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A333865 Number of simple graphs on n nodes with vertex count > edge count + 1. 0
0, 1, 2, 4, 8, 18, 40, 100, 256, 705, 2057, 6370, 20803, 71725, 259678, 985244, 3905022, 16124936, 69188809, 307765510, 1416146859, 6727549181, 32938379216, 165942445714, 859020421012, 4563322971706, 24847598243116, 138533012486423, 790075521708603, 4605183081182354 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

These graphs correspond to "trivially ungraceful" graphs that do not have enough integers less than or equal to the edge count to cover all the vertices.

LINKS

Table of n, a(n) for n=1..30.

Eric Weisstein's World of Mathematics, Graceful Graph

Eric Weisstein's World of Mathematics, Simple Graph

Eric Weisstein's World of Mathematics, Ungraceful Graph

FORMULA

a(n) <= A308556(n).

a(n) = Sum_{k=0..n-2} A008406(n, k). - Andrew Howroyd, Apr 08 2020

MATHEMATICA

Get["Combinatorica`"] // Quiet;

Table[Total[Take[CoefficientList[GraphPolynomial[n, x], x], n - 1]], {n, 20}]

PROG

(PARI)

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, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c-1)\2)*if(c%2, 1, t(c/2)))}

a(n)={my(s=0); if(n>1, forpart(p=n, s+=permcount(p)*polcoef(edges(p, i->1 + x^i + O(x^(n-1)))/(1-x), n-2) )); s/n!} \\ Andrew Howroyd, Apr 08 2020

CROSSREFS

Cf. A008406.

Cf. A308556 (number of simple ungraceful graphs on n nodes).

Sequence in context: A330052 A317787 A019231 * A274547 A112483 A151381

Adjacent sequences:  A333862 A333863 A333864 * A333866 A333867 A333868

KEYWORD

nonn

AUTHOR

Eric W. Weisstein, Apr 08 2020

EXTENSIONS

Terms a(11) and beyond from Andrew Howroyd, Apr 08 2020

STATUS

approved

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Last modified March 7 09:46 EST 2021. Contains 341869 sequences. (Running on oeis4.)