|
|
A327071
|
|
Number of labeled simple connected graphs with n vertices and at least one bridge, or graphs with spanning edge-connectivity 1.
|
|
27
|
|
|
0, 0, 1, 3, 28, 475, 14736, 818643, 82367552, 15278576679, 5316021393280, 3519977478407687, 4487518206535452672, 11116767463976825779115, 53887635281876408097483776, 513758302006787897939587736715, 9668884580476067306398361085853696
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
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).
The spanning edge-connectivity of a graph is the minimum number of edges that must be removed (without removing incident vertices) to obtain a disconnected or empty graph.
|
|
LINKS
|
|
|
FORMULA
|
|
|
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]]]]]]]]];
spanEdgeConn[vts_, eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds], Union@@#!=vts||Length[csm[#]]!=1&];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]], spanEdgeConn[Range[n], #]==1&]], {n, 0, 4}]
|
|
CROSSREFS
|
Connected graphs without bridges are A007146.
The enumeration of labeled connected graphs by number of bridges is A327072.
Connected graphs with exactly one bridge are A327073.
Graphs with non-spanning edge-connectivity 1 are A327079.
BII-numbers of set-systems with spanning edge-connectivity 1 are A327111.
Covering set-systems with spanning edge-connectivity 1 are A327145.
Graphs with spanning edge-connectivity 2 are A327146.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|