cp's OEIS Frontend

A319371 Numbers k such that the characteristic polynomial of a wheel graph of k nodes has exactly one monomial with vanishing coefficient.

%I A319371 #36 Nov 08 2020 23:15:50
%S A319371 1,2,3,4,6,7,8,10,11,12,14,15,16,18,19,20,22,23,24,26,27,28,30,31,32,
%T A319371 34,35,36,38,39,40,42,43,44,46,47,48,50,51,52,54,55,56,58,59,60,62,63,
%U A319371 64,66,67,68,70,71,72,74,75,76,78,79,80,82,83,84,86,87,88
%N A319371 Numbers k such that the characteristic polynomial of a wheel graph of k nodes has exactly one monomial with vanishing coefficient.
%C A319371 For the spectrum of W_n see, e.g., the Wikipedia link. - _Wolfdieter Lang_, Oct 30 2018
%H A319371 Wikipedia, <a href="https://en.wikipedia.org/wiki/Wheel_graph">Wheel graph</a>
%F A319371 Conjecture: a(n) = A004772(n) for n> 1. [clarified by _Michel Marcus_, Apr 16 2019]
%F A319371 Conjectures from _Colin Barker_, Nov 02 2020: (Start)
%F A319371 G.f.: x*(1 + x + x^2 + x^4) / ((1 - x)^2*(1 + x + x^2)).
%F A319371 a(n) = a(n-1) + a(n-3) - a(n-4) for n>5.
%F A319371 (End)
%e A319371 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.
%e A319371 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
%o A319371 (Sage)
%o A319371 def how_many_zeros(v):
%o A319371     t=0
%o A319371     for el in v:
%o A319371         if el==0: t += 1
%o A319371     return t
%o A319371 r=""
%o A319371 for i in range(1,100):
%o A319371         p = graphs.WheelGraph(i)
%o A319371         cp=p.characteristic_polynomial()
%o A319371         vcp=(cp.coefficients(sparse=False))
%o A319371         if how_many_zeros(vcp)==1:
%o A319371             r=r+","+str(i)
%o A319371 print(r)
%Y A319371 Cf. A004772.
%K A319371 nonn
%O A319371 1,2
%A A319371 _Pierandrea Formusa_, Sep 17 2018