

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

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, ...).
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



FORMULA



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]]];
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

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);
(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

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.


KEYWORD

nonn


AUTHOR

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


EXTENSIONS



STATUS

approved



