A327073 -id:A327073 - cp's OEIS frontend

A327071 Number of labeled simple connected graphs with n vertices and at least one bridge, or graphs with spanning edge-connectivity 1.

0, 0, 1, 3, 28, 475, 14736, 818643, 82367552, 15278576679, 5316021393280, 3519977478407687, 4487518206535452672, 11116767463976825779115, 53887635281876408097483776, 513758302006787897939587736715, 9668884580476067306398361085853696

Offset: 0

Gus Wiseman, Aug 24 2019

nonn

A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Connected graphs with no bridges are counted by A095983 (2-edge-connected graphs).

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.

Jean-François Alcover and Vaclav Kotesovec, Table of n, a(n) for n = 0..82 [using A001187 and b-file from A095983]
Eric Weisstein's World of Mathematics, Bridged Graph

Column k = 1 of A327069.
The unlabeled version is A052446.
Connected graphs without bridges are A007146.
The enumeration of labeled connected graphs by number of bridges is A327072.
Connected graphs with exactly one bridge are A327073.
Graphs with non-spanning edge-connectivity 1 are A327079.
BII-numbers of set-systems with spanning edge-connectivity 1 are A327111.
Covering set-systems with spanning edge-connectivity 1 are A327145.
Graphs with spanning edge-connectivity 2 are A327146.
Cf. A001187, A001349, A006125, A059166, A322395, A327071, A327077, A327099, 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]]]]]]]]];
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],#]==1&]],{n,0,4}]

a(1) = 0; a(n > 1) = A001187(n) - A095983(n).

A327069 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and spanning edge-connectivity k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 4, 3, 1, 0, 26, 28, 9, 1, 0, 296, 475, 227, 25, 1, 0, 6064, 14736, 10110, 1782, 75, 1, 0

Offset: 0

Gus Wiseman, Aug 23 2019

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.

We consider a graph with one vertex and no edges to be disconnected.

			Triangle begins:
    1
    1   0
    1   1   0
    4   3   1   0
   26  28   9   1   0
  296 475 227  25   1   0

Row sums are A006125.
Column k = 0 is A054592, if we assume A054592(1) = 1.
Column k = 1 is A327071.
Column k = 2 is A327146.
The unlabeled version (except with offset 1) is A263296.
Cf. A001187, A095983, A259862, A322338, A326787, A327070, A327072, A327073.

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]]]]]]]]];
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],#]==k&]],{n,0,5},{k,0,n}]

a(21)-a(27) from Robert Price, May 25 2021

A327079 Number of labeled simple connected graphs covering n vertices with at least one bridge that is not an endpoint/leaf (non-spanning edge-connectivity 1).

Original entry on oeis.org

0, 0, 1, 0, 12, 180, 4200, 157920, 9673664, 1011129840, 190600639200, 67674822473280, 46325637863907072, 61746583700640860736, 161051184122415878112640, 824849999242893693424992000, 8317799170120961768715123118080

Offset: 0

Gus Wiseman, Aug 25 2019

nonn

A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Graphs with no bridges are counted by A095983 (2-edge-connected graphs).

Also labeled simple connected graphs covering n vertices with non-spanning edge-connectivity 1, where 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.

Column k = 1 of A327149.
The non-covering version is A327231.
Connected bridged graphs (spanning edge-connectivity 1) are A327071.
BII-numbers of graphs with non-spanning edge-connectivity 1 are A327099.
Covering set-systems with non-spanning edge-connectivity 1 are A327129.
Cf. A001187, A006129, A052446, A059166, A322395, A327072, A327073, A327148.

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[#]==1&]],{n,0,4}]

a(n) = A001187(n) - A322395(n) for n > 2. - Andrew Howroyd, Aug 27 2019

Inverse binomial transform of A327231.

Terms a(6) and beyond from Andrew Howroyd, Aug 27 2019

A327077 Triangle read by rows where T(n,k) is the number of unlabeled simple connected graphs with n vertices and k bridges.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 3, 1, 0, 2, 0, 11, 4, 3, 0, 3, 0, 60, 25, 14, 7, 0, 6, 0, 502, 197, 91, 34, 18, 0, 11, 0, 7403, 2454, 826, 267, 100, 44, 0, 23, 0, 197442, 48201, 11383, 2800, 831, 259, 117, 0, 47, 0

Offset: 0

Gus Wiseman, Aug 26 2019

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

			Triangle begins:
     1
     1    0
     0    1   0
     1    0   1   0
     3    1   0   2   0
    11    4   3   0   3  0
    60   25  14   7   0  6  0
   502  197  91  34  18  0 11  0
  7403 2454 826 267 100 44  0 23 0
  ...

Gus Wiseman, Unlabeled connected graphs with 5 vertices and k bridges.

