A284707 Number of maximal independent vertex sets in the n-hypercube graph Q_n.

1, 2, 2, 6, 42, 1670, 1281402

Offset

0,2

Links

Table of n, a(n) for n=0..6.

Dwight Duffus, Peter Frankl, and Vojtěch Rödl, Maximal independent sets in bipartite graphs obtained from Boolean lattices, European Journal of Combinatorics 32.1 (2011): 1-9.

Dwight Duffus, Peter Frankl, and Vojtěch Rödl, Maximal independent sets in the covering graph of the cube, Discrete Applied Mathematics 161.9 (2013): 1203-1208.

Dmitry I. Ignatov, On the Maximal Independence Polynomial of the Covering Graph of the Hypercube up to n = 6, Int'l Conf. Formal Concept Analysis, 2023.

Liviu Ilinca and Jeff Kahn, Counting maximal antichains and independent sets, arXiv:1202.4427 [math.CO], Feb 2012; Order 30.2 (2013): 427-435.

Jeff Kahn and Jinyoung Park, The number of maximal independent sets in the Hamming cube, arXiv:1909.04283 [math.CO], 2019.

Eric Weisstein's World of Mathematics, Hypercube Graph

Eric Weisstein's World of Mathematics, Independent Vertex Set

Eric Weisstein's World of Mathematics, Maximal Independent Vertex Set

Formula

a(n) ~ 2*n*2^(N/4) where N = 2^n [Kahn and Park]. - N. J. A. Sloane, Sep 11 2019

Mathematica

Table[Length @ FindIndependentVertexSet[HypercubeGraph[n], Infinity, All], {n, 0, 6}] (* Eric W. Weisstein, Jan 01 2024 *)

Prog

(Python)
from networkx import empty_graph, find_cliques
def A284707(n):
    k = 1<<n
    G = empty_graph(list(range(k)))
    G.add_edges_from((a, b) for a in range(k) for b in range(a) if (lambda m: (m&-m)^m if m else 1)(a^b))
    return sum(1 for c in find_cliques(G)) # Chai Wah Wu, Jan 11 2024

Crossrefs

Cf. A027624 (not necessarily maximal), A366425 (non-isomorphic).

Sequence in context: A290957 A032117 A137244 * A174589 A326942 A247943

Adjacent sequences: A284704 A284705 A284706 * A284708 A284709 A284710

Keyword

nonn,more

Author

Eric W. Weisstein, Apr 01 2017

Status

approved