with(linalg);

st:={}: L:=4:
for i from 0 to L-1 do for j from i+1 to L do
st:=st union {[i,j]}: od: od:

H:=array(1..nops(st),1..nops(st),sparse);

for i from 1 to nops(st) do for j from 1 to nops(st) do
rr:=st[j]: ll:=st[i];
x1r:=rr[1]: x2r:=rr[2]:
x1l:=ll[1]: x2l:=ll[2]:
if (x1r=x1l) then
   if abs(x2r-x2l)=1 then H[i,j]:=1:
   else H[i,j]:=0: fi:
elif (x2r=x2l) then
   if abs(x1r-x1l)=1 then H[i,j]:=1:
   else H[i,j]:=0: fi:
else H[i,j]:=0: fi: od: od:

eigenvals(H);