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Decimal expansion of the zero x1 of the cubic polynomial x^3 - 2*x^2 - 10*x - 6.
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%I #15 Nov 17 2020 21:22:52

%S 4,5,1,1,4,0,4,6,6,4,2,2,6,7,5,8,1,2,3,3,3,9,2,2,1,4,9,6,8,1,3,1,6,9,

%T 5,7,4,0,2,1,8,4,3,6,1,6,4,5,0,8,8,5,7,4,6,3,5,1,7,4,8,6,8,6,1,2,7,9,

%U 5,8,3,4,4,8,2,1,6,4,9,2,5,1,5,8,9,6,7,5,8,2,7,1,7,4,3,2,5,5,3,3

%N Decimal expansion of the zero x1 of the cubic polynomial x^3 - 2*x^2 - 10*x - 6.

%C This cubic polynomial P3(x) = x^3 - 2*x^2 - 10*x - 6 is a factor of the characteristic polynomial F(x) of degree 7 of the 7 X 7 adjacency matrix F7 of the Fano graph with nodes (vertices) of degree 6, 5, 5, 5, 3, 3, 3. See the links for the Fano plane. The graph is in fact planar.

%C The adjacency matrix is F7 = Matrix([[0, 1, 1, 1, 1, 1, 1], [1, 0, 1, 1, 1, 1, 0], [1, 1, 0, 1, 0, 1, 1], [1, 1, 1, 0, 1, 0, 1], [1, 1, 0, 1, 0, 0, 0], [1, 1, 1, 0, 0, 0, 0], [1, 0, 1, 1, 0, 0, 0]]).

%C The determinant of F7 is 6. The characteristic polynomial is F(x) = x^7 - 15*x^5 - 26*x^4 + 3*x^3 + 24*x^2 + 2*x - 6 = P3(x)*(x^2 + x - 1)^2. The zeros of F(x) (the eigenvalues or spectrum of F7) are: x1, x2 = -A335863 = -1.752517821..., x3 = -A335864 = -0.7588868422..., twice -1 + phi = 0.618033988..., and twice -phi, where phi = A001622.

%C For the bipartite incidence graph see the links for the Heawood graph.

%H Wolfdieter Lang, <a href="https://www.itp.kit.edu/~wl/EISpub/A333852.pdf">A list of representative simple difference sets of the Singer type for small orders m</a>, Karlsruher Institut für Technologie (Karlsruhe, Germany 2020).

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FanoPlane.html">Fano plane</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeawoodGraph.html">Heawood graph</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fano_plane">Fano plane</a>

%H Wikipedia, <a href="https://de.wikipedia.org/wiki/Heawood-Graph">Heawood graph</a>

%F x1 = (1/3)*(2 + (179 + 3*sqrt(3)*sqrt(269)*i)^(1/3) + ( 179 - 3*sqrt(3)*sqrt(269)*i)^(1/3)), where i is the imaginary unit.

%e x1 = 4.5114046642267581233392214968131695740218436164...

%t With[{k = 3 Sqrt[3] Sqrt[269] I}, First@ RealDigits[Re[(1/3) (2 + (179 + k)^(1/3) + (179 - k)^(1/3))], 10, 100]] (* _Michael De Vlieger_, Nov 17 2020 *)

%Y Cf. A001622, A335863 (-x2), A335864 (-x3).

%K nonn,cons,easy

%O 1,1

%A _Wolfdieter Lang_, Jun 29 2020