

A327079


Number of labeled simple connected graphs covering n vertices with at least one bridge that is not an endpoint/leaf (nonspanning edgeconnectivity 1).


14



0, 0, 1, 0, 12, 180, 4200, 157920, 9673664, 1011129840, 190600639200, 67674822473280, 46325637863907072, 61746583700640860736, 161051184122415878112640, 824849999242893693424992000, 8317799170120961768715123118080
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OFFSET

0,5


COMMENTS

A bridge is an edge that, if removed without removing any incident vertices, disconnects the graph. Graphs with no bridges are counted by A095983 (2edgeconnected graphs).
Also labeled simple connected graphs covering n vertices with nonspanning edgeconnectivity 1, where the nonspanning edgeconnectivity of a graph is the minimum number of edges that must be removed (along with any noncovered vertices) to obtain a disconnected or empty graph.


LINKS

Table of n, a(n) for n=0..16.


FORMULA

a(n) = A001187(n)  A322395(n) for n > 2.  Andrew Howroyd, Aug 27 2019
Inverse binomial transform of A327231.


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[[1]]], Union@@s[[c[[1]]]]]]]]];
eConn[sys_]:=If[Length[csm[sys]]!=1, 0, Length[sys]Max@@Length/@Select[Union[Subsets[sys]], Length[csm[#]]!=1&]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], Union@@#==Range[n]&&eConn[#]==1&]], {n, 0, 4}]


CROSSREFS

Column k = 1 of A327149.
The noncovering version is A327231.
Connected bridged graphs (spanning edgeconnectivity 1) are A327071.
BIInumbers of graphs with nonspanning edgeconnectivity 1 are A327099.
Covering setsystems with nonspanning edgeconnectivity 1 are A327129.
Cf. A001187, A006129, A052446, A059166, A322395, A327072, A327073, A327148.
Sequence in context: A318245 A051609 A001814 * A013924 A145560 A332960
Adjacent sequences: A327076 A327077 A327078 * A327080 A327081 A327082


KEYWORD

nonn


AUTHOR

Gus Wiseman, Aug 25 2019


EXTENSIONS

Terms a(6) and beyond from Andrew Howroyd, Aug 27 2019


STATUS

approved



