The question I am wondering about is:

Can the discrete Laplacian have complex eigenvalues on a graph?

Clearly, there are two cases where it is obvious that this is impossible.

1.) The graph is finite 2.) The underlying space is $\ell^2$, since then the discrete Laplacian is self-adjoint.

Thus, my question requires us to look at an infinite graph and a large space.

Hence: **Does there exist an infinite graph such that the discrete Laplacian on $\ell^{\infty}$ has complex eigenvalues?**

Thank you very much

BTW: My casual use of complex in the above text refers to $\mathbb C \backslash \mathbb R$

finitedirected graphs with complex eigenvalues, so something is not right with your "clearly this is impossible for finite graphs". $\endgroup$ – M. Winter Aug 22 at 12:17