In 1836, Sturm (see [1]) discovered remarkable properties of eigenvalues and eigenfunctions of second-order differential operators of special form. These properties, called oscillatory, describe general harmonic properties of oscillations of elastic systems. One of the main tools for substantiating oscillatory properties is theorems on the distribution of zeros of solutions to a differential equation. These theorems and their various generalizations are very important for diverse applications to the theory of elastic oscillations and the corresponding spectral problems [2]–[5].

In this paper, we show that the Sturm separation theorem on the alternation of the zeros of solutions can be extended to fourth-order equations on a geometric graph Γ that arise in modeling rod systems. We mention at once that the problem of studying the oscillatory properties of the eigenfunctions of fourth-order differential operators is substantially more complicated than the similar problem for the Sturm–Liouville operators [6]–[10]. Moreover, matrix methods involving sign-regular kernels do not apply to operators on graphs. In this paper, we follow the approach of [5],[11]–[13].