Counting simple, connected, labeled graphs with N vertices and K edges











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Given the number of vertices $n$ and the number of edges $k$, I need to calculate the number of possible non-isomorphic, simple, connected, labelled graphs.



My question is very similar to this one. Quoting the accepted answer:




This sequence of numbers is A001187 in the On-Line Encyclopedia of Integer Sequences. If $d_n$ is the number of labelled, connected, simple graphs on $n$ vertices, the numbers $d_n$ satisfy the recurrence
$$sum_kbinom{n}kkd_k2^{binom{n-k}2}=n2^{binom{n}2};$$



from which it’s possible to calculate $d_n$ for small values of $n$. This recurrence is derived as formula (3.10.2) in Herbert S. Wilf, generatingfunctionology, 2nd edition, which is available for free download here.




Example



Let's have $n=4$ (vertices) and $k=3$ (edges). Given my limited knowledge, I'm unable to actually calculate the number of graphs by using the formula above.



Can somebody please walk me through it? I was able to manually count 16 different graphs for this example.



Edit #1



I've found the formula for calculating the total number of simple labelled graphs:
$$binom{binom{n}2}m$$
(Handbook of Discrete and Combinatorial Mathematics, available here, Page 580)



The problem is that the number of graphs given by this formula (for the above example is $20$) also includes all the disconnected graphs.










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  • This should be what I'm looking for: oeis.org/A144161
    – Babicaa
    Dec 18 '14 at 13:26












  • No https: oeis.org/A144161
    – dspyz
    Aug 17 '15 at 20:28










  • It would appear that this MSE link is relevant.
    – Marko Riedel
    Dec 21 '17 at 21:59















up vote
9
down vote

favorite
4












Given the number of vertices $n$ and the number of edges $k$, I need to calculate the number of possible non-isomorphic, simple, connected, labelled graphs.



My question is very similar to this one. Quoting the accepted answer:




This sequence of numbers is A001187 in the On-Line Encyclopedia of Integer Sequences. If $d_n$ is the number of labelled, connected, simple graphs on $n$ vertices, the numbers $d_n$ satisfy the recurrence
$$sum_kbinom{n}kkd_k2^{binom{n-k}2}=n2^{binom{n}2};$$



from which it’s possible to calculate $d_n$ for small values of $n$. This recurrence is derived as formula (3.10.2) in Herbert S. Wilf, generatingfunctionology, 2nd edition, which is available for free download here.




Example



Let's have $n=4$ (vertices) and $k=3$ (edges). Given my limited knowledge, I'm unable to actually calculate the number of graphs by using the formula above.



Can somebody please walk me through it? I was able to manually count 16 different graphs for this example.



Edit #1



I've found the formula for calculating the total number of simple labelled graphs:
$$binom{binom{n}2}m$$
(Handbook of Discrete and Combinatorial Mathematics, available here, Page 580)



The problem is that the number of graphs given by this formula (for the above example is $20$) also includes all the disconnected graphs.










share|cite|improve this question
























  • This should be what I'm looking for: oeis.org/A144161
    – Babicaa
    Dec 18 '14 at 13:26












  • No https: oeis.org/A144161
    – dspyz
    Aug 17 '15 at 20:28










  • It would appear that this MSE link is relevant.
    – Marko Riedel
    Dec 21 '17 at 21:59













up vote
9
down vote

favorite
4









up vote
9
down vote

favorite
4






4





Given the number of vertices $n$ and the number of edges $k$, I need to calculate the number of possible non-isomorphic, simple, connected, labelled graphs.



My question is very similar to this one. Quoting the accepted answer:




This sequence of numbers is A001187 in the On-Line Encyclopedia of Integer Sequences. If $d_n$ is the number of labelled, connected, simple graphs on $n$ vertices, the numbers $d_n$ satisfy the recurrence
$$sum_kbinom{n}kkd_k2^{binom{n-k}2}=n2^{binom{n}2};$$



from which it’s possible to calculate $d_n$ for small values of $n$. This recurrence is derived as formula (3.10.2) in Herbert S. Wilf, generatingfunctionology, 2nd edition, which is available for free download here.




