A370064 -id:A370064 - cp's OEIS frontend

A325111 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes with k articulation vertices, (0 <= k <= n).

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 3, 2, 1, 0, 0, 10, 7, 3, 1, 0, 0, 56, 33, 17, 5, 1, 0, 0, 468, 244, 101, 32, 7, 1, 0, 0, 7123, 2792, 890, 242, 60, 9, 1, 0, 0, 194066, 52448, 11468, 2461, 527, 97, 12, 1, 0, 0, 9743542, 1690206, 239728, 35839, 6056, 1029, 155, 15, 1, 0, 0

Offset: 0

Andrew Howroyd, Sep 05 2019

Articulation vertices are also called cutpoints. These are vertices that when removed increase the component count of the graph.

			Triangle begins:
     1;
     1     0;
     1,    0,   0;
     1,    1,   0,   0;
     3,    2,   1,   0,  0;
    10,    7,   3,   1,  0, 0;
    56,   33,  17,   5,  1, 0, 0;
   468,  244, 101,  32,  7, 1, 0, 0;
  7123, 2792, 890, 242, 60, 9, 1, 0, 0;
  ...

Eric Weisstein's World of Mathematics, Articulation Vertex

Columns k=0..5 are A002218(n>1), A241767, A241768, A241769, A241770, A241771.
Row sums are A001349.
Cf. A327077, A370064 (labeled version).

Diagonal for k = n inserted by Andrew Howroyd, Feb 25 2024

A013923 Number of labeled connected graphs with n vertices and 1 cutpoint.

Original entry on oeis.org

3, 16, 250, 8496, 540568, 61672192, 12608406288, 4697459302400, 3256012245850496, 4276437400678311936, 10796431791679078528256, 52955364458428847588956160, 508511231062550463852707804160, 9611398894866376672902234634977280

Offset: 3

Stanley Selkow (sms(AT)owl.WPI.EDU)

nonn

Andrew Howroyd, Table of n, a(n) for n = 3..50
S. Selkow, The enumeration of labeled graphs by number of cutpoints, Discr. Math. 185 (1998), 183-191.

Column k=1 of A370064.

E.g.f.: x * (exp(B'(x)) - B'(x) - 1) where B(x) is the e.g.f. for A013922. - Sean A. Irvine, Aug 28 2018

More terms from Sean A. Irvine, Aug 28 2018

A013924 Number of labeled connected graphs with n nodes and 2 cutpoints.

Original entry on oeis.org

12, 180, 4560, 211680, 17186624, 2416430016, 597615868800, 266262716016000, 218583901063537152, 336744209796848156160, 987653716558634390487040, 5576385411303897176201779200, 61117320790343064985205192540160

Offset: 4

Stanley Selkow (sms(AT)owl.WPI.EDU)

nonn

Andrew Howroyd, Table of n, a(n) for n = 4..50
S. Selkow, The enumeration of labeled graphs by number of cutpoints, Discr. Math. 185 (1998), 183-191.

Column k=2 of A370064.

More terms from Sean A. Irvine, Aug 29 2018

A013925 Number of labeled connected graphs with n nodes and 3 cutpoints.

Original entry on oeis.org

60, 1920, 75600, 4663680, 469336896, 79132032000, 23121510192000, 12082931084928000, 11564306171310537216, 20625639730671895535616, 69501783433272242637312000, 447403458171641201324598067200, 5550030610876067133522251137105920

Offset: 5

Stanley Selkow (sms(AT)owl.WPI.EDU)

nonn

Selkow paper has typographical error a(9) = 469336898. - Sean A. Irvine, Aug 29 2018

Andrew Howroyd, Table of n, a(n) for n = 5..50
S. Selkow, The enumeration of labeled graphs by number of cutpoints, Discr. Math. 185 (1998), 183-191.

Column k=3 of A370064.

a(9) corrected and more terms from Sean A. Irvine, Aug 29 2018

A370065 Triangle read by rows: T(n,k) is the number of simple graphs on n labeled nodes with k articulation vertices, (0 <= k <= n).

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 5, 3, 0, 0, 24, 28, 12, 0, 0, 334, 390, 240, 60, 0, 0, 13262, 10776, 6090, 2280, 360, 0, 0, 1106862, 615860, 255570, 92820, 23520, 2520, 0, 0, 175376048, 66625504, 19275424, 5446560, 1429680, 262080, 20160, 0, 0, 52257938968, 13210716600, 2592577512, 520122456, 112145040, 22649760, 3144960, 181440, 0, 0

Offset: 0

Andrew Howroyd, Feb 25 2024

			Triangle begins:
        1;
        1,      0;
        2,      0,      0;
        5,      3,      0,     0;
       24,     28,     12,     0,     0;
      334,    390,    240,    60,     0,    0;
    13262,  10776,   6090,  2280,   360,    0, 0;
  1106862, 615860, 255570, 92820, 23520, 2520, 0, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
S. Selkow, The enumeration of labeled graphs by number of cutpoints, Discr. Math. 185 (1998), 183-191.

Row sums are A006125.
Column k=0 is A370066.
Cf. A188588, A370064 (connected).

\\ Needs G, J defined in A370064.
T(n)={my(v=Vec( ((y-1)*x + serreverse(x/((1-y) + y*exp(G(n)))))/y ), w=Vec(serlaplace(exp(sum(k=1, n, Polrev(J(v[k],k),y)*x^k, O(x*x^n)) )))); vector(#w, n, Vecrev(w[n],n))}
{ my(A=T(8)); for(i=1, #A, print(A[i])) }

Exponential transform of A370064.

T(n+2, n) = A188588(n + 1).

cp's OEIS Frontend

A325111 Triangle read by rows: T(n,k) is the number of simple connected graphs on n unlabeled nodes with k articulation vertices, (0 <= k <= n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A013923 Number of labeled connected graphs with n vertices and 1 cutpoint.

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A013924 Number of labeled connected graphs with n nodes and 2 cutpoints.

Views

Author

Keywords

Links

Crossrefs

Extensions

A013925 Number of labeled connected graphs with n nodes and 3 cutpoints.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A370065 Triangle read by rows: T(n,k) is the number of simple graphs on n labeled nodes with k articulation vertices, (0 <= k <= n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula