A182294
Number of connected labeled graphs with n nodes and n+9 edges.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 20349, 21426300, 8956859646, 2352103292070, 470090359867986, 79002015147719136, 11836068369346126698, 1640443794179544776604, 215598057543037336382670, 27336005392867324870778880, 3385297472808136707459580488, 413211903044379104303226531072
Offset: 1
- Andrew Howroyd, Table of n, a(n) for n = 1..100
- Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
- S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.
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N:=20: [seq(coeff(op(i,[seq(coeff(taylor(log(add(x^i*(1+y)^(binomial(i,2))/i!,i=0..N)),x=0,N+1),x,i)*i!,i=1..N)]),y,i-1+10),i=1..N)];
Offset corrected and terms a(16) and beyond from
Andrew Howroyd, Apr 16 2021
A182295
Number of connected labeled graphs with n nodes and n+10 edges.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 5985, 13112470, 8535294180, 3096620034795, 800118566011380, 166591475854153740, 30012638793107746776, 4892304538906805158775, 743352352817243899253160, 107478174967432322995403280, 15008321493306766503800761840, 2046331888629918743459557040544
Offset: 1
- Andrew Howroyd, Table of n, a(n) for n = 1..100
- Steven R. Finch, An exceptional convolutional recurrence, arXiv:2408.12440 [math.CO], 22 Aug 2024.
- S. Janson, D. E. Knuth, T. Łuczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233-358.
-
N:=20: [seq(coeff(op(i,[seq(coeff(taylor(log(add(x^i*(1+y)^(binomial(i,2))/i!,i=0..N)),x=0,N+1),x,i)*i!,i=1..N)]),y,i-1+11),i=1..N)];
Offset corrected and terms a(16) and beyond from
Andrew Howroyd, Apr 16 2021
A182371
Number of connected labeled graphs with n nodes and n+11 edges.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 1330, 6905220, 7279892361, 3717889913655, 1255470137209650, 326123611416074340, 70993993399632155710, 13659118629343706026053, 2405832308811599670396135, 397496768417871214784702640, 62693059156926401902640364120, 9561367292987041683030275944320
Offset: 1
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N:=20: [seq(coeff(op(i, [seq(coeff(taylor(log(add(x^i*(1+y)^(binomial(i, 2))/i!, i=0..N)), x=0, N+1), x, i)*i!, i=1..N)]), y, i-1+12), i=1..N)];
Offset corrected and terms a(17) and beyond from
Andrew Howroyd, Apr 16 2021
A218696
Number of components over all graphs on n labeled nodes with unicyclic components (graphs counted by A137916).
Original entry on oeis.org
1, 15, 222, 3680, 69345, 1477182, 35234220, 932070708, 27109785510, 860394764515, 29600058300780, 1097511032533500, 43637308561557074, 1852311640075120980, 83612841417061582320, 3999611090385007608840, 202111299843794061251580, 10758947714752854861908379
Offset: 3
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nn=22;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];Drop[Range[0,nn]!CoefficientList[ Series[D[Exp[y(Log[1/(1-t)]/2-t/2-t^2/4)],y]/.y->1,{x,0,nn}],x],3]
A372153
Irregular triangular array read by rows. T(n,k) is the number of simple labeled graphs on [n] with circuit rank equal to k, n >= 1, 0 <= k <= binomial(n-1,2).
Original entry on oeis.org
1, 2, 7, 1, 38, 19, 6, 1, 291, 317, 235, 125, 45, 10, 1, 2932, 5582, 7120, 6915, 5215, 3057, 1371, 455, 105, 15, 1, 36961, 108244, 207130, 306775, 368046, 364539, 300342, 205940, 116910, 54362, 20356, 5985, 1330, 210, 21, 1, 561948, 2331108, 6176387, 12709760
Offset: 1
Triangle T(n,k) begins:
1;
2;
7, 1;
38, 19, 6, 1;
291, 317, 235, 125, 45, 10, 1;
2932, 5582, 7120, 6915, 5215, 3057, 1371, 455, 105, 15, 1;
...
- R. Diestel, Graph Theory, Springer, 2017, pp. 23-27.
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Needs["Combinatorica`"]; Map[Select[#, # > 0 &] &, Transpose[ Table[ Table[ Total[ Map[#[[1]] &,Select[Table[{n!/GraphData[{n, i}, "AutomorphismCount"], GraphData[{n, i}, "CyclomaticNumber"]}, {i, 1, NumberOfGraphs[n]}], #[[2]] == k &]]], {n, 1, 7}], {k, 0, 15}]]] // Grid
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T(n)={[Vecrev(p)| p<-Vec(-1+serlaplace(exp(y*log(sum(k=0, n, (1+y)^binomial(k,2)*x^k/k!/y^k, O(x*x^n))))))]}
{ foreach(T(7), row, print(row)) } \\ Andrew Howroyd, Jun 09 2025
A331563
Number of labeled cyclic graphs with n edges and 2n vertices.
Original entry on oeis.org
0, 0, 20, 1610, 129654, 11688369, 1194822915, 137766789810, 17758192128830, 2535895233070628, 397875362655895761, 68087081506276861665, 12626853606957534296975, 2523446241515288646389325
Offset: 1
a(4) = 1610 since we have 3 non-isomorphic cyclic graphs with 4 edges and 8 nodes. (See illustration below.)
To compute a(4) we can consult A057500, which provides the number of labeled connected unicycles. Because A057500(4)=15, and knowing that there are 3 labeled squares, we have 15-3 = 12 Paw Graphs [see Weisstein link]. So graph 1 is labeled in 12 * C(8,4) = 840 ways. Graph 2 is labeled in 3* C(8,4) = 210 ways. A105599 gives 10 as the number of labeled forests with 5 nodes and 4 components, so graph 3 is labeled in 10 * C(8,3) = 560 ways. We have 840 + 210 + 560 = 1610.
.
graph 1 graph 2 graph 3 (triangle + forest with
5 nodes and 4 components)
*--* *--* *--* *
| /| | | | / |
|/ | | | |/ |
* * *--* * *
* * * * * * * * * * *
- Eric Weisstein's World of Mathematics, Paw Graph
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