A322395 -id:A322395 - cp's OEIS frontend

A327073 Number of labeled simple connected graphs with n vertices and exactly one bridge.

0, 0, 1, 0, 12, 200, 7680, 506856, 58934848, 12205506096, 4595039095680, 3210660115278000, 4240401342141499392, 10743530775519296581944, 52808688280248604235191296, 507730995579614277599205009240, 9603347831901155679455061048606720, 358743609478638769812094362544644831968

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).

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 12 graphs with exactly one bridge.

Column k = 1 of A327072.
The unlabeled version is A327074.
Connected graphs with no bridges are A007146.
Connected graphs whose bridges are all leaves are A322395.
Connected graphs with at least one bridge are A327071.
Cf. A001187, A006129, A052446, A095983, A327069, A327077, A327108, A327111, A327145, A327146.

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[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[csm[#]]==1&&Count[Table[Length[Union@@Delete[#,i]]1,{i,Length[#]}],True]==1&]],{n,0,5}]

\\ See A095983.
seq(n)={my(p=x*deriv(log(sum(k=0, n-1, 2^binomial(k, 2) * x^k / k!) + O(x^n)))); Vec(serlaplace(log(x/serreverse(x*exp(p)))^2/2), -(n+1)) } \\ Andrew Howroyd, Dec 28 2020

E.g.f.: (x + Sum_{k>=2} A095983(k)*x^k/(k-1)!)^2/2. - Andrew Howroyd, Aug 25 2019

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

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.

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

A327200 Number of labeled graphs with n vertices and non-spanning edge-connectivity >= 2.

Original entry on oeis.org

0, 0, 0, 4, 42, 718, 26262, 1878422, 256204460, 67525498676, 34969833809892, 35954978661632864, 73737437034063350534, 302166248212488958298674, 2475711390267267917290354410, 40563960064630744031043287569378, 1329219366981359393514586291328267704

Offset: 0

Gus Wiseman, Sep 01 2019

nonn

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.

Gus Wiseman, The a(4) = 42 graphs with non-spanning edge-connectivity >= 2.

Row sums of A327148 if the first two columns are removed.
BII-numbers of set-systems with non-spanning edge-connectivity >= 2 are A327102.
Graphs with non-spanning edge-connectivity 1 are A327231.
Cf. A001187, A006129, A095983, A182100, A322395, A326787, A327076, A327079, A327097, A327099, 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]]]]]]]]];
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}]],eConn[#]>=2&]],{n,0,5}]

Binomial transform of A322395, if we assume A322395(0) = A322395(1) = A322395(2) = 0.

A327362 Number of labeled connected graphs covering n vertices with at least one endpoint (vertex of degree 1).

Original entry on oeis.org

0, 0, 1, 3, 28, 475, 14646, 813813, 82060392, 15251272983, 5312295240010, 3519126783483377, 4487168285715524124, 11116496280631563128723, 53887232400918561791887118, 513757147287101157620965656285, 9668878162669182924093580075565776

Offset: 0

Gus Wiseman, Sep 04 2019

nonn

A graph is covering if the vertex set is the union of the edge set, so there are no isolated vertices.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 28 connected covering graphs with at least one endpoint.

The non-connected version is A327227.
The non-covering version is A327364.
Graphs with endpoints are A245797.
Connected covering graphs are A001187.
Connected graphs with bridges are A327071.
Cf. A004110, A059166, A006125, A006129, A059166, A100743, A141580, A322395, A327105, A327229, A327230, A327366, A327369, A327377.

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}]],Union@@#==Range[n]&&Length[csm[#]]==1&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}]

seq(n)={Vec(serlaplace(-x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k! + O(x*x^n))) - log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x + O(x^n)))^k/k!))), -(n+1))} \\ Andrew Howroyd, Sep 11 2019

Inverse binomial transform of A327364.

a(n) = A001187(n) - A059166(n). - Andrew Howroyd, Sep 11 2019

Terms a(7) and beyond from Andrew Howroyd, Sep 11 2019

A322396 Number of unlabeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.

Original entry on oeis.org

1, 1, 1, 2, 5, 18, 98, 779, 10589, 255790, 11633297, 1004417286, 163944008107, 50324877640599, 29001521193534445, 31396727025729968365, 63969154112074956299242, 245871360738448777028919520, 1787330701747389106609369225312, 24636017249593067184544456944967278

Offset: 0

Gus Wiseman, Dec 06 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..25
Eric Weisstein's World of Mathematics, Graph Bridge
Eric Weisstein's World of Mathematics, Endpoint
Gus Wiseman, The a(5) = 18 simple connected graphs whose bridges are all leaves.

Cf. A001187, A006125, A007146, A013922, A054921, A095983, A322338, A322394, A322395.

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

a(6)-a(10) from Andrew Howroyd, Dec 08 2018

Terms a(11) and beyond from Andrew Howroyd, Dec 31 2020

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.

A327364 Number of labeled simple graphs with n vertices, a connected edge-set, and at least one endpoint (vertex of degree 1).

Original entry on oeis.org

0, 0, 1, 6, 46, 655, 17991, 927416, 89009740, 16020407709, 5468601546685, 3578414666656214, 4529751815161579194, 11175105490563109463875, 54043272967471942825421219, 514566625051705610110588073460, 9677104749727084630538798805505880

Offset: 0

Gus Wiseman, Sep 04 2019

nonn

			The a(4) = 46 edge-sets:
  {12}  {12,13}  {12,13,14}  {12,13,14,23}
  {13}  {12,14}  {12,13,24}  {12,13,14,24}
  {14}  {12,23}  {12,13,34}  {12,13,14,34}
  {23}  {12,24}  {12,14,23}  {12,13,23,24}
  {24}  {13,14}  {12,14,34}  {12,13,23,34}
  {34}  {13,23}  {12,23,24}  {12,14,23,24}
        {13,34}  {12,23,34}  {12,14,24,34}
        {14,24}  {12,24,34}  {12,23,24,34}
        {14,34}  {13,14,23}  {13,14,23,34}
        {23,24}  {13,14,24}  {13,14,24,34}
        {23,34}  {13,23,24}  {13,23,24,34}
        {24,34}  {13,23,34}  {14,23,24,34}
                 {13,24,34}
                 {14,23,24}
                 {14,23,34}
                 {14,24,34}

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

The covering case is A327362.
Graphs with endpoints are A245797.
Graphs with connected edge-set are A287689.
Connected graphs with bridges are A327071.
Covering graphs with endpoints are A327227.
Cf. A001187, A059167, A141580, A322395, A327148, A327335, A327369.

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&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}]

seq(n)={my(x=x + O(x*x^n)); Vec(serlaplace(exp(x)*(-x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!)) - log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x))^k/k!)))), -(n+1))} \\ Andrew Howroyd, Sep 11 2019

Binomial transform of A327362.

Terms a(7) and beyond from Andrew Howroyd, Sep 11 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.

cp's OEIS Frontend

A327073 Number of labeled simple connected graphs with n vertices and exactly one bridge.

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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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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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A327200 Number of labeled graphs with n vertices and non-spanning edge-connectivity >= 2.

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A327362 Number of labeled connected graphs covering n vertices with at least one endpoint (vertex of degree 1).

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A322396 Number of unlabeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.

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

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