Example



Let's have $n=4$ (vertices) and $k=3$ (edges). Given my limited knowledge, I'm unable to actually calculate the number of graphs by using the formula above.



Can somebody please walk me through it? I was able to manually count 16 different graphs for this example.



Edit #1



I've found the formula for calculating the total number of simple labelled graphs:
$$binom{binom{n}2}m$$
(Handbook of Discrete and Combinatorial Mathematics, available here, Page 580)



The problem is that the number of graphs given by this formula (for the above example is $20$) also includes all the disconnected graphs.










share|cite|improve this question















Given the number of vertices $n$ and the number of edges $k$, I need to calculate the number of possible non-isomorphic, simple, connected, labelled graphs.



My question is very similar to this one. Quoting the accepted answer:




This sequence of numbers is A001187 in the On-Line Encyclopedia of Integer Sequences. If $d_n$ is the number of labelled, connected, simple graphs on $n$ vertices, the numbers $d_n$ satisfy the recurrence
$$sum_kbinom{n}kkd_k2^{binom{n-k}2}=n2^{binom{n}2};$$



from which it’s possible to calculate $d_n$ for small values of $n$. This recurrence is derived as formula (3.10.2) in Herbert S. Wilf, generatingfunctionology, 2nd edition, which is available for free download here.




Example



Let's have $n=4$ (vertices) and $k=3$ (edges). Given my limited knowledge, I'm unable to actually calculate the number of graphs by using the formula above.



Can somebody please walk me through it? I was able to manually count 16 different graphs for this example.



Edit #1



I've found the formula for calculating the total number of simple labelled graphs:
$$binom{binom{n}2}m$$
(Handbook of Discrete and Combinatorial Mathematics, available here, Page 580)



The problem is that the number of graphs given by this formula (for the above example is $20$) also includes all the disconnected graphs.







graph-theory






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edited Apr 13 '17 at 12:21









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asked Dec 18 '14 at 0:14









Babicaa

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  • This should be what I'm looking for: oeis.org/A144161
    – Babicaa
    Dec 18 '14 at 13:26












  • No https: oeis.org/A144161
    – dspyz
    Aug 17 '15 at 20:28










  • It would appear that this MSE link is relevant.
    – Marko Riedel
    Dec 21 '17 at 21:59


















  • This should be what I'm looking for: oeis.org/A144161
    – Babicaa
    Dec 18 '14 at 13:26












  • No https: oeis.org/A144161
    – dspyz
    Aug 17 '15 at 20:28










  • It would appear that this MSE link is relevant.
    – Marko Riedel
    Dec 21 '17 at 21:59
















This should be what I'm looking for: oeis.org/A144161
– Babicaa
Dec 18 '14 at 13:26






This should be what I'm looking for: oeis.org/A144161
– Babicaa
Dec 18 '14 at 13:26














No https: oeis.org/A144161
– dspyz
Aug 17 '15 at 20:28




No https: oeis.org/A144161
– dspyz
Aug 17 '15 at 20:28












It would appear that this MSE link is relevant.
– Marko Riedel
Dec 21 '17 at 21:59




It would appear that this MSE link is relevant.
– Marko Riedel
Dec 21 '17 at 21:59










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There is no analytic formula for the number of connected graphs given $n$ labeled nodes and $N$ edges. It has been known, however, since around $1860$ that for $n$ labeled nodes with $n-1$ edges (e.g. $4$ nodes and $3$ edges) the number of connected graphs is $n^{n-2}$.



For any graph of $n$ labeled nodes and $kgtbinom {n-1}2$ edges the graph is always connected and thus the number of connected graphs in that case is equal to the total number of graphs.



For any graph of $n$ labeled nodes and $Klt(n-1)$ edges the graph is never connected. So the number of connected graphs in that case is always $0$.



It is the connected graphs of $n$ labeled nodes with $(n-1)le klebinom{n-1}2$ edges that are non trivial. Carl Wilhelm Borchardt found the formula that we use for the case of $k=n-1$. Cayley popularized it by "expanding" the result.



