A052447 -id:A052447 - cp's OEIS frontend

A052446 Number of unlabeled simple connected bridged graphs on n nodes.

0, 1, 1, 3, 10, 52, 351, 3714, 63638, 1912203, 103882478, 10338614868, 1892863194064, 639799762452639, 400857034314325045, 467526363203064793081, 1019286659457016864347582, 4170114225096278323394128049, 32130213534058019378134295287305

Offset: 1

Eric W. Weisstein, May 08 2000

These are unlabeled connected graphs with spanning edge-connectivity 1, where the spanning edge-connectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty graph. - Gus Wiseman, Sep 02 2019

Jean-François Alcover, Table of n, a(n) for n = 1..22
Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, Discr. Appl. Math. 201 (2016) 172-181
Eric Weisstein's World of Mathematics, k-Edge-Connected Graph
Eric Weisstein's World of Mathematics, Bridged Graph
Eric Weisstein's World of Mathematics, Connected Graph
Gus Wiseman, The a(2) = 1 through a(5) = 10 connected bridged graphs

Cf. other k-edge-connected unlabeled graph sequences A052446, A052447, A052448, A241703, A241704, A241705.
Cf. A001349 (number of simple connected graphs).
Cf. A007146 (number of simple connected bridgeless graphs).
Cf. A263914 (number of simple bridgeless graphs).
Cf. A263915 (number of simple bridged graphs).
Column k = 1 of A263296.
Row sums of A327077 if the first column is removed.
BII-numbers of set-systems with spanning edge-connectivity 1 are A327111.
The labeled version is A327071.
Cf. A002494, A327069, A327074, A327109, A327144.

A001349 = Cases[Import["https://oeis.org/A001349/b001349.txt", "Table"], {, }][[All, 2]];
A007146 = Cases[Import["https://oeis.org/A007146/b007146.txt", "Table"], {, }][[All, 2]] ;
a[n_] := A001349[[n + 1]] - A007146[[n]];
Array[a, 22] (* Jean-François Alcover, Nov 09 2019 *)

a(n) = A001349(n) - A007146(n).

a(8) and a(9) and better description by Eric W. Weisstein, Nov 07 2010
a(10) from the Encyclopedia of Finite Graphs by Travis Hoppe and Anna Petrone, Apr 22 2014
Additional terms from A001349 and A007146 by Eric W. Weisstein, Oct 29 2015
a(18)-a(22) from A001349 and A007146 by Jean-François Alcover, Nov 09 2019

A263296 Triangle read by rows: T(n,k) is the number of graphs with n vertices with edge connectivity k.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 5, 3, 2, 1, 13, 10, 8, 2, 1, 44, 52, 41, 15, 3, 1, 191, 351, 352, 121, 25, 3, 1, 1229, 3714, 4820, 2159, 378, 41, 4, 1, 13588, 63638, 113256, 68715, 14306, 1095, 65, 4, 1, 288597, 1912203, 4602039, 3952378, 1141575, 104829, 3441, 100, 5, 1

Offset: 1

Christian Stump, Oct 13 2015

This is spanning edge-connectivity. The spanning edge-connectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a graph that is disconnected or covers fewer vertices. The non-spanning edge-connectivity of a graph (A327236) is the minimum number of edges that must be removed to obtain a graph whose edge-set is disconnected or empty. Compare to vertex-connectivity (A259862). - Gus Wiseman, Sep 03 2019

			Triangle begins:
     1;
     1,    1;
     2,    1,    1;
     5,    3,    2,    1;
    13,   10,    8,    2,   1;
    44,   52,   41,   15,   3,  1;
   191,  351,  352,  121,  25,  3, 1;
  1229, 3714, 4820, 2159, 378, 41, 4, 1;
  ...

Georg Grasegger, Table of n, a(n) for n = 1..78 (rows 1..12)
FindStat - Combinatorial Statistic Finder, The edge connectivity of a graph.
Jens M. Schmidt, Combinatorial Data
Gus Wiseman, Unlabeled graphs with 5 vertices organized by spanning edge-connectivity (isolated vertices not shown).

Row sums give A000088, n >= 1.
Columns k=0..10 are A000719, A052446, A052447, A052448, A241703, A241704, A241705, A324096, A324097, A324098, A324099.
Number of graphs with edge connectivity at least k for k=1..10 are A001349, A007146, A324226, A324227, A324228, A324229, A324230, A324231, A324232, A324233.
The labeled version is A327069.
Cf. A002494, A095983, A259862, A327076, A327108, A327109, A327111, A327144, A327145, A327147, A327236.

a(22)-a(55) added by Andrew Howroyd, Aug 11 2019

A052448 Number of simple unlabeled n-node graphs of edge-connectivity 3.

Original entry on oeis.org

0, 0, 0, 1, 2, 15, 121, 2159, 68715, 3952378, 389968005, 65161587084

Offset: 1

Eric W. Weisstein, May 08 2000

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Jens M. Schmidt, Combinatorial data.
Eric Weisstein's World of Mathematics, k-Edge Connected Graph.

