A355227 -id:A355227 - cp's OEIS frontend

A290888 Number of independent vertex sets and vertex covers in the n-folded cube graph.

Original entry on oeis.org

3, 5, 31, 393, 177347, 2932100733

Offset: 2

Andrew Howroyd, Aug 15 2017

The independence number of the n-folded cube graphs is given by A058622(n-1).

Eric Weisstein's World of Mathematics, Folded Cube Graph.
Eric Weisstein's World of Mathematics, Independent Vertex Set.
Eric Weisstein's World of Mathematics, Vertex Cover.

Row sums of A355227.

Cf. A027624, A058622, A301460.

A355559 The independence polynomial of the n-folded cube graph evaluated at -1.

Original entry on oeis.org

-1, -3, -1, 9, 131, 253, 25607

Offset: 2

Christopher Flippen, Jul 06 2022

The independence number alpha(G) of a graph is the cardinality of the largest independent vertex set. The n-folded cube has alpha(G) = A058622(n-1). The independence polynomial for the n-folded cube is given by Sum_{k=0..alpha(G)} A355227(n,k)*t^k, meaning that a(n) is the alternating sum of row n of A355227.

			Row 5 of A355227 is 1, 16, 80, 160, 120, 16. This means the 5-folded cube graph has independence polynomial 1 + 16*t + 80*t^2 + 160*t^3 + 120*t^4 + 16*t^5. Taking the alternating row sum of row 5, or evaluating the polynomial at -1, gives us 1 - 16 + 80 - 160 + 120 - 16 = 9 = a(5).

Eric Weisstein's World of Mathematics, Independence polynomial
Eric Weisstein's World of Mathematics, Folded cube graph

Cf. A058622, A355227, A290888.

from sage.graphs.independent_sets import IndependentSets
def a(n):
    icount=0
    for Iset in IndependentSets(graphs.FoldedCubeGraph(n)):
        if len(Iset) % 2 == 0:
            icount += 1
        else:
            icount += -1
    return icount

cp's OEIS Frontend

A290888 Number of independent vertex sets and vertex covers in the n-folded cube graph.

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