

A095983


Number of 2edgeconnected labeled graphs on n nodes.


38



0, 0, 0, 1, 10, 253, 11968, 1047613, 169181040, 51017714393, 29180467201536, 32121680070545657, 68867078000231169536, 290155435185687263172693, 2417761175748567327193407488, 40013922635723692336670167608181, 1318910073755307133701940625759574016
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OFFSET

0,5


COMMENTS

From Falk Hüffner, Jun 28 2018: (Start)
Essentially the same sequence arises as the number of connected bridgeless labeled graphs (graphs that are kedge connected for k >= 2, starting elements of this sequence are 1, 1, 0, 1, 10, 253, 11968, ...).
Labeled version of A007146. (End)
The spanning edgeconnectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a graph that is disconnected or covers fewer vertices. This sequence counts graphs with spanning edgeconnectivity >= 2, which, for n > 1, are connected graphs with no bridges.  Gus Wiseman, Sep 20 2019


LINKS

Jinyuan Wang, Table of n, a(n) for n = 0..82
P. Hanlon and R. W. Robinson, Counting bridgeless graphs, J. Combin. Theory, B 33 (1982), 276305.
Gus Wiseman, The a(4) = 10 labeled graphs with spanning edgeconnectivity >= 2.
Gus Wiseman, Nonisomorphic representatives of the a(5) = 253 graphs with spanning edgeconnectivity >= 2.


FORMULA

a(n) = A001187(n)  A327071(n).  Gus Wiseman, Sep 20 2019


MATHEMATICA

seq[n_] := Module[{v, p, q, c}, v[_] = 0; p = x*D[#, x]& @ Log[ Sum[ 2^Binomial[k, 2]*x^k/k!, {k, 0, n}] + O[x]^(n+1)]; q = x*E^p; p = q; For[k = 3, k <= n, k++, c = Coefficient[p, x, k]; v[k] = c*(k1)!; p = c*q^k]; Join[{0}, Array[v, n]]];
seq[16] (* JeanFrançois Alcover, Aug 13 2019, after Andrew Howroyd *)
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]]]]]]]]];
spanEdgeConn[vts_, eds_]:=Length[eds]Max@@Length/@Select[Subsets[eds], Union@@#!=vtsLength[csm[#]]!=1&];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], spanEdgeConn[Range[n], #]>=2&]], {n, 0, 5}] (* Gus Wiseman, Sep 20 2019 *)


PROG

(PARI) \\ here p is initially A053549, q is A198046 as e.g.f.s.
seq(n)={my(v=vector(n));
my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n))));
my(q=x*exp(p)); p=q;
for(k=3, n, my(c=polcoeff(p, k)); v[k]=c*(k1)!; p=c*q^k);
concat([0], v)} \\ Andrew Howroyd, Jun 18 2018
(PARI) seq(n)={my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x*x^n)))); Vec(serlaplace(log(x/serreverse(x*exp(p)))/x1), (n+1))} \\ Andrew Howroyd, Dec 28 2020


CROSSREFS

Cf. A053549, A198046.
The unlabeled version is A007146.
Row sums of A327069 if the first two columns are removed.
BIInumbers of setsystems with spanning edgeconnectivity >= 2 are A327109.
Graphs with spanning edgeconnectivity 2 are A327146.
Graphs with nonspanning edgeconnectivity >= 2 are A327200.
2vertexconnected graphs are A013922.
Graphs without endpoints are A059167.
Graphs with spanning edgeconnectivity 1 are A327071.
Cf. A001187, A002218, A052446, A059166, A100743, A322395, A327079, A327144.
Sequence in context: A001536 A114450 A178689 * A059166 A100743 A251588
Adjacent sequences: A095980 A095981 A095982 * A095984 A095985 A095986


KEYWORD

nonn


AUTHOR

Yifei Chen (yifei(AT)mit.edu), Jul 17 2004


EXTENSIONS

Name corrected and extended by Pavel Irzhavski, Nov 01 2014
Offset corrected by Falk Hüffner, Jun 17 2018
a(12)a(16) from Andrew Howroyd, Jun 18 2018


STATUS

approved



