Higher order Dirichlet-to-Neumann maps on graphs and their eigenvalues

In this paper, we first introduce higher order Dirichlet-to-Neumann maps on graphs which can be viewed as a discrete analogue of the corresponding Dirichlet-to-Neumann maps on compact Riemannian manifolds with boundary and a higher order generalization of the Dirichlet-to-Neumann map on graphs introduced by Hua-Huang-Wang\cite{HHW} and Hassannezhad-Miclo \cite{HM}. Then, some Raulot-Savo-type estimates on the eigenvalues of the DtN maps introduced are derived.