A005195 -id:A005195 - cp's OEIS frontend

A372168 Number of triangle-free simple labeled graphs covering n vertices.

1, 0, 1, 3, 22, 237, 3961, 99900, 3757153, 208571691, 16945953790, 1999844518737, 340422874696873, 83041703920313712, 28850117307732482737, 14191512425207950473867, 9829313296102303971441502

Offset: 0

Gus Wiseman, Apr 23 2024

The unlabeled version is A372169.

			The a(4) = 22 graphs are:
  12-34
  13-24
  14-23
  12-13-14
  12-13-24
  12-13-34
  12-14-23
  12-14-34
  12-23-24
  12-23-34
  12-24-34
  13-14-23
  13-14-24
  13-23-24
  13-23-34
  13-24-34
  14-23-24
  14-23-34
  14-24-34
  12-13-24-34
  12-14-23-34
  13-14-23-24

Eric Weisstein's World of Mathematics, Triangle-Free Graph

Dominated by A006129, unlabeled A002494.
For all cycles (not just triangles) we have A105784, unlabeled A144958.
Covering case of A213434 (column k = 0 of A372170, unlabeled A263340).
The connected case is A345218, unlabeled A024607.
Column k = 0 of A372167, unlabeled A372173.
The unlabeled version is A372169.
For a unique triangle we have A372171, non-covering A372172.
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A054548 counts covering graphs by number of edges, unlabeled A370167.
Cf. A000272, A053530, A121251, A367862, A367863, A372191, A372174, A372175.

cys[y_]:=Select[Subsets[Union@@y,{3}],MemberQ[y,{#[[1]],#[[2]]}] && MemberQ[y,{#[[1]],#[[3]]}] && MemberQ[y,{#[[2]],#[[3]]}]&];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Union@@#==Range[n]&&Length[cys[#]]==0&]],{n,0,5}]

Binomial transform is A213434.

A372172 Number of labeled simple graphs on n vertices with exactly one triangle.

Original entry on oeis.org

0, 0, 0, 1, 16, 290, 6980, 235270, 11298056, 777154308, 76560083040

Offset: 0

Gus Wiseman, Apr 24 2024

The unlabeled version is A372194.

			The a(4) = 16 graphs:
  12,13,23
  12,14,24
  13,14,34
  23,24,34
  12,13,14,23
  12,13,14,24
  12,13,14,34
  12,13,23,24
  12,13,23,34
  12,14,23,24
  12,14,24,34
  12,23,24,34
  13,14,23,34
  13,14,24,34
  13,23,24,34
  14,23,24,34

For no triangles we have A213434, covering A372168 (unlabeled A372169).
Column k = 1 of A372170, unlabeled A263340.
The covering case is A372171, unlabeled A372174.
For all cycles (not just triangles) we have A372193, covering A372195.
The unlabeled version is A372194.
A001858 counts acyclic graphs, unlabeled A005195.
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494
A054548 counts labeled covering graphs by edges, unlabeled A370167.
A372167 counts covering graphs by triangles, unlabeled A372173.
Cf. A000272, A105784, A121251, A137916, A236570, A372176.

cys[y_]:=Select[Subsets[Union@@y,{3}],MemberQ[y,{#[[1]],#[[2]]}] && MemberQ[y,{#[[1]],#[[3]]}] && MemberQ[y,{#[[2]],#[[3]]}]&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[cys[#]]==1&]],{n,0,5}]

Binomial transform of A372171.

a(8)-a(10) from Andrew Howroyd, Aug 01 2024

A372174 Number of unlabeled simple graphs covering n vertices with a unique triangle.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 16, 79, 424, 3098, 28616

Offset: 0

Gus Wiseman, Apr 24 2024

The labeled version is A372171.

Gus Wiseman, The a(3) = 1 through a(6) = 16 covering graphs with a unique triangle.

