In this article, we study a fourth-order differential operator L λ on a graph, depending on a real parameter λ. The operator L λ is connected with a model of the Euler–Bernoulli beam system. Our consideration is devoted to the study of the set of values of the spectral parameter λ for which the operator L λ is inverse positive. It is shown that the operator L λ is positively invertible if and only if there exists a fundamental system of solutions to the corresponding homogeneous equation, consisting of functions that are positive on the graph. We establish necessary and sufficient conditions for the differential operator L λ to be inverse-positive for all positive values of the spectral parameter less than the lowest eigenvalue of the differential operator L 0 corresponding to λ = 0 . In this way, we establish the positivity of eigenvalues and formulate comparison theorems for eigenvalues of the spectral problem. Next, we formulate maximum principles for a fourth-order differential inequalities on a graph.