OFFSET
1,1
COMMENTS
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.
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]]).
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.
For the bipartite incidence graph see the links for the Heawood graph.
LINKS
Wolfdieter Lang, A list of representative simple difference sets of the Singer type for small orders m, Karlsruher Institut für Technologie (Karlsruhe, Germany 2020).
Eric Weisstein's World of Mathematics, Fano plane
Eric Weisstein's World of Mathematics, Heawood graph
Wikipedia, Fano plane
Wikipedia, Heawood graph
FORMULA
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.
EXAMPLE
x1 = 4.5114046642267581233392214968131695740218436164...
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Jun 29 2020
STATUS
approved