# Spectrum of Combinatorial Laplacian

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?

-
Do Fourier transform – Alexander Chervov Jan 16 '13 at 19:45
Alexander is perfectly correct, let me just give a hint: the spectrum is the set of values of the function $6-2\cos(x)-2\cos(y)-2\cos(x+y)$ on the 2-torus. I'll leave the rest to you! – Alain Valette Jan 16 '13 at 21:00