I wonder if there are some results which guarantee that the largest eigenvalue of a Hashimoto matrix of a graph is always real (Hashmioto matrices are square but not symmetric, so in general there are usually some complex eigenvalues). I checked on some examples and it seems to be true, but I would like to know for sure.
If possible, I would like to know this for two cases: directed and undirected graphs.
Yes. This is shown using Perron-Frobenius theory.
For example, the introduction of this paper shows that the radius of convergence of the Ihara zeta function coincides with the leading eigenvalue of the Hashimoto matrix.
Alternatively, you could try to apply the Perron-Frobenius theorem to the Hashimoto matrix yourself. It works because the (extended version of) Perron-Frobenius theorem applies to any non-negative matrix.
This works whether the graph is undirected, directed, or contains multi-edges or self-loops.