|
|
A327079
|
|
Number of labeled simple connected graphs covering n vertices with at least one bridge that is not an endpoint/leaf (non-spanning edge-connectivity 1).
|
|
14
|
|
|
0, 0, 1, 0, 12, 180, 4200, 157920, 9673664, 1011129840, 190600639200, 67674822473280, 46325637863907072, 61746583700640860736, 161051184122415878112640, 824849999242893693424992000, 8317799170120961768715123118080
(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. Graphs with no bridges are counted by A095983 (2-edge-connected graphs).
Also labeled simple connected graphs covering n vertices with non-spanning edge-connectivity 1, where the non-spanning edge-connectivity of a graph is the minimum number of edges that must be removed (along with any non-covered vertices) to obtain a disconnected or empty graph.
|
|
LINKS
|
|
|
FORMULA
|
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
|
The non-covering version is A327231.
Connected bridged graphs (spanning edge-connectivity 1) are A327071.
BII-numbers of graphs with non-spanning edge-connectivity 1 are A327099.
Covering set-systems with non-spanning edge-connectivity 1 are A327129.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|