The non-covering version is column k = 1 of A263340, labeled A372170.
Case of A370167 with a unique triangle, labeled A054548.
For no triangles we have A372169, labeled A372168 (non-covering A213434).
The labeled version is A372171, column k = 1 of A372167.
Column k = 1 of A372173, labeled A372167.
For cycles (not just triangles) we have A372191, labeled A372195.
The non-covering version is A372194, labeled A372172.
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A002494 counts unlabeled covering graphs, labeled A006129.
A372176 counts labeled graphs by directed cycles, covering A372175.
Cf. A000055, A053530, A137917, A137918, A140637, A144958, A370169.

First differences of A372194.

A372193 Number of labeled simple graphs on n vertices with a unique cycle of length > 2.

Original entry on oeis.org

0, 0, 0, 1, 19, 317, 5582, 108244, 2331108, 55636986, 1463717784, 42182876763, 1323539651164, 44955519539963, 1644461582317560, 64481138409909506, 2698923588248208224, 120133276796015812548, 5667351458582453925696, 282496750694780020437765, 14837506263979393796687088

Offset: 0

Gus Wiseman, Apr 25 2024

nonn

An undirected cycle in a graph is a sequence of distinct vertices, up to rotation and reversal, such that there are edges between all consecutive elements, including the last and the first.

			The a(4) = 19 graphs:
  12,13,23
  12,14,24
  13,14,34
  23,24,34
  12,13,14,23
  12,13,14,24
  12,13,14,34
  12,13,23,24
  12,13,23,34
  12,13,24,34
  12,14,23,24
  12,14,23,34
  12,14,24,34
  12,23,24,34
  13,14,23,24
  13,14,23,34
  13,14,24,34
  13,23,24,34
  14,23,24,34

Andrew Howroyd, Table of n, a(n) for n = 0..200

For no cycles we have A001858 (covering A105784), unlabeled A005195 (covering A144958).
Counting triangles instead of cycles gives A372172 (non-covering A372171), unlabeled A372194 (non-covering A372174).
The unlabeled version is A236570, non-covering A372191.
The covering case is A372195, column k = 1 of A372175.
A000088 counts unlabeled graphs, labeled A006125.
A002807 counts cycles in a complete graph.
A006129 counts labeled graphs, unlabeled A002494.
A372167 counts graphs by triangles, non-covering A372170.
A372173 counts unlabeled graphs by triangles, non-covering A263340.
Cf. A000272, A054548, A057500, A121251, A137916, A213434, A322661, A372169, A372176.

cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations /@ Subsets[Union@@y,{k}],And @@ Table[MemberQ[Sort/@y,Sort[{#[[i]],#[[If[i==k,1,i+1]]]}]],{i,k}]&], {k,3,Length[y]}],Min@@#==First[#]&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[cyc[#]]==2&]],{n,0,5}]

seq(n)={my(w=lambertw(-x+O(x*x^n))); Vec(serlaplace(exp(-w-w^2/2)*(-log(1+w)/2 + w/2 - w^2/4)), -n-1)} \\ Andrew Howroyd, Jul 31 2024

E.g.f.: B(x)*C(x) where B(x) is the e.g.f. of A057500 and C(x) is the e.g.f. of A001858. - Andrew Howroyd, Jul 31 2024

a(7) onwards from Andrew Howroyd, Jul 31 2024

A372195 Number of labeled simple graphs covering n vertices with a unique undirected cycle of length > 2.

Original entry on oeis.org

0, 0, 0, 1, 15, 232, 3945, 75197, 1604974, 38122542, 1000354710, 28790664534, 902783451933, 30658102047787, 1121532291098765, 43985781899812395, 1841621373756094796, 82002075703514947236, 3869941339069299799884, 192976569550677042208068, 10139553075163838030949495

Offset: 0

Gus Wiseman, Apr 25 2024

nonn

An undirected cycle in a graph is a sequence of distinct vertices, up to rotation and reversal, such that there are edges between all consecutive elements, including the last and the first.

