On the ground state for quantum graphs

Ground-state eigenfunctions of Schrodinger operators can often be chosen positive. We analyse to which extent this is true for quantum graphs-differential operators on metric graphs. It is shown that the theorem holds in the case of generalised delta couplings at the vertices-a new class of vertex conditions introduced in the paper. It is shown that this class of vertex conditions is optimal. Relations to positivity preserving and positivity improving semigroups are clarified.