OFFSET
1,2
COMMENTS
For the spectrum of W_n see, e.g., the Wikipedia link. - Wolfdieter Lang, Oct 30 2018
LINKS
Wikipedia, Wheel graph
FORMULA
Conjecture: a(n) = A004772(n) for n> 1. [clarified by Michel Marcus, Apr 16 2019]
Conjectures from Colin Barker, Nov 02 2020: (Start)
G.f.: x*(1 + x + x^2 + x^4) / ((1 - x)^2*(1 + x + x^2)).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>5.
(End)
EXAMPLE
4 is a term as the characteristic polynomial of the wheel graph of 4 nodes is x^4 - 6*x^2 - 8*x - 3, in which the monomial of x^3 has null coefficient and no other ones, so this polynomial has exactly one monomial with vanishing coefficient.
5 is not member of this sequence because the eigenvalues of A(W_5) (the adjacency matrix of W_5) has eigenvalues 0, 0, 2, 1 + sqrt(5), 1 - sqrt(5), and the monic characteristic polynomial is x^5 - 8*x^3 - 8*x^2 with three missing monomials x^0, x^1 and x^4. - Wolfdieter Lang, Oct 30 2018
PROG
(Sage)
def how_many_zeros(v):
t=0
for el in v:
if el==0: t += 1
return t
r=""
for i in range(1, 100):
p = graphs.WheelGraph(i)
cp=p.characteristic_polynomial()
vcp=(cp.coefficients(sparse=False))
if how_many_zeros(vcp)==1:
r=r+", "+str(i)
print(r)
CROSSREFS
KEYWORD
nonn
AUTHOR
Pierandrea Formusa, Sep 17 2018
STATUS
approved