V[α_, p_, q_, ky_] = 
        2 Cos[((2  π p α)/q)  -  
                ky] - λ;
M[kx_, ky_, 
          2] = {{V[1, 1, 2, ky], 
            1 + (E^(I kx 2))}, {1 + (E^(-I kx 2)), 
            V[2, 1, 2, ky]}};
\(\(M[kx_, ky_, p_, q_] := 
        Block[{}, 
Table[Δ = 
              Abs[i - j]; 
If[Δ == 0, 
              V[i, p, q, ky], 
	
              If[Δ == 1, 
                1, 
		If[Δ == \ 
                    q - 1, E\^\(I\ kx\ q\ Sign[i - j]\), 0]]]
, {i, 1, q}, {j, 1, 
              q}]
];\)\)

Eigenvalues[M[kx, ky, 3]] // Simplify

MM[kx_, 
        ky_] =   M[kx, ky, 2, 5] /. λ -> 
          0
t = 
        Table[Re[Eigenvalues[MM[kx, ky]]], {kx, 0, 
            2  π,  .01*2  π}, {ky, 0, 
            2  π,  .01*2  π}];
e = Flatten[t];
l = Length[e];
x  =  Table[1, {l}];
tt = Transpose[{x, e}];
ListPlot[tt]

Det[M[kx, ky, 1, 3]] // Simplify

f[λ_] = 
        (1/2) λ (-35 + 5 (Sqrt[5]) + 20 (λ^2) - 
              2 (λ^4)) ;

Plot[f[λ], {λ, -10, 10}, 
      PlotRange -> {{-10, 10}, {-10, 10}}]