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

Abstract

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