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 A327073 Number of labeled simple connected graphs with n vertices and exactly one bridge. 8
 0, 0, 1, 0, 12, 200, 7680, 506856, 58934848, 12205506096, 4595039095680, 3210660115278000, 4240401342141499392, 10743530775519296581944, 52808688280248604235191296, 507730995579614277599205009240, 9603347831901155679455061048606720, 358743609478638769812094362544644831968 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 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). LINKS Andrew Howroyd, Table of n, a(n) for n = 0..50 Gus Wiseman, The a(4) = 12 graphs with exactly one bridge. FORMULA E.g.f.: (x + Sum_{k>=2} A095983(k)*x^k/(k-1)!)^2/2. - Andrew Howroyd, Aug 25 2019 MATHEMATICA 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[]], Union@@s[[c[]]]]]]]]; Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&Length[csm[#]]==1&&Count[Table[Length[Union@@Delete[#, i]]1, {i, Length[#]}], True]==1&]], {n, 0, 5}] PROG (PARI) \\ See A095983 seq(n)={my(r=x+O(x^n)); my(p=x*deriv(log(sum(k=0, n, 2^binomial(k, 2) * x^k / k!) + O(x^n)))); my(q=x*exp(p)); p-=q; for(k=3, n-1, my(c=polcoeff(p, k)); r+=c*x^k; p-=c*q^k); Vec(serlaplace(r^2/2), -(n+1))} \\ Andrew Howroyd, Aug 25 2019 CROSSREFS Column k = 1 of A327072. The unlabeled version is A327074. Connected graphs with no bridges are A007146. Connected graphs whose bridges are all leaves are A322395. Connected graphs with at least one bridge are A327071. Cf. A001187, A006129, A052446, A095983, A327069, A327077, A327108, A327111, A327145, A327146. Sequence in context: A036240 A292056 A277311 * A133242 A141836 A083932 Adjacent sequences:  A327070 A327071 A327072 * A327074 A327075 A327076 KEYWORD nonn AUTHOR Gus Wiseman, Aug 24 2019 EXTENSIONS Terms a(6) and beyond from Andrew Howroyd, Aug 25 2019 STATUS approved

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Last modified July 15 02:31 EDT 2020. Contains 335762 sequences. (Running on oeis4.)