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A355558
The independence polynomial of the n-halved cube graph evaluated at -1.
1
0, -1, -3, -3, 25, -135, -2079, 1879969
OFFSET
1,3
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.
LINKS
Eric Weisstein's World of Mathematics, Independence polynomial
Eric Weisstein's World of Mathematics, Halved cube graph
EXAMPLE
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).
PROG
(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
CROSSREFS
KEYWORD
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AUTHOR
Christopher Flippen, Jul 06 2022
STATUS
approved