A327200 -id:A327200 - cp's OEIS frontend

A095983 Number of 2-edge-connected labeled graphs on n nodes.

0, 0, 0, 1, 10, 253, 11968, 1047613, 169181040, 51017714393, 29180467201536, 32121680070545657, 68867078000231169536, 290155435185687263172693, 2417761175748567327193407488, 40013922635723692336670167608181, 1318910073755307133701940625759574016

Offset: 0

Yifei Chen (yifei(AT)mit.edu), Jul 17 2004

nonn

From Falk Hüffner, Jun 28 2018: (Start)
Essentially the same sequence arises as the number of connected bridgeless labeled graphs (graphs that are k-edge connected for k >= 2, starting elements of this sequence are 1, 1, 0, 1, 10, 253, 11968, ...).
Labeled version of A007146. (End)
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. This sequence counts graphs with spanning edge-connectivity >= 2, which, for n > 1, are connected graphs with no bridges. - Gus Wiseman, Sep 20 2019

Jinyuan Wang, Table of n, a(n) for n = 0..82
P. Hanlon and R. W. Robinson, Counting bridgeless graphs, J. Combin. Theory, B 33 (1982), 276-305.
Gus Wiseman, The a(4) = 10 labeled graphs with spanning edge-connectivity >= 2.
Gus Wiseman, Non-isomorphic representatives of the a(5) = 253 graphs with spanning edge-connectivity >= 2.

Cf. A053549, A198046.
The unlabeled version is A007146.
Row sums of A327069 if the first two columns are removed.
BII-numbers of set-systems with spanning edge-connectivity >= 2 are A327109.
Graphs with spanning edge-connectivity 2 are A327146.
Graphs with non-spanning edge-connectivity >= 2 are A327200.
2-vertex-connected graphs are A013922.
Graphs without endpoints are A059167.
Graphs with spanning edge-connectivity 1 are A327071.
Cf. A001187, A002218, A052446, A059166, A100743, A322395, A327079, A327144.

seq[n_] := Module[{v, p, q, c}, v[_] = 0; p = x*D[#, x]& @ Log[ Sum[ 2^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^(n+1)]; q = x*E^p; p -= q; For[k = 3, k <= n, k++, c = Coefficient[p, x, k]; v[k] = c*(k-1)!; p -= c*q^k]; Join[{0}, Array[v, n]]];
seq[16] (* Jean-François Alcover, Aug 13 2019, after Andrew Howroyd *)
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]]]]]]]]];
spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],spanEdgeConn[Range[n],#]>=2&]],{n,0,5}] (* Gus Wiseman, Sep 20 2019 *)

\\ here p is initially A053549, q is A198046 as e.g.f.s.
seq(n)={my(v=vector(n));
my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))));
my(q=x*exp(p)); p-=q;
for(k=3, n, my(c=polcoeff(p,k)); v[k]=c*(k-1)!; p-=c*q^k);
concat([0],v)} \\ Andrew Howroyd, Jun 18 2018

seq(n)={my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n)))); Vec(serlaplace(log(x/serreverse(x*exp(p)))/x-1), -(n+1))} \\ Andrew Howroyd, Dec 28 2020

a(n) = A001187(n) - A327071(n). - Gus Wiseman, Sep 20 2019

Name corrected and more terms from Pavel Irzhavski, Nov 01 2014
Offset corrected by Falk Hüffner, Jun 17 2018
a(12)-a(16) from Andrew Howroyd, Jun 18 2018

A007146 Number of unlabeled simple connected bridgeless graphs with n nodes.

Original entry on oeis.org

1, 0, 1, 3, 11, 60, 502, 7403, 197442, 9804368, 902818087, 153721215608, 48443044675155, 28363687700395422, 30996524108446916915, 63502033750022111383196, 244852545022627009655180986, 1783161611023802810566806448531, 24603891215865809635944516464394339

Offset: 1

N. J. A. Sloane

Also unlabeled simple graphs with spanning edge-connectivity >= 2. The spanning edge-connectivity of a set-system is the minimum number of edges that must be removed (without removing incident vertices) to obtain a set-system that is disconnected or covers fewer vertices. - Gus Wiseman, Sep 02 2019

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 1..40 (terms 1..22 from R. J. Mathar)
P. Hanlon and R. W. Robinson, Counting bridgeless graphs, J. Combin. Theory, B 33 (1982), 276-305, Table III.
Eric Weisstein's World of Mathematics, Bridgeless Graph
Eric Weisstein's World of Mathematics, Connected Graph
Eric Weisstein's World of Mathematics, Simple Graph
Gus Wiseman, The a(3) = 1 through a(5) = 11 connected bridgeless graphs.

