Clear[liens,cparmodcom]

l=Length[liens];
ll=Length[cparmodcom];
rat=29*Sqrt[2];
Mat=Table[0,{x,ll},{y,ll}];
cut=0.5;

Do[{dx=cparmodcom[[liens[[m,1]],1]]-cparmodcom[[liens[[m,2]],1]];dy=cparmodcom[[liens[[m,1]],2]]-cparmodcom[[liens[[m,2]],2]];If[(Abs[dx]>cut)&&(Abs[dy]>cut),{Mat[[liens[[m,1]],liens[[m,2]]]]=-Exp[I*(kx*rat*(dx-Sign[dx])+ky*rat*(dy-Sign[dy]))],Mat[[liens[[m,2]],liens[[m,1]]]]=-Exp[-I*rat*(kx*(dx-Sign[dx])+ky*(dy-Sign[dy]))]},If[(Abs[dx]>cut),{Mat[[liens[[m,1]],liens[[m,2]]]]=-Exp[I*rat*(kx*(dx-Sign[dx])+ky*(dy))],Mat[[liens[[m,2]],liens[[m,1]]]]=-Exp[-I*rat*(kx*(dx-Sign[dx])+ky*(dy))]},If[(Abs[dy]>cut),{Mat[[liens[[m,1]],liens[[m,2]]]]=-Exp[I*rat*(kx*(dx)+ky*(dy-Sign[dy]))],Mat[[liens[[m,2]],liens[[m,1]]]]=-Exp[-I*rat*(kx*(dx)+ky*(dy-Sign[dy]))]},{Mat[[liens[[m,1]],liens[[m,2]]]]=-Exp[I*rat*(kx*(dx)+ky*(dy))],Mat[[liens[[m,2]],liens[[m,1]]]]=-Exp[-I*rat*(kx*(dx)+ky*(dy))]}]]]},{m,1,l}];

spx3={};
Do[AppendTo[spx3,Flatten[{Pi*n/(1000*12*Sqrt[2]),Eigenvalues[Mat/.kx->Pi*n/(1000*12*Sqrt[2])/.ky->0][[1]]}]],{n,-10,10,1}];
spy3={};
Do[AppendTo[spy3,Flatten[{Pi*n/(1000*12*Sqrt[2]),Eigenvalues[Mat/.ky->Pi*n/(1000*12*Sqrt[2])/.kx->0][[1]]}]],{n,-10,10,1}];
spxy3={};
Do[AppendTo[spxy3,Flatten[{Pi*n/(1000*12*Sqrt[2]),Eigenvalues[Mat/.kx->Pi*n/(1000*12*Sqrt[2])/.ky->Pi*n/(1000*12*Sqrt[2])][[1]]}]],{n,-10,10,1}];

Fit[spx3,{1,xx,(xx^(2  ))},xx]

Fit[spy3,{1,xx,(xx^(2  ))},xx]
Fit[spxy3,{1,xx,(xx^(2  ))},xx]

alphax3=2*1.4732800095038943` ;
alphay3=2*1.47328001134617`;
alphaxy3=2.9498355827351004`-1.4732800095038943`-1.47328001134617`;
Sqrt[alphax3*alphay3-(alphaxy3^2)]

l

spx4={};
Do[AppendTo[spx4,Flatten[{Pi*n/(1000*29*Sqrt[2]),Eigenvalues[Mat/.kx->Pi*n/(1000*29*Sqrt[2])/.ky->0][[1]]}]],{n,-10,10,1}];
spy4={};
Do[AppendTo[spy4,Flatten[{Pi*n/(1000*29*Sqrt[2]),Eigenvalues[Mat/.ky->Pi*n/(1000*29*Sqrt[2])/.kx->0][[1]]}]],{n,-10,10,1}];
spxy4={};
Do[AppendTo[spxy4,Flatten[{Pi*n/(1000*29*Sqrt[2]),Eigenvalues[Mat/.kx->Pi*n/(1000*29*Sqrt[2])/.ky->Pi*n/(1000*29*Sqrt[2])][[1]]}]],{n,-10,10,1}];

Fit[spx4,{1,xx,(xx^(2  ))},xx]
Fit[spy4,{1,xx,(xx^(2  ))},xx]
Fit[spxy4,{1,xx,(xx^(2  ))},xx]

alphax4=2*1.45885124369709` ;
alphay4=2*1.4588512463869403`;
alphaxy4=2.9182633239157574`-1.45885124369709`-1.4588512463869403`;
Sqrt[alphax4*alphay4-(alphaxy4^2)]

Length[Mat]