cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A232699 Number of labeled point-determining bipartite graphs on n vertices.

Original entry on oeis.org

1, 1, 1, 3, 15, 135, 1875, 38745, 1168545, 50017905, 3029330745, 257116925835, 30546104308335, 5065906139629335, 1172940061645387035, 379092680506164049425, 171204492289446788997825, 108139946568584292606269025, 95671942593719946611454522225
Offset: 0

Views

Author

Justin M. Troyka, Nov 27 2013

Keywords

Comments

A graph is point-determining if no two vertices have the same set of neighbors. This kind of graph is also called a mating graph.
a(n) is always odd.
For every prime p > 2, a(n) is divisible by p for all n >= p. It follows that, if m is odd and squarefree with largest prime factor q, then a(n) is divisible by m for all n >= q. A similar property appears to hold for odd prime powers, in which case it would hold for all odd numbers.

Examples

			Consider n = 3. The triangle graph is point-determining, but it is not bipartite, so it is not counted in a(3). The graph 1--2--3 is bipartite, but it is not point-determining (the vertices on the two ends have the same neighborhood), so it is also not counted in a(3). The only graph counted in a(3) is the graph *--*  * (with three possible labelings). - _Justin M. Troyka_, Nov 27 2013

Links

Crossrefs

Cf. A006024, A004110 (labeled and unlabeled point-determining graphs).
Cf. A092430, A004108 (labeled and unlabeled connected point-determining graphs).
Cf. A218090 (unlabeled point-determining bipartite graphs).
Cf. A232700, A088974 (labeled and unlabeled connected point-determining bipartite graphs).

Programs

  • Mathematica
    terms = 20;
CoefficientList[Sqrt[Sum[((1+x)^2^k Log[1+x]^k)/k!, {k, 0, terms}]] + O[x]^terms, x] Range[0, terms-1]! (* Jean-François Alcover, Sep 13 2018, after Andrew Howroyd *)
  • PARI
    seq(n)={my(A=log(1+x+O(x*x^n))); Vec(serlaplace(sqrt(sum(k=0, n, exp(2^k*A)*A^k/k!))))} \\ Andrew Howroyd, Sep 09 2018

Formula

From Andrew Howroyd, Sep 09 2018: (Start)
a(n) = Sum_{k=0..n} Stirling1(n,k)*A047864(k).
E.g.f: sqrt(Sum_{k=0..n} exp(2^k*log(1+x))*log(1+x)^k/k!). (End)

A232700 Number of labeled connected point-determining bipartite graphs on n vertices.

Original entry on oeis.org

1, 1, 0, 12, 60, 1320, 26880, 898800, 40446000, 2568736800, 225962684640, 27627178692960, 4686229692144000, 1104514965434200320, 361988888631722352000, 165271302775469812521600, 105278651889065640047462400, 93750696652129931568573619200
Offset: 1

Views

Author

Justin M. Troyka, Nov 27 2013

Keywords

Comments

A graph is point-determining if no two vertices have the same set of neighbors. This kind of graph is also called a mating graph.
For all n >= 3, a(n) is even. For every prime p > 2, a(n) is divisible by p for all n >= p. It follows that, if m is squarefree with largest prime factor q, then a(n) is divisible by m for all n >= q. A similar property appears to hold for prime powers, in which case it would hold for all numbers.

Examples

			Consider n = 4.  There are 12 connected point-determining bipartite graphs on 4 vertices: the graph *--*--*--*, with 12 possible labelings. - _Justin M. Troyka_, Nov 27 2013

Links

Crossrefs

Cf. A006024, A004110 (labeled and unlabeled point-determining graphs).
Cf. A092430, A004108 (labeled and unlabeled connected point-determining graphs).
Cf. A232699, A218090 (labeled and unlabeled point-determining bipartite graphs).
Cf. A088974 (unlabeled connected point-determining bipartite graphs).

Programs

  • Mathematica
    terms = 18;
G[x_] = Sqrt[Sum[((1 + x)^2^k*Log[1 + x]^k)/k!, {k, 0, terms+1}]] + O[x]^(terms+1);
A[x_] = x + Log[1 + (G[x] - x - 1)/(1 + x)];
(CoefficientList[A[x], x]*Range[0, terms]!) // Rest (* Jean-François Alcover, Sep 13 2018, after Andrew Howroyd *)
  • PARI
    seq(n)={my(A=log(1+x+O(x*x^n))); my(p=sqrt(sum(k=0, n, exp(2^k*A)*A^k/k!))); Vec(serlaplace(x + log(1+(p-x-1)/(1+x))))} \\ Andrew Howroyd, Sep 09 2018

Formula

E.g.f.: x + log(1 + (G(x)-x-1)/(1+x)) where G(x) is the e.g.f. of A232699. - Andrew Howroyd, Sep 09 2018

A088974 Number of (nonisomorphic) connected bipartite graphs with minimum degree at least 2 and with n vertices.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 9, 45, 160, 1018, 6956, 67704, 830392, 13539344, 288643968, 8112651795, 300974046019, 14796399706863, 967194378235406, 84374194347669628, 9856131011755992817, 1546820212559671605395
Offset: 1

Views

Author

Felix Goldberg (felixg(AT)tx.technion.ac.il), Oct 30 2003

Keywords

Comments

The terms were computed using the program Nauty.
As shown in the Hardt et al. reference, this sequence (for n >= 3) also enumerates the connected point-determining bipartite graphs. - Justin M. Troyka, Nov 27 2013

Examples

			Consider n = 4.  There is one connected bipartite graph with minimum degree at least 2: the square graph.  Also there is one connected point-determining bipartite graph: the graph *--*--*--*. - _Justin M. Troyka_, Nov 27 2013

Links

Crossrefs

Cf. A006024, A004110 (labeled and unlabeled point-determining graphs [the latter is also unlabeled graphs w/ min. degree >= 2]).
Cf. A059167 (labeled graphs w/ min. degree >= 2).
Cf. A092430, A004108 (labeled and unlabeled connected point-determining graphs [the latter is also unlabeled connected graphs w/ min. degree >= 2]).
Cf. A059166 (labeled connected graphs w/ min. degree >= 2).
Cf. A232699, A218090 (labeled and unlabeled point-determining bipartite graphs).
Cf. A232700 (labeled connected point-determining bipartite graphs).

Extensions

More terms from Andy Hardt, Oct 31 2012
Showing 1-3 of 3 results.

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