Cf. A005470 (number of simple graphs).
Cf. A007145 (number of simple connected rooted bridgeless graphs).
Cf. A052446 (number of simple connected bridged graphs).
Cf. A263914 (number of simple bridgeless graphs).
Cf. A263915 (number of simple bridged graphs).
The labeled version is A095983.
Row sums of A263296 if the first two columns are removed.
BII-numbers of set-systems with spanning edge-connectivity >= 2 are A327109.
Graphs with non-spanning edge-connectivity >= 2 are A327200.
2-vertex-connected graphs are A013922.
Cf. A000719, A001349, A002494, A261919, A327069, A327071, A327074, A327075, A327077, A327109, A327144, A327146.

\\ Translation of theorem 3.2 in Hanlon and Robinson reference. See A004115 for graphsSeries and A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(gc=sLog(graphsSeries(n)), gcr=sPoint(gc)); sSolve( gc + gcr^2/2 - sRaise(gcr,2)/2, x*sv(1)*sExp(gcr) )}
NumUnlabeledObjsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 31 2020

a(n) = A001349(n) - A052446(n). - Gus Wiseman, Sep 02 2019

Reference gives first 22 terms.

A327148 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of labeled simple graphs with n vertices and non-spanning edge-connectivity k.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 4, 18, 27, 14, 1, 56, 250, 402, 240, 65, 10, 1, 1031, 5475, 11277, 9620, 4282, 921, 146, 15, 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 isolated vertices) to obtain a disconnected or empty graph.

			Triangle begins:
   1
   1
   1   1
   1   3   3   1
   4  18  27  14   1
  56 250 402 240  65  10   1

Row sums are A006125.
Column k = 0 is A327199.
Column k = 1 is A327231.
The corresponding triangle for vertex-connectivity is A327125.
The corresponding triangle for spanning edge-connectivity is A327069.
The covering version is A327149.
The unlabeled version is A327236, with covering version A327201.
Cf. A001187, A263296, A322338, A322395, A326787, A327079, A327097, A327099, A327102, A327126, A327144, A327196, A327200, A327201.

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]]]]]]]]];
edgeConnSys[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}]],edgeConnSys[#]==k&]],{n,0,4},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}

T(n,k) = Sum_{m = 0..n} binomial(n,m) A327149(m,k). In words, column k is the binomial transform of column k of A327149.

a(20)-a(28) from Robert Price, May 25 2021

A327236 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs with n vertices whose edge-set has non-spanning edge-connectivity k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 1, 4, 5, 10, 8, 5, 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
  1
  1  1
  1  1  1  1
  2  2  3  3  1
  4  5 10  8  5  1  1

Gus Wiseman, Unlabeled graphs with 5 vertices, organized by non-spanning edge-connectivity (isolated vertices not shown).

Row sums are A000088.
Column k = 0 is A327235.
The labeled version is A327148.
The covering version is A327201.
Spanning edge-connectivity is A263296.
Vertex-connectivity is A259862.
Cf. A322338, A322396, A326787, A327069, A327077, A327097, A327099, A327102, A327200, A327231.

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]]]]]]]]];
edgeConnSys[sys_]:=If[Length[csm[sys]]!=1,0,Length[sys]-Max@@Length/@Select[Union[Subsets[sys]],Length[csm[#]]!=1&]];
Table[Length[Union[normclut/@Select[Subsets[Subsets[Range[n],{2}]],edgeConnSys[#]==k&]]],{n,0,5},{k,0,Binomial[n,2]}]//.{foe___,0}:>{foe}

A327199 Number of labeled simple graphs with n vertices whose edge-set is not connected.

Original entry on oeis.org

1, 1, 1, 1, 4, 56, 1031, 27189, 1165424, 89723096, 13371146135, 3989665389689, 2388718032951812, 2852540291841718752, 6768426738881535155247, 31870401029679493862010949, 297787425565749788134314214272

Offset: 0

Gus Wiseman, Sep 01 2019

nonn

Also graphs with non-spanning edge-connectivity 0.

			The a(4) = 4 edge-sets: {}, {12,34}, {13,24}, {14,23}.

Column k = 0 of A327148.
The covering case is A327070.
The unlabeled version is A327235.
Cf. A001187, A006129, A322395, A326787, A327075, A327076, A327079, A327129, A327200, A327201, A327231, A327236.

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]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[csm[#]]!=1&]],{n,0,5}]

Binomial transform of A327070.

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.

cp's OEIS Frontend

A095983 Number of 2-edge-connected labeled graphs on n nodes.

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A007146 Number of unlabeled simple connected bridgeless graphs with n nodes.

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A327148 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of labeled simple graphs with n vertices and non-spanning edge-connectivity k.

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A327236 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of unlabeled simple graphs with n vertices whose edge-set has non-spanning edge-connectivity k.

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A327199 Number of labeled simple graphs with n vertices whose edge-set is not connected.

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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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