Upper Bounds for the Eigenvalue Multiplicities of a Fourth-Order Differential Operator on a Graph

The paper studies a model of a planar beam structure described by a fourth-orderboundary value problem on a geometric graph. Elastic-hinge joint conditions are posed at theinterior vertices of the graph. We study the properties of the spectral points of the correspondingspectral problem, prove upper bounds for the eigenvalue multiplicities, and show that theeigenvalue multiplicities depend on the graph structure (the number of boundary vertices, cycles,etc.). We give an example showing that our estimates are sharp.