			The a(4) = 15 graphs:
  12,13,14,23
  12,13,14,24
  12,13,14,34
  12,13,23,24
  12,13,23,34
  12,13,24,34
  12,14,23,24
  12,14,23,34
  12,14,24,34
  12,23,24,34
  13,14,23,24
  13,14,23,34
  13,14,24,34
  13,23,24,34
  14,23,24,34

Andrew Howroyd, Table of n, a(n) for n = 0..200

For no cycles we have A105784 (for triangles A372168, non-covering A213434), unlabeled A144958 (for triangles A372169).
Counting triangles instead of cycles gives A372171 (non-covering A372172), unlabeled A372174 (non-covering A372194).
The unlabeled version is A372191, non-covering A236570.
The non-covering version is A372193, column k = 1 of A372176.
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A002807 counts cycles in a complete graph.
A006129 counts labeled graphs, unlabeled A002494.
A322661 counts covering loop-graphs, unlabeled A322700.
A372167 counts covering graphs by triangles (non-covering A372170), unlabeled A372173 (non-covering A263340).
Cf. A000169, A000272, A053530, A054548, A137916, A137917, A367863, A368597.

cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations/@Subsets[Union@@y,{k}],And@@Table[MemberQ[Sort/@y,Sort[{#[[i]],#[[If[i==k,1,i+1]]]}]],{i,k}]&],{k,3,Length[y]}],Min@@#==First[#]&];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[cyc[#]]==2&]],{n,0,5}]

seq(n)={my(w=lambertw(-x+O(x*x^n))); Vec(serlaplace(exp(-w-w^2/2-x)*(-log(1+w)/2 + w/2 - w^2/4)), -n-1)} \\ Andrew Howroyd, Jul 31 2024

Inverse binomial transform of A372193. - Andrew Howroyd, Jul 31 2024

a(7) onwards from Andrew Howroyd, Jul 31 2024

A185334 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 4.

Original entry on oeis.org

1, 0, 0, 1, 2, 6, 23, 112, 801, 7840, 97723, 1436873, 23791155, 432878091, 8544173926, 181519645163, 4127569521160

Offset: 0

Jason Kimberley, Feb 15 2011

The null graph on 0 vertices is vacuously 3-regular; since it is acyclic, it has infinite girth.

Jason Kimberley, Not necessarily connected k-regular graphs with girth at least 4
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g

3-regular simple graphs with girth at least 4: A014371 (connected), A185234 (disconnected), this sequence (not necessarily connected).
Not necessarily connected k-regular simple graphs with girth at least 4: A185314 (any k), A185304 (triangle); specified degree k: A008484 (k=2), this sequence (k=3), A185344 (k=4), A185354 (k=5), A185364 (k=6).
Not necessarily connected 3-regular simple graphs with girth *at least* g: A005638 (g=3), this sequence (g=4), A185335 (g=5), A185336 (g=6).
Not necessarily connected 3-regular simple graphs with girth *exactly* g: A185133 (g=3), A185134 (g=4), A185135 (g=5), A185136 (g=6).

A014371 = Cases[Import["https://oeis.org/A014371/b014371.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A014371] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transformation of A014371.

A165626 Number of 5-regular graphs (quintic graphs) on 2n vertices.

Original entry on oeis.org

1, 0, 0, 1, 3, 60, 7849, 3459386, 2585136741, 2807105258926, 4221456120848125, 8516994772686533749, 22470883220896245217626, 75883288448434648617038134, 322040154712674550886226182668

Offset: 0

Jason Kimberley, Sep 22 2009

Because the triangle A051031 is symmetric, a(n) is also the number of (2n-6)-regular graphs on 2n vertices.

Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146.
N. J. A. Sloane, Transforms

5-regular simple graphs: A006821 (connected), A165655 (disconnected), this sequence (not necessarily connected).

Regular graphs A005176 (any degree), A051031 (triangular array), specified degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), this sequence (k=5), A165627 (k=6), A165628 (k=7), A180260 (k=8).

A006821 = Cases[Import["https://oeis.org/A006821/b006821.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006821] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transform of A006821.

