A327335 -id:A327335 - cp's OEIS frontend

A327230 Number of non-isomorphic set-systems covering n vertices with at least one endpoint/leaf.

0, 1, 3, 14, 198

Offset: 0

Gus Wiseman, Sep 01 2019

A set-system is a finite set of finite nonempty sets. Elements of a set-system are sometimes called edges. A leaf is an edge containing a vertex that does not belong to any other edge, while an endpoint is a vertex belonging to only one edge.

Also covering set-systems with minimum vertex-degree 1.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 14 set-systems:
  {{1}}  {{1,2}}      {{1,2,3}}
         {{1},{2}}    {{1},{2,3}}
         {{2},{1,2}}  {{1},{2},{3}}
                      {{1,3},{2,3}}
                      {{3},{1,2,3}}
                      {{1},{3},{2,3}}
                      {{2,3},{1,2,3}}
                      {{2},{1,3},{2,3}}
                      {{2},{3},{1,2,3}}
                      {{3},{1,3},{2,3}}
                      {{1},{2},{3},{2,3}}
                      {{3},{2,3},{1,2,3}}
                      {{2},{3},{1,3},{2,3}}
                      {{2},{3},{2,3},{1,2,3}}

Unlabeled covering set-systems are A055621.
The labeled version is A327229.
The non-covering version is A327335 (partial sums).
Cf. A002494, A245797, A261919, A283877, A327103, A327105, A327197, A327227, A327228.

A294217 Triangle read by rows: T(n,k) is the number of graphs with n vertices and minimum vertex degree k, (0 <= k < n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 4, 4, 2, 1, 11, 12, 8, 2, 1, 34, 60, 43, 15, 3, 1, 156, 378, 360, 121, 25, 3, 1, 1044, 3843, 4869, 2166, 378, 41, 4, 1, 12346, 64455, 113622, 68774, 14306, 1095, 65, 4, 1, 274668, 1921532, 4605833, 3953162, 1141597, 104829, 3441, 100, 5, 1

Offset: 1

Eric W. Weisstein, Oct 25 2017

Terms may be computed without generating each graph by enumerating the number of graphs by degree sequence. A PARI program showing this technique for graphs with labeled vertices is given in A327366. Burnside's lemma can be used to extend this method to the unlabeled case. - Andrew Howroyd, Mar 10 2020

			Triangle begins:
    1;
    1,   1;
    2,   1,   1;
    4,   4,   2,   1;
   11,  12,   8,   2,  1;
   34,  60,  43,  15,  3, 1;
  156, 378, 360, 121, 25, 3, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..210 (first 20 rows)
Eric Weisstein's World of Mathematics, Minimum Vertex Degree

Row sums are A000088 (simple graphs on n nodes).
Columns k=0..2 are A000088(n-1), A324693, A324670.
Cf. A263293 (triangle of n-node maximum vertex degree counts).
The labeled version is A327366.
Cf. A002494, A004110, A261919, A327227, A327230, A327335, A327372.

T(n, 0) = A000088(n-1).
T(n, n-2) = A004526(n) for n > 1.
T(n, n-1) = 1.
T(n, k) = A263293(n, n-1-k). - Andrew Howroyd, Sep 03 2019

A141580 Number of unlabeled non-mating graphs with n vertices.

Original entry on oeis.org

0, 1, 2, 6, 18, 78, 456, 4299, 68754, 1990286, 106088988, 10454883132, 1904236651216, 641859005526860, 401547534010157680, 467956331904669136874, 1019785644052109276678788, 4171197546082606538129623140

Offset: 1

Tanya Khovanova, Aug 19 2008

nonn

a(n) is the difference between A000088 (number of graphs on n unlabeled nodes) and A004110 (number of n-node graphs without endpoints)
A non-mating graph has two vertices with an identical set of neighbors.
The adjacency matrix of a non-mating graph is degenerate.
Also the number of unlabeled graphs with n vertices and at least one endpoint. - Gus Wiseman, Sep 11 2019

			A cycle with 4 vertices is a non-mating graph. In the standard ordering of vertices, vertices 1 and 3 are both connected to vertices 2 an 4, thus having an identical sets of neighbors.
From _Gus Wiseman_, Sep 11 2019: (Start)
Non-isomorphic representatives of the a(2) = 1 through a(5) non-mating graph edge-sets:
  {12}  {12}     {12}           {12}
        {13,23}  {12,34}        {12,34}
                 {13,23}        {13,23}
                 {13,24,34}     {12,35,45}
                 {14,24,34}     {13,24,34}
                 {14,23,24,34}  {14,24,34}
                                {12,34,35,45}
                                {13,24,35,45}
                                {14,23,24,34}
                                {14,25,35,45}
                                {15,25,35,45}
                                {12,25,34,35,45}
                                {14,25,34,35,45}
                                {15,23,24,35,45}
                                {15,25,34,35,45}
                                {13,24,25,34,35,45}
                                {15,24,25,34,35,45}
                                {15,23,24,25,34,35,45}
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..50
Ronald C. Read, The enumeration of mating-type graphs, Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.
Gus Wiseman, The a(1) = 1 through a(5) = 18 non-mating graphs (isolated vertices not shown).
Gus Wiseman, The a(1) = 1 through a(5) = 18 graphs with at least one endpoint (isolated vertices not shown).

The labeled version is A327379.

Cf. A000088, A004110, A028242, A059167, A245797, A327335, A327371.

k = {}; For[i = 1, i < 8, i++, lg = ListGraphs[i] ; len = Length[lg]; k = Append[k, Length[Select[Range[len], Length[Union[ToAdjacencyMatrix[lg[[ # ]]]]] != i &]]]]; k

a(n) = A000088(n) - A004110(n).

Extended by R. J. Mathar, Sep 12 2008

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

A324693 Number of simple graphs on n unlabeled nodes with minimum degree exactly 1.

Original entry on oeis.org

0, 1, 1, 4, 12, 60, 378, 3843, 64455, 1921532, 104098702, 10348794144, 1893781768084, 639954768875644, 400905675004630820, 467554784370658979194, 1019317687720204607541914, 4170177760438554428852944352, 32130458453030025927403299167172

Offset: 1

Andrew Howroyd, Sep 03 2019

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Minimum Vertex Degree
Gus Wiseman, The a(2) = 1 through a(5) = 12 unlabeled graphs with minimum degree 1.

Column k = 1 of A294217.
A diagonal of A263293.
The labeled version is A327227.
The generalization to set-systems is A327335, with covering case A327230.
Unlabeled covering graphs are A002494.
Cf. A000088, A004110, A100743, A141580, A245797, A261919, A327105, A327362, A327364, A327366, A327372.

a(n) = A002494(n) - A261919(n).

First differences of A141580. - Andrew Howroyd, Jan 11 2021

cp's OEIS Frontend

A327230 Number of non-isomorphic set-systems covering n vertices with at least one endpoint/leaf.

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A294217 Triangle read by rows: T(n,k) is the number of graphs with n vertices and minimum vertex degree k, (0 <= k < n).

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A141580 Number of unlabeled non-mating graphs with n vertices.

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

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A324693 Number of simple graphs on n unlabeled nodes with minimum degree exactly 1.

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