Column k=3 of A263296.
Cf. other edge-connectivity unlabeled graph sequences A052446, A052447, A241703, A241704, A241705.
Cf. A338511, A338583, A338593, A338594, A338604, A339072.

a(8), a(9), a(10) from the Encyclopedia of Finite Graphs by Travis Hoppe and Anna Petrone, Apr 22 2014
a(11) by Jens M. Schmidt, Feb 18 2019
a(12) from Jens M. Schmidt's web page, Jan 10 2021

A241703 Number of simple unlabeled n-node graphs of edge-connectivity 4.

Original entry on oeis.org

0, 0, 0, 0, 1, 3, 25, 378, 14306, 1141575, 164245876, 39637942895

Offset: 1

Travis Hoppe and Anna Petrone, Apr 27 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Jens M. Schmidt, Combinatorial data.
Eric Weisstein's World of Mathematics, k-Edge Connected Graph.

Column k=4 of A263296.

Cf. other edge-connectivity unlabeled graph sequences A052446, A052447, A052448, A241704, A241705.

a(11) by Jens M. Schmidt, Feb 18 2019

a(12) from Jens M. Schmidt's web page, Jan 10 2021

A241704 Number of simple unlabeled n-node graphs of edge-connectivity 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 3, 41, 1095, 104829, 21981199, 8077770931

Offset: 1

Travis Hoppe and Anna Petrone, Apr 27 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Jens M. Schmidt, Data files in graph6 format
Eric Weisstein's World of Mathematics, k-Edge Connected Graph.

Column k=5 of A263296.

Cf. other edge-connectivity unlabeled graph sequences A052446, A052447, A052448, A241703, A241705.

a(11)-a(12) by Jens M. Schmidt, Feb 18 2019

A241705 Number of simple unlabeled n-node graphs of edge-connectivity 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 4, 65, 3441, 857365, 487560158, 466534106494

Offset: 1

Travis Hoppe and Anna Petrone, Apr 27 2014

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
Jens M. Schmidt, Data files in graph6 format
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, k-Edge Connected Graph.

Column k=6 of A263296.

Cf. other edge-connectivity unlabeled graph sequences A052446, A052447, A052448, A241703, A241704.

a(11)-a(13) by Jens M. Schmidt, Feb 20 2019

A324227 Number of simple 4-edge-connected non-isomorphic n-vertex graphs.

Original entry on oeis.org

0, 0, 0, 0, 1, 4, 29, 424, 15471, 1249951, 187095386, 48211082866

Offset: 1

Jens M. Schmidt, Feb 18 2019

Jens M. Schmidt, Data files in graph6 format

Cf. A007146, A052447.

a(12) added by Georg Grasegger, Jan 07 2025

A290011 Number of ways to connect n nodes with n+1 edges to form a 2-edge-connected graph.

Original entry on oeis.org

6, 85, 900, 9450, 104160, 1224720, 15422400, 207900000, 2993760000, 45924278400, 748280332800, 12913284384000, 235381386240000, 4520194398720000, 91233825306624000, 1931115968990208000, 42778526977105920000, 989887004576870400000, 23885015465274163200000

Offset: 4

Eugene Y. Q. Shen, Jul 17 2017

Robert Israel, Table of n, a(n) for n = 4..448

Cf. A001072, A052447, A054596, A095983

seq((n^2 + 2 *n - 18)* n!/24, n=6..30); # Robert Israel, Jul 19 2017

Table[(n - 4) (n!/8) + (n (n - 1)/2 - 3) (n!/12), {n, 4, 22}] (* Michael De Vlieger, Jul 18 2017 *)

a(n) = (n - 4)*(n!/8) + (n*(n - 1)/2 - 3)*(n!/12); \\ Michel Marcus, Jul 18 2017

a(n) = (n - 4)*(n!/8) + (n*(n - 1)/2 - 3)*(n!/12) = (n^2 + 2 n - 18)*(n!/24).

E.g.f.: x^4*(3*x^2+x-6)/(24*(x-1)^3). - Robert Israel, Jul 19 2017

A324096 Number of simple non-isomorphic n-vertex graphs of edge-connectivity 7.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 4, 100, 10790, 7772555, 12294282710

Offset: 1

Jens M. Schmidt, Feb 18 2019

Jens M. Schmidt, Data files in graph6 format

Column k=7 of A263296.

Cf. A052447, A007146.

A324097 Number of simple non-isomorphic n-vertex graphs of edge-connectivity 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 150, 36095, 77299066, 348666442245

Offset: 1

Jens M. Schmidt, Feb 18 2019

Jens M. Schmidt, Data files in graph6 format

Column k=8 of A263296.

Cf. A052447, A007146.

cp's OEIS Frontend

A052446 Number of unlabeled simple connected bridged graphs on n nodes.

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A263296 Triangle read by rows: T(n,k) is the number of graphs with n vertices with edge connectivity k.

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A052448 Number of simple unlabeled n-node graphs of edge-connectivity 3.

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A241703 Number of simple unlabeled n-node graphs of edge-connectivity 4.

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A241704 Number of simple unlabeled n-node graphs of edge-connectivity 5.

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A241705 Number of simple unlabeled n-node graphs of edge-connectivity 6.

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A324227 Number of simple 4-edge-connected non-isomorphic n-vertex graphs.

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A290011 Number of ways to connect n nodes with n+1 edges to form a 2-edge-connected graph.

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A324096 Number of simple non-isomorphic n-vertex graphs of edge-connectivity 7.

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A324097 Number of simple non-isomorphic n-vertex graphs of edge-connectivity 8.

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