Thanks for your answer! I am now looking into your references. Although I was aware of the symmetry of the spectrum and, in general, of the fact that there is a wealth of information about the eigenvalues of bipartites graphs, my impression is that there aren't so many results concerning the eigenvectors (beyond the mirror symmetry that relates eigenvectors for eigenvalues $\lambda$ and $-\lambda$), and my question is specifically about these median eigenvectors (I didn't know this key word btw; I'll use it now).

I really don't know. What happens? Anyway, in the problem I am concerned with the matrices are much larger.

Thanks for the reference and the nice explanation, Pedro.

Kiitos, Joonas. I'll check these things out.

Yes, I am sorry, I meant Helgason's support theorem (The microlocal version in terms of the Wave Front Set)