Oscillatory properties of the forth-order differential operator on a metric graph

We consider a boundary value problem for a network beam equation with a forcing term. The main aim of the paper is to study oscillatory property of the corresponding forth-order differential operator. The definition of the oscillation of an operator is given in terms of a special fundamental system of solutions of the corresponding homogeneous equation on a graph. We study sign-regularity properties of solutions of the corresponding homogeneous equations on a graph and show that the positivity of a special system of solutions implies disconjugacy of the operator on each edge of the graph.