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.

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A287511 Number of simple connected perfect non-bipartite graphs on n vertices.

Original entry on oeis.org

0, 0, 1, 3, 15, 88, 680, 7623, 126047, 3118189, 112367111, 5736020864
Offset: 1

Views

Author

Eric W. Weisstein, May 26 2017

Keywords

Links

Crossrefs

Cf. A005142, A052433.

Formula

a(n) = A052433(n) - A005142(n), since all bipartite graphs are perfect. - Falk Hüffner, Aug 10 2017

Extensions

a(11)-a(12) from formula by Falk Hüffner, Aug 10 2017

A332964 Triangle read by rows: T(n,k) is the number of unlabeled simple graphs on n nodes with exactly k bipartite connected components, n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 3, 4, 2, 1, 1, 16, 9, 5, 2, 1, 1, 96, 37, 13, 6, 2, 1, 1, 812, 162, 46, 14, 6, 2, 1, 1, 10957, 1120, 194, 50, 15, 6, 2, 1, 1, 260494, 12675, 1219, 204, 51, 15, 6, 2, 1, 1, 11713772, 276758, 13099, 1254, 208, 52, 15, 6, 2, 1, 1
Offset: 0

Views

Author

Geoffrey Critzer, Mar 04 2020

Keywords

Comments

T(n,k) is the number of graphs on n nodes with incidence matrix of rank n-k, where the incidence matrix is defined as in Godsil-Royle reference below.

Examples

			Triangle T(n,k) begins:
    1;
    0,   1;
    0,   1,   1;
    1,   1,   1,  1;
    3,   4,   2,  1,  1;
   16,   9,   5,  2,  1, 1;
   96,  37,  13,  6,  2, 1, 1;
  812, 162,  46, 14,  6, 2, 1, 1;
  ...

References

  • C. Godsil and G. Royle, Algebraic Graph Theory, Springer, 2001, page 166.

Crossrefs

Cf. A157051 (column k=0 for n>0), A000088 (row sums), A157015, A005142.

Programs

  • Mathematica
    Needs["Combinatorica`"];
Table[Table[Count[Prepend[Flatten[Table[g = {n, k};b = GraphData[g,"IncidenceMatrix"]; {n - MatrixRank[b]}, {k,2, NumberOfGraphs[n]}]], n], i], {i, 0, n}], {n, 0,7}] // Grid

Formula

G.f.: Product_{i>=1} (1/(1-x^i))^A157051(i)*(1/(1-y*x^i))^A005142(i).

A382348 Number of connected bipartite graphs with n edges.

Original entry on oeis.org

1, 1, 2, 4, 7, 17, 36, 94, 237, 658, 1845, 5527, 16809, 53357, 173298, 580331, 1988935, 6991328, 25124511, 92325353, 346401296, 1326493369
Offset: 1

Views

Author

Sergey Pupyrev, May 29 2025

Keywords

Examples

			a(3) = 2 since there are two connected bipartite graphs with three edges: a path and a star "Y".

Links

Crossrefs

Cf. A002905, A005142.

Programs

  • nauty
    for (( i=$(bc <<< "sqrt(4*${n})"); i<=$((n+1)); i++ )) do ~/nauty2_8_9/geng -c -b ${i}:${i} ${n} -u; done

Extensions

a(19)-a(22) from Sean A. Irvine, Jun 09 2025
Previous Showing 31-33 of 33 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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