cp's OEIS Frontend

A372193 Number of labeled simple graphs on n vertices with a unique cycle of length > 2.

%I A372193 #15 Aug 01 2024 00:10:15
%S A372193 0,0,0,1,19,317,5582,108244,2331108,55636986,1463717784,42182876763,
%T A372193 1323539651164,44955519539963,1644461582317560,64481138409909506,
%U A372193 2698923588248208224,120133276796015812548,5667351458582453925696,282496750694780020437765,14837506263979393796687088
%N A372193 Number of labeled simple graphs on n vertices with a unique cycle of length > 2.
%C A372193 An undirected cycle in a graph is a sequence of distinct vertices, up to rotation and reversal, such that there are edges between all consecutive elements, including the last and the first.
%H A372193 Andrew Howroyd, <a href="/A372193/b372193.txt">Table of n, a(n) for n = 0..200</a>
%F A372193 E.g.f.: B(x)*C(x) where B(x) is the e.g.f. of A057500 and C(x) is the e.g.f. of A001858. - _Andrew Howroyd_, Jul 31 2024
%e A372193 The a(4) = 19 graphs:
%e A372193   12,13,23
%e A372193   12,14,24
%e A372193   13,14,34
%e A372193   23,24,34
%e A372193   12,13,14,23
%e A372193   12,13,14,24
%e A372193   12,13,14,34
%e A372193   12,13,23,24
%e A372193   12,13,23,34
%e A372193   12,13,24,34
%e A372193   12,14,23,24
%e A372193   12,14,23,34
%e A372193   12,14,24,34
%e A372193   12,23,24,34
%e A372193   13,14,23,24
%e A372193   13,14,23,34
%e A372193   13,14,24,34
%e A372193   13,23,24,34
%e A372193   14,23,24,34
%t A372193 cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations /@ Subsets[Union@@y,{k}],And @@ Table[MemberQ[Sort/@y,Sort[{#[[i]],#[[If[i==k,1,i+1]]]}]],{i,k}]&], {k,3,Length[y]}],Min@@#==First[#]&];
%t A372193 Table[Length[Select[Subsets[Subsets[Range[n],{2}]], Length[cyc[#]]==2&]],{n,0,5}]
%o A372193 (PARI) seq(n)={my(w=lambertw(-x+O(x*x^n))); Vec(serlaplace(exp(-w-w^2/2)*(-log(1+w)/2 + w/2 - w^2/4)), -n-1)} \\ _Andrew Howroyd_, Jul 31 2024
%Y A372193 For no cycles we have A001858 (covering A105784), unlabeled A005195 (covering A144958).
%Y A372193 Counting triangles instead of cycles gives A372172 (non-covering A372171), unlabeled A372194 (non-covering A372174).
%Y A372193 The unlabeled version is A236570, non-covering A372191.
%Y A372193 The covering case is A372195, column k = 1 of A372175.
%Y A372193 A000088 counts unlabeled graphs, labeled A006125.
%Y A372193 A002807 counts cycles in a complete graph.
%Y A372193 A006129 counts labeled graphs, unlabeled A002494.
%Y A372193 A372167 counts graphs by triangles, non-covering A372170.
%Y A372193 A372173 counts unlabeled graphs by triangles, non-covering A263340.
%Y A372193 Cf. A000272, A054548, A057500, A121251, A137916, A213434, A322661, A372169, A372176.
%K A372193 nonn
%O A372193 0,5
%A A372193 _Gus Wiseman_, Apr 25 2024
%E A372193 a(7) onwards from _Andrew Howroyd_, Jul 31 2024