On the multiplicity of the eigenvalues of discrete tori

It is well known that the standard flat torus ${\mathbb{T}}^2={\mathbb{R}}^2\mathrm{/}{\mathbb{Z}}^2$ has arbitrarily large Laplacian-eigenvalue multiplicities. We prove, however, that $24$ is the optimal upper bound for the multiplicities of the nonzero eigenvalues of a $2$-dimensional discrete torus. For general higher dimension discrete tori, we characterize the eigenvalues with large multiplicities. As consequences, we get uniform boundedness results of the multiplicity for a long range and an optimal global bound for the multiplicity. Our main tool of proof is the theory of vanishing sums of roots of unity.