%I #14 May 26 2021 02:14:33

%S 1,1,0,1,1,0,4,3,1,0,26,28,9,1,0,296,475,227,25,1,0,6064,14736,10110,

%T 1782,75,1,0

%N Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and spanning edge-connectivity k.

%C 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.

%C We consider a graph with one vertex and no edges to be disconnected.

%e Triangle begins:

%e 1

%e 1 0

%e 1 1 0

%e 4 3 1 0

%e 26 28 9 1 0

%e 296 475 227 25 1 0

%t 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 spanEdgeConn[vts_,eds_]:=Length[eds]-Max@@Length/@Select[Subsets[eds],Union@@#!=vts||Length[csm[#]]!=1&];

%t Table[Length[Select[Subsets[Subsets[Range[n],{2}]],spanEdgeConn[Range[n],#]==k&]],{n,0,5},{k,0,n}]

%Y Row sums are A006125.

%Y Column k = 0 is A054592, if we assume A054592(1) = 1.

%Y Column k = 1 is A327071.

%Y Column k = 2 is A327146.

%Y The unlabeled version (except with offset 1) is A263296.

%Y Cf. A001187, A095983, A259862, A322338, A326787, A327070, A327072, A327073.

%K nonn,tabl,more

%O 0,7

%A _Gus Wiseman_, Aug 23 2019

%E a(21)-a(27) from _Robert Price_, May 25 2021