Nonreal eigenvalues for second order differential operators on networks with circuits ✩

It is shown that on every finite network with at least one circuit there exist second order differential operators having an infinite number of nonreal eigenvalues. The presence of nonreal eigenvalues implies that these operators cannot be selfadjoint with respect to any metric. These eigenvalues reveal also the existence of oscillatory solutions for the corresponding time-dependent partial differential equations.  2002 Elsevier Science (USA). All rights reserved.