A004110 Number of n-node unlabeled graphs without endpoints (i.e., no nodes of degree 1).

1, 1, 1, 2, 5, 16, 78, 588, 8047, 205914, 10014882, 912908876, 154636289460, 48597794716736, 28412296651708628, 31024938435794151088, 63533059372622888758054, 244916078509480823407040988, 1783406527599529094009748567708, 24605674623474428415849066062642456

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n) is also the number of unlabeled mating graphs with n nodes. A mating graph has no two vertices with identical sets of neighbors. - Tanya Khovanova, Oct 23 2008

References

• F. Harary, Graph Theory, Wiley, 1969. See illustrations in Appendix 1.
• F. Harary and E. Palmer, Graphical Enumeration, (1973), compare formula (8.7.11).
• R. W. Robinson, personal communication.
• R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..26 from R. W. Robinson)
• David Cook II, Nested colourings of graphs, arXiv preprint arXiv:1306.0140 [math.CO], 2013.
• Ira M. Gessel and Ji Li, Enumeration of point-determining graphs, J. Combinatorial Theory Ser. A 118 (2011), 591-612.
• Hemanshu Kaul and Jeffrey A. Mudrock, Counting List Colorings of Unlabeled Graphs, arXiv:2409.06063 [math.CO], 2024. See p. 6.
• R. J. Mathar, Illustrations for n=1..5 nodes.
• Ronald C. Read, The enumeration of mating-type graphs, Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.
• R. W. Robinson, Graphs without endpoints - computer printout.
• N. J. A. Sloane, Illustration of a(0)-a(5).

Crossrefs

Row sums of A123551.

Cf. A059166 (n-node connected labeled graphs without endpoints), A059167 (n-node labeled graphs without endpoints), A004108 (n-node connected unlabeled graphs without endpoints), A006024 (number of labeled mating graphs with n nodes), A129584 (bi-point-determining graphs).

If isolated nodes are forbidden, see A261919.

Cf. A000088.

Programs

Mathematica

permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t * k; s += t]; s!/m];
edges[v_] := Sum[GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[Quotient[v, 2]];
a[n_] := Sum[permcount[p] * 2^edges[p] * Coefficient[Product[1 - x^p[[i]], {i, 1, Length[p]}], x, n - k]/k!, {k, 1, n}, {p, IntegerPartitions[k]}]; a[0] = 1;
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 27 2018, after Andrew Howroyd *)

PARI

\\ Compare A000088.
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]\2)}
a(n) = {my(s=0); sum(k=1, n, forpart(p=k, s+=permcount(p) * 2^edges(p) * polcoef(prod(i=1, #p, 1-x^p[i]), n-k)/k!)); s} \\ Andrew Howroyd, Sep 09 2018