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A327073 Number of labeled simple connected graphs with n vertices and exactly one bridge.

%I A327073 #14 Dec 28 2020 17:31:30
%S A327073 0,0,1,0,12,200,7680,506856,58934848,12205506096,4595039095680,
%T A327073 3210660115278000,4240401342141499392,10743530775519296581944,
%U A327073 52808688280248604235191296,507730995579614277599205009240,9603347831901155679455061048606720,358743609478638769812094362544644831968
%N A327073 Number of labeled simple connected graphs with n vertices and exactly one bridge.
%C A327073 A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Connected graphs with no bridges are counted by A095983 (2-edge-connected graphs).
%H A327073 Andrew Howroyd, <a href="/A327073/b327073.txt">Table of n, a(n) for n = 0..50</a>
%H A327073 Gus Wiseman, <a href="/A327073/a327073.png">The a(4) = 12 graphs with exactly one bridge.</a>
%F A327073 E.g.f.: (x + Sum_{k>=2} A095983(k)*x^k/(k-1)!)^2/2. - _Andrew Howroyd_, Aug 25 2019
%t A327073 csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A327073 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[csm[#]]==1&&Count[Table[Length[Union@@Delete[#,i]]<n||Length[csm[Delete[#,i]]]>1,{i,Length[#]}],True]==1&]],{n,0,5}]
%o A327073 (PARI) \\ See A095983.
%o A327073 seq(n)={my(p=x*deriv(log(sum(k=0, n-1, 2^binomial(k, 2) * x^k / k!) + O(x^n)))); Vec(serlaplace(log(x/serreverse(x*exp(p)))^2/2), -(n+1)) } \\ _Andrew Howroyd_, Dec 28 2020
%Y A327073 Column k = 1 of A327072.
%Y A327073 The unlabeled version is A327074.
%Y A327073 Connected graphs with no bridges are A007146.
%Y A327073 Connected graphs whose bridges are all leaves are A322395.
%Y A327073 Connected graphs with at least one bridge are A327071.
%Y A327073 Cf. A001187, A006129, A052446, A095983, A327069, A327077, A327108, A327111, A327145, A327146.
%K A327073 nonn
%O A327073 0,5
%A A327073 _Gus Wiseman_, Aug 24 2019
%E A327073 Terms a(6) and beyond from _Andrew Howroyd_, Aug 25 2019