Regular graphs cross-references edited by Jason Kimberley, Nov 07 2009
a(9) from Jason Kimberley, Nov 24 2009
a(10)-a(14) from Andrew Howroyd, Mar 10 2020

A165627 Number of 6-regular graphs (sextic graphs) on n vertices.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 1, 4, 21, 266, 7849, 367860, 21609301, 1470293676, 113314233813, 9799685588961, 945095823831333, 101114579937196179, 11945375659140003692, 1551593789610531820695, 220716215902794066709555, 34259321384370735003091907, 5782740798229835127025560294

Offset: 0

Jason Kimberley, Sep 22 2009

Because the triangle A051031 is symmetric, a(n) is also the number of (n-7)-regular graphs on n vertices.

Georg Grasegger, Hakan Guler, Bill Jackson, Anthony Nixon, Flexible circuits in the d-dimensional rigidity matroid, arXiv:2003.06648 [math.CO], 2020.
Jason Kimberley, Index of sequences counting not necessarily connected k-regular simple graphs with girth at least g
M. Meringer, Tables of Regular Graphs
M. Meringer, Fast generation of regular graphs and construction of cages, J. Graph Theory 30 (2) (1999) 137-146.
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Regular Graph
Eric Weisstein's World of Mathematics, Sextic Graph

6-regular simple graphs: A006822 (connected), A165656 (disconnected), this sequence (not necessarily connected).

Regular graphs A005176 (any degree), A051031 (triangular array), chosen degrees: A000012 (k=0), A059841 (k=1), A008483 (k=2), A005638 (k=3), A033301 (k=4), A165626 (k=5), this sequence (k=6), A165628 (k=7), A180260 (k=8).

A006822 = Cases[Import["https://oeis.org/A006822/b006822.txt", "Table"], {, }][[All, 2]];
(* EulerTransform is defined in A005195 *)
EulerTransform[Rest @ A006822] (* Jean-François Alcover, Dec 04 2019, updated Mar 18 2020 *)

Euler transformation of A006822.

Cross-references edited by Jason Kimberley, Nov 07 2009 and Oct 17 2011
a(17) from Jason Kimberley, Dec 30 2010
a(18)-a(24) from Andrew Howroyd, Mar 07 2020

A342211 Largest number of maximal acyclic node-induced subgraphs of an n-node graph.

Original entry on oeis.org

1, 1, 3, 6, 10, 15, 22, 38, 64

Offset: 1

Pontus von Brömssen, Mar 05 2021

This sequence is log-superadditive, i.e., a(m+n) >= a(m)*a(n). By Fekete's subadditive lemma, it follows that the limit of a(n)^(1/n) exists and equals the supremum of a(n)^(1/n). - Pontus von Brömssen, Mar 03 2022

a(10) >= 105.

			All optimal graphs (i.e., graphs having n nodes and a(n) maximal acyclic subgraphs) for 1 <= n <= 9 are listed below. Here, FCB(n_1, ..., n_k) denotes the full cyclic braid graph with cluster sizes n_1, ..., n_k, as defined by Morrison and Scott (2017), i.e., the graph obtained by arranging complete graphs of orders n_1, ..., n_k (in that order) in a cycle, and joining all pairs of nodes in neighboring parts with edges. (The graph in the paper by Fomin, Gaspers, Pyatkin, and Razgon, which shows that a(10) >= 105, is FCB(2, 2, 2, 2, 2).)
        n = 1: the 1-node graph;
        n = 2: the complete graph and the empty graph;
  3 <= n <= 6: the complete graph;
        n = 7: FCB(1, 2, 2, 2);
        n = 8: FCB(1, 2, 1, 2, 2);
        n = 9: FCB(1, 2, 2, 1, 3).

Fedor V. Fomin, Serge Gaspers, Artem V. Pyatkin, and Igor Razgon, On the minimum feedback vertex set problem: exact and enumeration algorithms, Algorithmica 52 (2008), 293-307.
Natasha Morrison and Alex Scott, Maximising the number of induced cycles in a graph, Journal of Combinatorial Theory Series B 126 (2017), 24-61.

