A263296 -id:A263296 - cp's OEIS frontend

A327201 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs covering n vertices with non-spanning edge-connectivity k.

1, 0, 1, 0, 0, 1, 1, 1, 1, 2, 2, 1, 2, 3, 7, 5, 4, 1, 1

Offset: 0

Gus Wiseman, Sep 03 2019

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed to obtain a disconnected or empty graph, ignoring isolated vertices.

			Triangle begins:
  1
  {}
  0 1
  0 0 1 1
  1 1 2 2 1
  2 3 7 5 4 1 1

Gus Wiseman, Unlabeled graphs covering 5 vertices, organized by non-spanning edge-connectivity.

Row sums are A002494.
Column k = 0 is A327075.
The labeled version is A327149.
Spanning edge-connectivity is A263296.
The non-covering version is A327236 (partial sums).
Cf. A000088, A322338, A322396, A326787, A327076, A327077, A327079, A327126, A327129, A327148, A327235.

A327149 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of simple labeled graphs covering n vertices with non-spanning edge-connectivity k.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 1, 3, 12, 15, 10, 1, 40, 180, 297, 180, 60, 10, 1

Offset: 0

Gus Wiseman, Aug 27 2019

The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty graph.

			Triangle begins:
   1
   {}
   0   1
   0   0   3   1
   3  12  15  10   1
  40 180 297 180  60  10   1

Row sums are A006129.
Column k = 0 is A327070.
Column k = 1 is A327079.
The corresponding triangle for vertex-connectivity is A327126.
The corresponding triangle for spanning edge-connectivity is A327069.
The non-covering version is A327148.
The unlabeled version is A327201.
Cf. A001187, A263296, A322338, A322395, A326787, A327097, A327099, A327102, A327125, A327129, A327144.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&eConn[#]==k&]],{n,0,4},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}

A327148(n,k) = Sum_{m = 0..n} binomial(n,m) T(m,k). In words, column k is the inverse binomial transform of column k of A327148.

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

A327074 Number of unlabeled connected graphs with n vertices and exactly one bridge.

Original entry on oeis.org

0, 0, 1, 0, 1, 4, 25, 197, 2454, 48201, 1604016, 93315450, 9696046452, 1822564897453, 625839625866540, 395787709599238772, 464137745175250610865, 1015091996575508453655611, 4160447945769725861550193834, 32088553211819016484736085677320, 467409605282347770524641700949750858

Offset: 0

Gus Wiseman, Aug 24 2019

nonn

A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Unlabeled graphs with no bridges are counted by A007146 (unlabeled graphs with spanning edge-connectivity >= 2).

Gus Wiseman, The a(5) = 4 unlabeled connected graphs with n vertices and exactly one bridge.

The labeled version is A327073.
Unlabeled graphs with at least one bridge are A052446.
The enumeration of unlabeled connected graphs by number of bridges is A327077.
BII-numbers of set-systems with spanning edge-connectivity >= 2 are A327109.
Cf. A001349, A002494, A007146, A263296, A327075, A327111, A327144, A327145.

A007145 = Cases[Import["https://oeis.org/A007145/b007145.txt", "Table"], {, }][[All, 2]];
f[x_] = A007145 . x^Range[lg = Length[A007145]];
(f[x]^2 + f[x^2])/2 + O[x]^lg // CoefficientList[#, x]& (* Jean-François Alcover, Sep 23 2019 *)

G.f.: (f(x)^2 + f(x^2))/2 where f(x) is the g.f. of A007145. - Andrew Howroyd, Aug 25 2019

Terms a(6) and beyond from Andrew Howroyd, Aug 25 2019

A327196 Number of connected set-systems with n vertices and at least one bridge that is not an endpoint (non-spanning edge-connectivity 1).

Original entry on oeis.org

0, 1, 4, 44, 2960

Offset: 0

Gus Wiseman, Aug 31 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. The non-spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty set-system.

			Non-isomorphic representatives of the a(3) = 44 set-systems:
  {{1}}
  {{1,2}}
  {{1,2,3}}
  {{1},{2},{1,2}}
  {{1},{1,2},{2,3}}
  {{1},{2},{1,2,3}}
  {{1},{2,3},{1,2,3}}
  {{1},{2},{1,2},{1,3}}
  {{1},{2},{1,3},{2,3}}
  {{1},{2},{3},{1,2,3}}
  {{1},{2},{1,3},{1,2,3}}
  {{1},{2},{3},{1,2},{1,3}}
  {{1},{2},{3},{1,2},{1,2,3}}

The covering version is A327129.
The BII-numbers of these set-systems are A327099.
The restriction to simple graphs is A327231.
Set-systems with spanning edge-connectivity 1 are A327145.
Cf. A263296, A322395, A327079, A327111, A327148, A327149, A327200.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],eConn[#]==1&]],{n,0,3}]

Binomial transform of A327129.

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

A327201 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs covering n vertices with non-spanning edge-connectivity k.

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A327149 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of simple labeled graphs covering n vertices with non-spanning 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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A327074 Number of unlabeled connected graphs with n vertices and exactly one bridge.

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A327196 Number of connected set-systems with n vertices and at least one bridge that is not an endpoint (non-spanning edge-connectivity 1).

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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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