05
wonderich
1,68821226
asked Aug 6 at 19:05
wonderich
1,68821226
asked Aug 6 at 19:05
wonderich
1,68821226
asked Aug 6 at 19:05
wonderich
1,68821226
1,68821226
2
The first integral does not integrate at $+infty$.
– ncmathsadist
Aug 6 at 19:08
Thanks, if you know about that, you may write the answer - I saw this integral from a seminar talk.
– wonderich
Aug 6 at 19:09
I double checked, and it was how the speaker wrote down the expression - I wrote it the same way as he did.
– wonderich
Aug 6 at 19:10
1
wellesley.edu/sites/default/files/assets/kayll-colloq.pdf
– saulspatz
Aug 6 at 19:15
Can you explain how the first integral implies that there is one trivalent graph on four vertices?
– Mike Earnest
Aug 6 at 19:33
 |Â
show 4 more comments
2
The first integral does not integrate at $+infty$.
– ncmathsadist
Aug 6 at 19:08
Thanks, if you know about that, you may write the answer - I saw this integral from a seminar talk.
– wonderich
Aug 6 at 19:09
I double checked, and it was how the speaker wrote down the expression - I wrote it the same way as he did.
– wonderich
Aug 6 at 19:10
1
wellesley.edu/sites/default/files/assets/kayll-colloq.pdf
– saulspatz
Aug 6 at 19:15
Can you explain how the first integral implies that there is one trivalent graph on four vertices?
– Mike Earnest
Aug 6 at 19:33
2
2
The first integral does not integrate at $+infty$.
– ncmathsadist
Aug 6 at 19:08
The first integral does not integrate at $+infty$.
– ncmathsadist
Aug 6 at 19:08
Thanks, if you know about that, you may write the answer - I saw this integral from a seminar talk.
– wonderich
Aug 6 at 19:09
Thanks, if you know about that, you may write the answer - I saw this integral from a seminar talk.
– wonderich
Aug 6 at 19:09
I double checked, and it was how the speaker wrote down the expression - I wrote it the same way as he did.
– wonderich
Aug 6 at 19:10
I double checked, and it was how the speaker wrote down the expression - I wrote it the same way as he did.
– wonderich
Aug 6 at 19:10
1
1
wellesley.edu/sites/default/files/assets/kayll-colloq.pdf
– saulspatz
Aug 6 at 19:15
wellesley.edu/sites/default/files/assets/kayll-colloq.pdf
– saulspatz
Aug 6 at 19:15
Can you explain how the first integral implies that there is one trivalent graph on four vertices?
– Mike Earnest
Aug 6 at 19:33
Can you explain how the first integral implies that there is one trivalent graph on four vertices?
– Mike Earnest
Aug 6 at 19:33
 |Â
show 4 more comments
1 Answer
1
active
oldest
votes
up vote
1
down vote
In paper I was reading recently, the author said "it is well known that many enumerative problems in graph theory can be solved using Gaussian integrals," and cited the following book:
Sergei K. Lando and Alexander K. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, vol. 141, Springer-Verlag, Berlin, 2004, With an appendix by Don B. Zagier, Low-Dimensional
Topology, II. MR 2036721. (Springer, or Springer Link).
Chapter 3 discusses matrix integral methods. It gives some examples, for instance a matrix integral for enumerating $1$-vertex graphs that fill a surface of genus $g$, or enumerating planar $4$-regular graphs that are the generic image of a circle. Perhaps the book describes the techniques well enough that you can figure out the construction of your given integrals.
A note to Proposition 3.2.10 mentions how you can disregard genus information by setting the matrix size to $1times 1$. This appears to be the transformation from your second integral to your first. Given what I've picked up so far, perhaps it ought to be "where $M$ is an $ntimes n$ Hermitian matrix" rather than "where $M$ is a rank-$n$ Hermitian matrix."
