Surgery transformations and eigenvalue estimates for quantum graphs with ${\delta }^{\mathrm{\prime }}$ vertex interactions

We extend the surgical tool box for quantum graphs to anti-standard and ${\delta }^{\mathrm{\prime }}$ vertex conditions. Monotonicity properties of eigenvalues of graph Laplacian with ${\delta }^{\mathrm{\prime }}$ interactions at vertices depend on the sign of vertex parameter. Using several interlacing inequalities between eigenvalues of graph Laplacian with different vertex conditions and surgery principles we obtain new upper and lower bounds on the eigenvalues of $\delta$ and ${\delta }^{\mathrm{\prime }}$ Laplacians.