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.

A290606 Number of maximal independent vertex sets (and minimal vertex covers) in the n-halved cube graph.

Original entry on oeis.org

1, 2, 4, 4, 40, 136, 8080, 17331120
Offset: 1

Views

Author

Eric W. Weisstein, Aug 07 2017

Keywords

Links

Crossrefs

Cf. A288943.

Programs

  • Mathematica
    Table[Length@FindIndependentVertexSet[GraphPower[HypercubeGraph[n - 1], 2], Infinity, All], {n, 7}]
  • Python
    from networkx import empty_graph, find_cliques, complement, power
def A290606(n):
    k = 1<Chai Wah Wu, Jan 11 2024

A355226 Irregular triangle read by rows where T(n,k) is the number of independent sets of size k in the n-halved cube graph.

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 8, 4, 1, 16, 40, 1, 32, 256, 480, 120, 1, 64, 1344, 11200, 36400, 40320, 13440, 1920, 240, 1, 128, 6336, 156800, 2104480, 15644160, 63672000, 136970880, 147748560, 76396800, 21087360, 4273920, 840000, 161280, 28800, 3840, 240
Offset: 1

Views

Author

Christopher Flippen, Jun 24 2022

Keywords

Comments

The independence number alpha(G) of a graph is the cardinality of the largest independent vertex set. The n-halved graph has alpha(G) = A005864(n). The independence polynomial for the n-halved cube is given by Sum_{k=0..alpha(G)} T(n,k)*t^k.
Since 0 <= k <= alpha(G), row n has length A005864(n) + 1.

Examples

			Triangle begins:
    k = 0   1   2
n = 1:  1,  1
n = 2:  1,  2
n = 3:  1,  4
n = 4:  1,  8,  4
n = 5:  1, 16, 40
The 4-halved cube graph has independence polynomial 1 + 8*t + 4*t^2.

Links

Crossrefs

Row sums are A288943.
Cf. A005864, A355558.

Programs

  • Sage
    from sage.graphs.independent_sets import IndependentSets
from collections import Counter
def row(n):
    if n == 1:
        g = graphs.CompleteGraph(1)
    else:
        g = graphs.HalfCube(n)
    setCounts = Counter()
    for Iset in IndependentSets(g):
        setCounts[len(Iset)] += 1
    outList = [0] * len(setCounts)
    for n in range(0,len(setCounts)):
        outList[n] = setCounts[n]
    return outList

A355558 The independence polynomial of the n-halved cube graph evaluated at -1.

Original entry on oeis.org

0, -1, -3, -3, 25, -135, -2079, 1879969
Offset: 1

Views

Author

Christopher Flippen, Jul 06 2022

Keywords

Comments

The independence number alpha(G) of a graph is the cardinality of the largest independent vertex set. The n-halved graph has alpha(G) = A005864(n). The independence polynomial for the n-halved cube is given by Sum_{k=0..alpha(G)} A355226(n,k)*t^k, meaning that a(n) is the alternating sum of row n of A355226.

Examples

			Row 5 of A355226 is 1, 16, 40. This means the 5-halved cube graph has independence polynomial 1 + 16*t + 40*t^2. Taking the alternating row sum of row 5, or evaluating the polynomial at -1, gives us 1 - 16 + 40 = 25 = a(5).

Links

Crossrefs

Cf. A005864, A355226, A288943.

Programs

  • Sage
    from sage.graphs.independent_sets import IndependentSets
def a(n):
    if n == 1:
        g = graphs.CompleteGraph(1)
    else:
        g = graphs.HalfCube(n)
    icount=0
    for Iset in IndependentSets(g):
        if len(Iset) % 2 == 0:
            icount += 1
        else:
            icount += -1
    return icount
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.