After looking at a paper by one of the book's authors,
A. Zvonkin, Matrix integrals and map enumeration: an accessible introduction, Mathematical and Computer
Modelling 26 (1997), no. 8-10, 281–304, Combinatorics and physics (Marseilles, 1995). MR 1492512
I think I can tell you what the terms of the integral represent. The first part is that a Gaussian measure on the space of $ntimes n$ Hermitian matrices is given by $dmu(M)=exp(operatornametr(-M^2/2))dM$, though in the paper equation (6) has some more normalization terms. The second part is $$exp(operatornametr(lambda M^3/3!))=sum_k=0^inftyoperatornametr(lambda M^3/3!)^k/k!=sum_k=0^infty(lambda/3!)^nkoperatornametr(M^3)^k/k!.$$
I do not understand what the $3!$ is doing in there, except perhaps to remove the dihedral symmetry of a $3$-valent vertex. The $k!$ is likely to remove the vertex-renaming symmetry. The paper says that $int operatornametr(M^3)^k,dmu$ expanded in terms of $N$ has coefficients giving the number of $3$-valent graphs with $k$ vertices in a particular genus. Integrating the infinite sum makes it so the coefficients of $N$ are power series in $lambda$, the coefficients of which give the number of graphs of a particular number of vertices.
answered Aug 9 at 20:39
Kyle Miller
7,054725
thanks +1, for the answer
– wonderich
Aug 9 at 21:10
@wonderich I was puzzled what the $M^2$ was doing in Witten's version of the integral, and it appears to be from the Gaussian measure on the space of Hermitian matrices. I added what I understand of this to the answer.
– Kyle Miller
Aug 9 at 21:50
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
In paper I was reading recently, the author said "it is well known that many enumerative problems in graph theory can be solved using Gaussian integrals," and cited the following book:
Sergei K. Lando and Alexander K. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, vol. 141, Springer-Verlag, Berlin, 2004, With an appendix by Don B. Zagier, Low-Dimensional
Topology, II. MR 2036721. (Springer, or Springer Link).
Chapter 3 discusses matrix integral methods. It gives some examples, for instance a matrix integral for enumerating $1$-vertex graphs that fill a surface of genus $g$, or enumerating planar $4$-regular graphs that are the generic image of a circle. Perhaps the book describes the techniques well enough that you can figure out the construction of your given integrals.
A note to Proposition 3.2.10 mentions how you can disregard genus information by setting the matrix size to $1times 1$. This appears to be the transformation from your second integral to your first. Given what I've picked up so far, perhaps it ought to be "where $M$ is an $ntimes n$ Hermitian matrix" rather than "where $M$ is a rank-$n$ Hermitian matrix."
After looking at a paper by one of the book's authors,
A. Zvonkin, Matrix integrals and map enumeration: an accessible introduction, Mathematical and Computer
Modelling 26 (1997), no. 8-10, 281–304, Combinatorics and physics (Marseilles, 1995). MR 1492512
I think I can tell you what the terms of the integral represent. The first part is that a Gaussian measure on the space of $ntimes n$ Hermitian matrices is given by $dmu(M)=exp(operatornametr(-M^2/2))dM$, though in the paper equation (6) has some more normalization terms. The second part is $$exp(operatornametr(lambda M^3/3!))=sum_k=0^inftyoperatornametr(lambda M^3/3!)^k/k!=sum_k=0^infty(lambda/3!)^nkoperatornametr(M^3)^k/k!.$$
I do not understand what the $3!$ is doing in there, except perhaps to remove the dihedral symmetry of a $3$-valent vertex. The $k!$ is likely to remove the vertex-renaming symmetry. The paper says that $int operatornametr(M^3)^k,dmu$ expanded in terms of $N$ has coefficients giving the number of $3$-valent graphs with $k$ vertices in a particular genus. Integrating the infinite sum makes it so the coefficients of $N$ are power series in $lambda$, the coefficients of which give the number of graphs of a particular number of vertices.
answered Aug 9 at 20:39
Kyle Miller
7,054725
thanks +1, for the answer
– wonderich
Aug 9 at 21:10
@wonderich I was puzzled what the $M^2$ was doing in Witten's version of the integral, and it appears to be from the Gaussian measure on the space of Hermitian matrices. I added what I understand of this to the answer.
– Kyle Miller
Aug 9 at 21:50
add a comment |Â
up vote
1
down vote
In paper I was reading recently, the author said "it is well known that many enumerative problems in graph theory can be solved using Gaussian integrals," and cited the following book:
Sergei K. Lando and Alexander K. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, vol. 141, Springer-Verlag, Berlin, 2004, With an appendix by Don B. Zagier, Low-Dimensional
Topology, II. MR 2036721. (Springer, or Springer Link).
Chapter 3 discusses matrix integral methods. It gives some examples, for instance a matrix integral for enumerating $1$-vertex graphs that fill a surface of genus $g$, or enumerating planar $4$-regular graphs that are the generic image of a circle. Perhaps the book describes the techniques well enough that you can figure out the construction of your given integrals.
