# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a327336 Showing 1-1 of 1 %I A327336 #8 Sep 11 2019 17:38:28 %S A327336 0,0,1,3,28,490,15336,851368,85010976,15615858960,5388679220480, %T A327336 3548130389657216,4507988483733389568,11145255551131555572992, %U A327336 53964198507018134569758720,514158235191699333805861463040,9672967865350359173180572164444160 %N A327336 Number of labeled simple graphs with vertex-connectivity 1. %C A327336 Same as A327114 except a(2) = 1. %C A327336 The vertex-connectivity of a graph is the minimum number of vertices that must be removed (along with any incident edges) to obtain a non-connected graph or singleton. %H A327336 Andrew Howroyd, Table of n, a(n) for n = 0..50 %e A327336 The a(2) = 1 through a(4) = 28 edge-sets: %e A327336 {12} {12,13} {12,13,14} %e A327336 {12,23} {12,13,24} %e A327336 {13,23} {12,13,34} %e A327336 {12,14,23} %e A327336 {12,14,34} %e A327336 {12,23,24} %e A327336 {12,23,34} %e A327336 {12,24,34} %e A327336 {13,14,23} %e A327336 {13,14,24} %e A327336 {13,23,24} %e A327336 {13,23,34} %e A327336 {13,24,34} %e A327336 {14,23,24} %e A327336 {14,23,34} %e A327336 {14,24,34} %e A327336 {12,13,14,23} %e A327336 {12,13,14,24} %e A327336 {12,13,14,34} %e A327336 {12,13,23,24} %e A327336 {12,13,23,34} %e A327336 {12,14,23,24} %e A327336 {12,14,24,34} %e A327336 {12,23,24,34} %e A327336 {13,14,23,34} %e A327336 {13,14,24,34} %e A327336 {13,23,24,34} %e A327336 {14,23,24,34} %t A327336 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]]; %t A327336 vertConnSys[vts_,eds_]:=Min@@Length/@Select[Subsets[vts],Function[del,Length[del]==Length[vts]-1||csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]; %t A327336 Table[Length[Select[Subsets[Subsets[Range[n],{2}]],vertConnSys[Range[n],#]==1&]],{n,0,4}] %Y A327336 Column k = 1 of A327334. %Y A327336 The unlabeled version is A052442. %Y A327336 Connected non-separable graphs are A013922. %Y A327336 Set-systems with vertex-connectivity 1 are A327128. %Y A327336 Labeled simple graphs with cut-connectivity 1 are A327114. %Y A327336 Cf. A006129, A054592, A322389, A322390, A326786, A327070, A327098, A327100, A327125, A327126. %K A327336 nonn %O A327336 0,4 %A A327336 _Gus Wiseman_, Sep 02 2019 %E A327336 Terms a(6) and beyond from _Andrew Howroyd_, Sep 11 2019 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE