The spectrum of the combinatorial laplacian is well understood for a square lattice. What about for other lattices?
In particular: let $ f: \mathbb{Z}^2 \rightarrow \mathbb{R} $. The usual combinatorial laplacian is:
$$ \Delta f(x,y) = 4 f(x,y) - f(x+1,y) - f(x-1,y) - f(x,y+1) - f(x,y-1) \; . $$
Suppose we also add the diagonals:
$$ \Delta f(x,y) = 6 f(x,y) - f(x+1,y) - f(x-1,y) - f(x,y+1) - f(x,y-1) - f(x+1,y+1) - f(x-1,y-1) \; . $$
(In some sense, this corresponds to a smooth laplacian with off diagonal metric element.) Does anybody know the spectrum for this?