A note to Proposition 3.2.10 mentions how you can disregard genus information by setting the matrix size to $1times 1$. This appears to be the transformation from your second integral to your first. Given what I've picked up so far, perhaps it ought to be "where $M$ is an $ntimes n$ Hermitian matrix" rather than "where $M$ is a rank-$n$ Hermitian matrix."
After looking at a paper by one of the book's authors,
A. Zvonkin, Matrix integrals and map enumeration: an accessible introduction, Mathematical and Computer
Modelling 26 (1997), no. 8-10, 281–304, Combinatorics and physics (Marseilles, 1995). MR 1492512
I think I can tell you what the terms of the integral represent. The first part is that a Gaussian measure on the space of $ntimes n$ Hermitian matrices is given by $dmu(M)=exp(operatornametr(-M^2/2))dM$, though in the paper equation (6) has some more normalization terms. The second part is $$exp(operatornametr(lambda M^3/3!))=sum_k=0^inftyoperatornametr(lambda M^3/3!)^k/k!=sum_k=0^infty(lambda/3!)^nkoperatornametr(M^3)^k/k!.$$
I do not understand what the $3!$ is doing in there, except perhaps to remove the dihedral symmetry of a $3$-valent vertex. The $k!$ is likely to remove the vertex-renaming symmetry. The paper says that $int operatornametr(M^3)^k,dmu$ expanded in terms of $N$ has coefficients giving the number of $3$-valent graphs with $k$ vertices in a particular genus. Integrating the infinite sum makes it so the coefficients of $N$ are power series in $lambda$, the coefficients of which give the number of graphs of a particular number of vertices.
answered Aug 9 at 20:39
Kyle Miller
7,054725
thanks +1, for the answer
– wonderich
Aug 9 at 21:10
@wonderich I was puzzled what the $M^2$ was doing in Witten's version of the integral, and it appears to be from the Gaussian measure on the space of Hermitian matrices. I added what I understand of this to the answer.
– Kyle Miller
Aug 9 at 21:50
add a comment |Â
up vote
1
down vote
up vote
1
down vote
In paper I was reading recently, the author said "it is well known that many enumerative problems in graph theory can be solved using Gaussian integrals," and cited the following book:
Sergei K. Lando and Alexander K. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, vol. 141, Springer-Verlag, Berlin, 2004, With an appendix by Don B. Zagier, Low-Dimensional
Topology, II. MR 2036721. (Springer, or Springer Link).
Chapter 3 discusses matrix integral methods. It gives some examples, for instance a matrix integral for enumerating $1$-vertex graphs that fill a surface of genus $g$, or enumerating planar $4$-regular graphs that are the generic image of a circle. Perhaps the book describes the techniques well enough that you can figure out the construction of your given integrals.
A note to Proposition 3.2.10 mentions how you can disregard genus information by setting the matrix size to $1times 1$. This appears to be the transformation from your second integral to your first. Given what I've picked up so far, perhaps it ought to be "where $M$ is an $ntimes n$ Hermitian matrix" rather than "where $M$ is a rank-$n$ Hermitian matrix."
After looking at a paper by one of the book's authors,
A. Zvonkin, Matrix integrals and map enumeration: an accessible introduction, Mathematical and Computer
Modelling 26 (1997), no. 8-10, 281–304, Combinatorics and physics (Marseilles, 1995). MR 1492512
I think I can tell you what the terms of the integral represent. The first part is that a Gaussian measure on the space of $ntimes n$ Hermitian matrices is given by $dmu(M)=exp(operatornametr(-M^2/2))dM$, though in the paper equation (6) has some more normalization terms. The second part is $$exp(operatornametr(lambda M^3/3!))=sum_k=0^inftyoperatornametr(lambda M^3/3!)^k/k!=sum_k=0^infty(lambda/3!)^nkoperatornametr(M^3)^k/k!.$$
I do not understand what the $3!$ is doing in there, except perhaps to remove the dihedral symmetry of a $3$-valent vertex. The $k!$ is likely to remove the vertex-renaming symmetry. The paper says that $int operatornametr(M^3)^k,dmu$ expanded in terms of $N$ has coefficients giving the number of $3$-valent graphs with $k$ vertices in a particular genus. Integrating the infinite sum makes it so the coefficients of $N$ are power series in $lambda$, the coefficients of which give the number of graphs of a particular number of vertices.
answered Aug 9 at 20:39
Kyle Miller
7,054725
In paper I was reading recently, the author said "it is well known that many enumerative problems in graph theory can be solved using Gaussian integrals," and cited the following book:
Sergei K. Lando and Alexander K. Zvonkin, Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, vol. 141, Springer-Verlag, Berlin, 2004, With an appendix by Don B. Zagier, Low-Dimensional
Topology, II. MR 2036721. (Springer, or Springer Link).
