The spectrum of the Hilbert space valued second derivative with general self-adjoint boundary conditions

We consider a large class of self-adjoint elliptic problems associated with the second derivative acting on a space of vector-valued functions. We present and survey several results that can be obtained by means of two different approaches to the study of the associated eigenvalues problems. The first, more general one allows to replace a secular equation (which is well known in some special cases) by an abstract rank condition. The second one, though available in general, seems to apply particularly well to a specific boundary condition, the sometimes dubbed anti-Kirchhoff condition in the literature, that arises in the theory of differential operators on graphs; it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities, in particular some results on the symmetry of the spectrum in either case.