OFFSET
0,4
COMMENTS
The independence number alpha(G) of a graph is the cardinality of the largest independent vertex set. The n-hypercube has alpha(G) = 1 for n = 0 and alpha(G) = 2^(n-1) for n >= 1. The independence polynomial for the n-hypercube is given by Sum_{k=0..alpha(G)} A354802(n,k)*t^k, meaning that a(n) is the alternating sum of row n of A354802.
Jenssen, Perkins and Potukuchi proved asymptotics for independent sets of given size.
It appears that this sequence remains positive for n > 3.
LINKS
M. Jenssen, W. Perkins and A. Potukuchi, Independent sets of a given size and structure in the hypercube, Combinatorics, Probability and Computing, 2022, 1-19; see also arXiv:2106.09709 [math.CO], 2021-2022.
Eric Weisstein's World of Mathematics, Hypercube graph
Eric Weisstein's World of Mathematics, Independence polynomial
EXAMPLE
Row 3 of A354802 is 1, 8, 16, 8, 2. This means the 3-hypercube cube graph has independence polynomial 1 + 8*t + 16*t^2 + 8*t^3 + 2*t^4. Taking the alternating row sum of row 3, or evaluating the polynomial at -1, gives us 1 - 8 + 16 - 8 + 2 = 3 = a(3).
PROG
(Sage) from sage.graphs.connectivity import connected_components
def recurse(g):
if g.order() == 0:
return 1
comp = g.connected_components()
if len(comp[-1]) == 1:
return 0
elif len(comp) > 1:
prod = 1
for c in comp:
if prod == 0:
return 0
else:
prod = prod*recurse(g.subgraph(vertices=c))
return prod
min_degree_vertex = g.vertices()[0]
for v in g.vertices():
if g.degree(v) < g.degree(min_degree_vertex):
min_degree_vertex = v
to_remove_edge = g.edges_incident(min_degree_vertex)[0]
to_remove_vertices = [to_remove_edge[0], to_remove_edge[1]]
to_remove_vertices.extend(g.neighbors(to_remove_edge[0]))
to_remove_vertices.extend(g.neighbors(to_remove_edge[1]))
to_remove_vertices = list(set(to_remove_vertices))
without_neighborhoods = copy(g)
without_edge = copy(g)
without_neighborhoods.delete_vertices(to_remove_vertices)
without_edge.delete_edge(to_remove_edge)
return recurse(without_edge) - recurse(without_neighborhoods)
def a(n):
if n == 0:
return recurse(graphs.CompleteGraph(1))
else:
return recurse(graphs.CubeGraph(n))
# Christopher Flippen and Scott Taylor, Jun 2022
CROSSREFS
KEYWORD
sign,more
AUTHOR
Christopher Flippen, Jun 05 2022
STATUS
approved