Chapter 3 discusses matrix integral methods. It gives some examples, for instance a matrix integral for enumerating $1$-vertex graphs that fill a surface of genus $g$, or enumerating planar $4$-regular graphs that are the generic image of a circle. Perhaps the book describes the techniques well enough that you can figure out the construction of your given integrals.
A note to Proposition 3.2.10 mentions how you can disregard genus information by setting the matrix size to $1times 1$. This appears to be the transformation from your second integral to your first. Given what I've picked up so far, perhaps it ought to be "where $M$ is an $ntimes n$ Hermitian matrix" rather than "where $M$ is a rank-$n$ Hermitian matrix."
After looking at a paper by one of the book's authors,
A. Zvonkin, Matrix integrals and map enumeration: an accessible introduction, Mathematical and Computer
Modelling 26 (1997), no. 8-10, 281–304, Combinatorics and physics (Marseilles, 1995). MR 1492512
I think I can tell you what the terms of the integral represent. The first part is that a Gaussian measure on the space of $ntimes n$ Hermitian matrices is given by $dmu(M)=exp(operatornametr(-M^2/2))dM$, though in the paper equation (6) has some more normalization terms. The second part is $$exp(operatornametr(lambda M^3/3!))=sum_k=0^inftyoperatornametr(lambda M^3/3!)^k/k!=sum_k=0^infty(lambda/3!)^nkoperatornametr(M^3)^k/k!.$$
I do not understand what the $3!$ is doing in there, except perhaps to remove the dihedral symmetry of a $3$-valent vertex. The $k!$ is likely to remove the vertex-renaming symmetry. The paper says that $int operatornametr(M^3)^k,dmu$ expanded in terms of $N$ has coefficients giving the number of $3$-valent graphs with $k$ vertices in a particular genus. Integrating the infinite sum makes it so the coefficients of $N$ are power series in $lambda$, the coefficients of which give the number of graphs of a particular number of vertices.
answered Aug 9 at 20:39
Kyle Miller
7,054725
edited Aug 9 at 21:49
answered Aug 9 at 20:39
Kyle Miller
7,054725
answered Aug 9 at 20:39
Kyle Miller
7,054725
answered Aug 9 at 20:39
Kyle Miller
7,054725
7,054725
thanks +1, for the answer
– wonderich
Aug 9 at 21:10
@wonderich I was puzzled what the $M^2$ was doing in Witten's version of the integral, and it appears to be from the Gaussian measure on the space of Hermitian matrices. I added what I understand of this to the answer.
– Kyle Miller
Aug 9 at 21:50
add a comment |Â
thanks +1, for the answer
– wonderich
Aug 9 at 21:10
@wonderich I was puzzled what the $M^2$ was doing in Witten's version of the integral, and it appears to be from the Gaussian measure on the space of Hermitian matrices. I added what I understand of this to the answer.
– Kyle Miller
Aug 9 at 21:50
thanks +1, for the answer
– wonderich
Aug 9 at 21:10
thanks +1, for the answer
– wonderich
Aug 9 at 21:10
@wonderich I was puzzled what the $M^2$ was doing in Witten's version of the integral, and it appears to be from the Gaussian measure on the space of Hermitian matrices. I added what I understand of this to the answer.
– Kyle Miller
Aug 9 at 21:50
@wonderich I was puzzled what the $M^2$ was doing in Witten's version of the integral, and it appears to be from the Gaussian measure on the space of Hermitian matrices. I added what I understand of this to the answer.
– Kyle Miller
Aug 9 at 21:50
add a comment |Â
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2
The first integral does not integrate at $+infty$.
– ncmathsadist
Aug 6 at 19:08
Thanks, if you know about that, you may write the answer - I saw this integral from a seminar talk.
– wonderich
Aug 6 at 19:09
I double checked, and it was how the speaker wrote down the expression - I wrote it the same way as he did.
– wonderich
Aug 6 at 19:10
1
wellesley.edu/sites/default/files/assets/kayll-colloq.pdf
– saulspatz
Aug 6 at 19:15
Can you explain how the first integral implies that there is one trivalent graph on four vertices?
– Mike Earnest
Aug 6 at 19:33