The labeled version is A327072.
Row sums are A001349.
Row sums without the k = 0 column are A052446.
Column k = 0 is A007146, if we assume A007146(0) = 1.
Column k = 1 is A327074.
Column k = n - 1 is A000055.
Cf. A002494, A327071, A327073, A327108, A327109, A327111, A327130, A327144, A327145, A327146.

a(21)-a(54) from Andrew Howroyd, Aug 28 2019

A327231 Number of labeled simple connected graphs covering a subset of {1..n} with at least one non-endpoint bridge (non-spanning edge-connectivity 1).

Original entry on oeis.org

0, 0, 1, 3, 18, 250, 5475, 191541, 11065572, 1104254964, 201167132805, 69828691941415, 47150542741904118, 62354150876493659118, 161919876753750972738791, 827272271567137357352991705, 8331016130913639432634637862600, 165634930763383717802534343776893928

Offset: 0

Gus Wiseman, Sep 01 2019

nonn

A bridge is an edge whose removal disconnected the graph, while an endpoint is a vertex belonging to only one edge. The non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed to obtain a graph whose edge-set is disconnected or empty.

			The a(2) = 1 through a(4) = 18 edge-sets:
  {12}  {12}  {12}
        {13}  {13}
        {23}  {14}
              {23}
              {24}
              {34}
              {12,13,24}
              {12,13,34}
              {12,14,23}
              {12,14,34}
              {12,23,34}
              {12,24,34}
              {13,14,23}
              {13,14,24}
              {13,23,24}
              {13,24,34}
              {14,23,24}
              {14,23,34}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Column k = 1 of A327148.
The covering version is A327079.
Connected bridged graphs (spanning edge-connectivity 1) are A327071.
BII-numbers of set-systems with non-spanning edge-connectivity 1 are A327099.
Covering set-systems with non-spanning edge-connectivity 1 are A327129.
Cf. A001187, A052446, A322395, A327072, A327073, A327102.

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[Select[Subsets[Subsets[Range[n],{2}]],edgeConnSys[#]==1&]],{n,0,4}]

Binomial transform of A327079.

Terms a(6) and beyond from Andrew Howroyd, Sep 11 2019

A327072 Triangle read by rows where T(n,k) is the number of labeled simple connected graphs with n vertices and exactly k bridges.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 1, 0, 3, 0, 10, 12, 0, 16, 0, 253, 200, 150, 0, 125, 0, 11968, 7680, 3600, 2160, 0, 1296, 0, 1047613, 506856, 190365, 68600, 36015, 0, 16807, 0, 169181040, 58934848, 16353792, 4695040, 1433600, 688128, 0, 262144, 0, 51017714393, 12205506096, 2397804444, 500828832, 121706550, 33067440, 14880348, 0, 4782969, 0

Offset: 0

Gus Wiseman, Aug 24 2019

A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Connected graphs with no bridges are counted by A095983 (2-edge-connected graphs).

Warning: In order to be consistent with A001187, we have treated the n = 0 and n = 1 cases in ways that are not consistent with A095983.

			Triangle begins:
    1
    1   0
    0   1   0
    1   0   3   0
   10  12   0  16   0
  253 200 150   0 125   0

Andrew Howroyd, Table of n, a(n) for n = 0..1325
Gus Wiseman, The 10 + 12 + 16 graphs counted in row n = 4.

Column k = 0 is A095983, if we assume A095983(0) = A095983(1) = 1.
Column k = 1 is A327073.
Column k = n - 1 is A000272.
Row sums are A001187.
The unlabeled version is A327077.
Row sums without the first column are A327071.
Cf. A001349, A007146, A052446, A054592, A059166, A322395, A327069, A327148.

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]]]]]]]]];
Table[If[n<=1&&k==0,1,Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[csm[#]]==1&&Count[Table[Length[Union@@Delete[#,i]]1,{i,Length[#]}],True]==k&]]],{n,0,4},{k,0,n}]

\\ p is e.g.f. of A053549.
T(n)={my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))), v=Vec(1+serreverse(serreverse(log(x/serreverse(x*exp(p))))/exp(x*y+O(x^n))))); vector(#v, k, max(0,k-2)!*Vecrev(v[k], k)) }
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Dec 28 2020

Terms a(21) and beyond from Andrew Howroyd, Dec 28 2020

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

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A327071 Number of labeled simple connected graphs with n vertices and at least one bridge, or graphs with spanning edge-connectivity 1.

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A327069 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and spanning edge-connectivity k.

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A327079 Number of labeled simple connected graphs covering n vertices with at least one bridge that is not an endpoint/leaf (non-spanning edge-connectivity 1).

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A327077 Triangle read by rows where T(n,k) is the number of unlabeled simple connected graphs with n vertices and k bridges.

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A327231 Number of labeled simple connected graphs covering a subset of {1..n} with at least one non-endpoint bridge (non-spanning edge-connectivity 1).

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A327072 Triangle read by rows where T(n,k) is the number of labeled simple connected graphs with n vertices and exactly k bridges.

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

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