Surgery Transformations and Spectral Estimates of \(\delta \) Beam Operators

We introduce \(\delta \) type vertex conditions for beam operators, the fourth-order differential operator, on finite, compact and connected metric graphs. Our study the effect of certain geometrical alterations (graph surgery) of the graph on their spectra. Results are obtained for a class of vertex conditions which can be seen as an analogue of \(\delta \)-conditions for graphs Laplacian. There are a number of possible candidates of \(\delta \) type conditions for beam operators. We develop surgery principles and record the monotonicity properties of their spectrum, keeping in view the possibility that vertex conditions may change within the same class after certain graph alterations. We also demonstrate the applications of surgery principles by obtaining several lower and upper estimates on the eigenvalues.