

A057500


Number of connected labeled graphs with n edges and n nodes.


24



0, 0, 1, 15, 222, 3660, 68295, 1436568, 33779340, 880107840, 25201854045, 787368574080, 26667815195274, 973672928417280, 38132879409281475, 1594927540549217280, 70964911709203684440, 3347306760024413356032, 166855112441313024389625, 8765006377126199463936000
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OFFSET

1,4


COMMENTS

Equivalently, number of connected unicyclic (i.e. containing one cycle) graphs on n labeled nodes.  Vladeta Jovovic, Oct 26 2004
a(n) is the number of trees on vertex set [n] = {1,2,...,n} rooted at 1 with one marked inversion (an inversion is a pair (i,j) with i>j and j a descendant of i in the tree). Here is a bijection from the title graphs (on [n]) to these marked trees. A title graph has exactly one cycle. There is a unique path from vertex 1 to this cycle, first meeting it at k, say (k may = 1). Let i and j be the two neighbors of k in the cycle, with i the larger of the two. Delete the edge k<>j thereby forming a tree (in which j is a descendant of i) and take (i,j) as the marked inversion. To reverse this map, create a cycle by joining the smaller element of the marked inversion to the parent of the larger element. a(n) = binom(n1,2)A129137(n). This is because, on the above marked trees, the marked inversion is uniformly distributed over 2element subsets of {2,3,...,n} and so a(n)/binom(n1,2) is the number of trees on [n] (rooted at 1) for which (3,2) is an inversion.  David Callan, Mar 30 2007


REFERENCES

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973.
C. L. Mallows, Letter to N. J. A. Sloane, 1980.
R. J. Riddell, Contributions to the theory of condensation, Dissertation, Univ. of Michigan, Ann Arbor, 1951.


LINKS

Washington G. Bomfim and Alois P. Heinz, Table of n, a(n) for n = 1..300 (terms n = 1..50 from Washington G. Bomfim)
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 133
S. Janson, D. E. Knuth, T. Luczak and B. Pittel, The Birth of the Giant Component, arXiv:math/9310236 [math.PR], 1993.
S. Janson, D. E. Knuth, T. Luczak and B. Pittel, The Birth of the Giant Component, Random Structures and Algorithms Vol. 4 (1993), 233358.
C. L. Mallows, Letter to N. J. A. Sloane, 1980
Marko Riedel et al., Nonisomorphic, connected, unicyclic graphs, Math Stackexchange, November 2018. (Proof of closed form by Cauchy Coefficient Formula / Lagrange Inversion.)


FORMULA

The number of labeled connected graphs with n nodes and m edges is Sum_{k=1..n} (1)^(k+1)/k*Sum_{n_1+n_2+..n_k=n, n_i>0} n!/(Product_{i=1..k} (n_i)!)* binomial(s, m), s=Sum_{i..k} binomial(n_i, 2).  Vladeta Jovovic, Apr 10 2001
E.g.f.: (1/2) Sum_{k>=3} T(x)^k/k, with T(x)= Sum_{n>=1} n^(n1)/n! x^n. R. J. Riddell's thesis contains a closedform expression for the number of connected graphs with m nodes and n edges. The present series applies to the special case m=n.
E.g.f.: 1/2*log(1+LambertW(x))+1/2*LambertW(x)1/4*LambertW(x)^2.  Vladeta Jovovic, Jul 09 2001
Asymptotic expansion (with xi=sqrt(2*Pi)): n^(n1/2)*[xi/47/6*n^(1/2)+xi/48* n^(1)+131/270*n^(3/2)+xi/1152*n^(2)+4/2835*n^(5/2)+O(n^(3))].  Keith Briggs (keith.briggs(AT)bt.com), Aug 16 2004
Row sums of A098909: a(n) = (n1)!*n^n/2*Sum_{k=3..n} 1/(n^k*(nk)!).  Vladeta Jovovic, Oct 26 2004
a(n) = Sum_{k=0..C(n1,2)} k*A052121(n,k).  Alois P. Heinz, Nov 29 2015
a(n) = (n^(n2)*(13*n)+exp(n)*Gamma(n+1,n)/n)/2.  Peter Luschny, Jan 27 2016


EXAMPLE

E.g., a(4)=15 because there are three different (labeled) 4cycles and 12 different labeled graphs with a 3cycle and an attached, external vertex.


MAPLE

egf:= 1/2*ln(1+LambertW(x)) +1/2*LambertW(x) 1/4*LambertW(x)^2:
a:= n> n!*coeff(series(egf, x, n+3), x, n):
seq(a(n), n=1..25); # Alois P. Heinz, Mar 27 2013


MATHEMATICA

nn=20; t=Sum[n^(n1) x^n/n!, {n, 1, nn}]; Drop[Range[0, nn]! CoefficientList[Series[Log[1/(1t)]/2t^2/4t/2, {x, 0, nn}], x], 1] (* Geoffrey Critzer, Oct 07 2012 *)
a[n_] := (n1)!*n^n/2*Sum[1/(n^k*(nk)!), {k, 3, n}]; Table[a[n], {n, 1, 20}] (* JeanFrançois Alcover, Jan 15 2014, after Vladeta Jovovic *)


PROG

(Sage)
# Warning: Floating point calculation. Adjust precision as needed!
from mpmath import *
mp.dps = 200; mp.pretty = True
for n in (1..100):
print chop((n^(n2)*(13*n)+exp(n)*gammainc(n+1, n)/n)/2) # Peter Luschny, Jan 27 2016


CROSSREFS

Cf. A000272 = labeled trees on n nodes; connected labeled graphs with n nodes and n+k edges for k=0..8: A057500 A061540 A061541 A061542 A061543 A096117 A061544 A096150 and A096224.
Cf. A001429 (unlabeled case), A052121.
Sequence in context: A027843 A027840 A279530 * A137916 A218696 A297669
Adjacent sequences: A057497 A057498 A057499 * A057501 A057502 A057503


KEYWORD

easy,nonn,changed


AUTHOR

QingHu Hou and David C. Torney (dct(AT)lanl.gov), Sep 01 2000


EXTENSIONS

More terms from Vladeta Jovovic, Jul 09 2001


STATUS

approved