I just recently found another case. I am currently writing the results now.






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    There is no analytic formula for the number of connected graphs given $n$ labeled nodes and $N$ edges. It has been known, however, since around $1860$ that for $n$ labeled nodes with $n-1$ edges (e.g. $4$ nodes and $3$ edges) the number of connected graphs is $n^{n-2}$.



    For any graph of $n$ labeled nodes and $kgtbinom {n-1}2$ edges the graph is always connected and thus the number of connected graphs in that case is equal to the total number of graphs.



    For any graph of $n$ labeled nodes and $Klt(n-1)$ edges the graph is never connected. So the number of connected graphs in that case is always $0$.



    It is the connected graphs of $n$ labeled nodes with $(n-1)le klebinom{n-1}2$ edges that are non trivial. Carl Wilhelm Borchardt found the formula that we use for the case of $k=n-1$. Cayley popularized it by "expanding" the result.



    I just recently found another case. I am currently writing the results now.






    share|cite|improve this answer



























      up vote
      0
      down vote













      There is no analytic formula for the number of connected graphs given $n$ labeled nodes and $N$ edges. It has been known, however, since around $1860$ that for $n$ labeled nodes with $n-1$ edges (e.g. $4$ nodes and $3$ edges) the number of connected graphs is $n^{n-2}$.



      For any graph of $n$ labeled nodes and $kgtbinom {n-1}2$ edges the graph is always connected and thus the number of connected graphs in that case is equal to the total number of graphs.



      For any graph of $n$ labeled nodes and $Klt(n-1)$ edges the graph is never connected. So the number of connected graphs in that case is always $0$.



      It is the connected graphs of $n$ labeled nodes with $(n-1)le klebinom{n-1}2$ edges that are non trivial. Carl Wilhelm Borchardt found the formula that we use for the case of $k=n-1$. Cayley popularized it by "expanding" the result.



      I just recently found another case. I am currently writing the results now.






      share|cite|improve this answer

























        up vote
        0
        down vote










        up vote
        0
        down vote









        There is no analytic formula for the number of connected graphs given $n$ labeled nodes and $N$ edges. It has been known, however, since around $1860$ that for $n$ labeled nodes with $n-1$ edges (e.g. $4$ nodes and $3$ edges) the number of connected graphs is $n^{n-2}$.



        For any graph of $n$ labeled nodes and $kgtbinom {n-1}2$ edges the graph is always connected and thus the number of connected graphs in that case is equal to the total number of graphs.



        For any graph of $n$ labeled nodes and $Klt(n-1)$ edges the graph is never connected. So the number of connected graphs in that case is always $0$.



        It is the connected graphs of $n$ labeled nodes with $(n-1)le klebinom{n-1}2$ edges that are non trivial. Carl Wilhelm Borchardt found the formula that we use for the case of $k=n-1$. Cayley popularized it by "expanding" the result.



        I just recently found another case. I am currently writing the results now.






        share|cite|improve this answer














        There is no analytic formula for the number of connected graphs given $n$ labeled nodes and $N$ edges. It has been known, however, since around $1860$ that for $n$ labeled nodes with $n-1$ edges (e.g. $4$ nodes and $3$ edges) the number of connected graphs is $n^{n-2}$.



        For any graph of $n$ labeled nodes and $kgtbinom {n-1}2$ edges the graph is always connected and thus the number of connected graphs in that case is equal to the total number of graphs.



        For any graph of $n$ labeled nodes and $Klt(n-1)$ edges the graph is never connected. So the number of connected graphs in that case is always $0$.



        It is the connected graphs of $n$ labeled nodes with $(n-1)le klebinom{n-1}2$ edges that are non trivial. Carl Wilhelm Borchardt found the formula that we use for the case of $k=n-1$. Cayley popularized it by "expanding" the result.



        I just recently found another case. I am currently writing the results now.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Mar 2 '17 at 22:41







        user409521

















        answered Mar 2 '17 at 22:22









        Kenyon L Coleman

        1




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