Spectral properties of a fourth-order differential operator on a network

In this paper, we study spectral properties of a fourth-order differential operator on a network, which is a model of the Euler-Bernoulli beam system. We propose a new approach to the development of oscillatory spectral theory of a fourth-order differential operator on a network. This approach is based on the concept of a sign-constant zone for a continuous function on a graph. We show that eigenvalues and eigenfunctions of the corresponding operator on a network have oscillatory properties. We establish a condition of simplicity of eigenvalues. Finally, we study the distribution of the zeros of the eigenfunctions. To that end, we introduce the Weyl solutions and study their dependence on the spectral parameter. We show that the k$$ k $$th eigenfunction has exactly k$$ k $$ zeros in the graph.