Cf. A000055, A005195.
Sequences of largest number of maximal induced subgraphs with a given property:
  A000792 (independent sets or cliques),
  this sequence (acyclic),
  A342212 (bipartite),
  A342213 (planar),
  A342324 (chordal),
  A352208 (3-colorable),
  A352209 (perfect),
  A352210 (2-degenerate),
  A352211 (cluster graphs),
  A352212 (triangle-free),
  A352213 (cographs),
  A352214 (claw-free),
  A352215 (C_4-free),
  A352216 (diamond-free).

a(m+n) >= a(m)*a(n).

1.5926... = 105^(1/10) <= lim_{n->oo} a(n)^(1/n) <= 1.8638. (Fomin, Gaspers, Pyatkin, and Razgon (2008)).

A372175 Irregular triangle read by rows where T(n,k) is the number of labeled simple graphs covering n vertices with exactly 2k directed cycles of length > 2.

Original entry on oeis.org

1, 0, 1, 3, 1, 19, 15, 0, 6, 0, 0, 0, 1, 155, 232, 15, 190, 0, 0, 70, 50, 0, 0, 0, 0, 30, 15, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Gus Wiseman, Apr 24 2024

A directed cycle in a simple (undirected) graph is a sequence of distinct vertices, up to rotation, such that there are edges between all consecutive elements, including the last and the first.

			Triangle begins (zeros shown as dots):
  1
  .
  1
  3 1
  19 15 . 6 ... 1
  155 232 15 190 .. 70 50 .... 30 15 .......... 10 .............. 1
Row n = 4 counts the following graphs:
  12,34     12,13,14,23  .  12,13,14,23,24  .  .  .  12,13,14,23,24,34
  13,24     12,13,14,24     12,13,14,23,34
  14,23     12,13,14,34     12,13,14,24,34
  12,13,14  12,13,23,24     12,13,23,24,34
  12,13,24  12,13,23,34     12,14,23,24,34
  12,13,34  12,13,24,34     13,14,23,24,34
  12,14,23  12,14,23,24
  12,14,34  12,14,23,34
  12,23,24  12,14,24,34
  12,23,34  12,23,24,34
  12,24,34  13,14,23,24
  13,14,23  13,14,23,34
  13,14,24  13,14,24,34
  13,23,24  13,23,24,34
  13,23,34  14,23,24,34
  13,24,34
  14,23,24
  14,23,34
  14,24,34

Row lengths are A002807 + 1.
Row sums are A006129, unlabeled A002494.
Column k = 0 is A105784 (for triangles A372168, non-covering A213434), unlabeled A144958 (for triangles A372169).
Counting triangles instead of cycles gives A372167 (non-covering A372170), unlabeled A372173 (non-covering A263340).
The non-covering version is A372176.
Column k = 1 is A372195 (non-covering A372193, for triangles A372171), unlabeled A372191 (non-covering A236570, for triangles A372174).
A000088 counts unlabeled graphs, labeled A006125.
A001858 counts acyclic graphs, unlabeled A005195.
A322661 counts covering loop-graphs, unlabeled A322700.
Cf. A000272, A053530, A121251, A137916, A137917, A372172, A372194.

cycles[g_]:=Join@@Table[Select[Join@@Permutations /@ Subsets[Union@@g,{k}],Min@@#==First[#]&&And@@Table[MemberQ[Sort/@g,Sort[{#[[i]], #[[If[i==k,1,i+1]]]}]],{i,k}]&],{k,3,Length[g]}];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Union@@#==Range[n]&&Length[cycles[#]]==2k&]], {n,0,5},{k,0,Length[cycles[Subsets[Range[n],{2}]]]/2}]

cp's OEIS Frontend

A372168 Number of triangle-free simple labeled graphs covering n vertices.

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A372172 Number of labeled simple graphs on n vertices with exactly one triangle.

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A372174 Number of unlabeled simple graphs covering n vertices with a unique triangle.

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A372193 Number of labeled simple graphs on n vertices with a unique cycle of length > 2.

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A372195 Number of labeled simple graphs covering n vertices with a unique undirected cycle of length > 2.

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A185334 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 4.

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A165626 Number of 5-regular graphs (quintic graphs) on 2n vertices.

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A165627 Number of 6-regular graphs (sextic graphs) on n vertices.

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