Eigenvalue estimates for the poly-Laplace operator on lattice subgraphs

We introduce the discrete poly-Laplace operator on a subgraph with Dirichlet boundary condition. We obtain upper and lower bounds for the sum of the first $k$ Dirichlet eigenvalues of the poly-Laplace operators on a finite subgraph of lattice graph $\mathbb{Z}^{d}$ extending classical results of Li-Yau and Kr\"oger. Moreover, we prove that the Dirichlet $2l$-order poly-Laplace eigenvalues are at least as large as the squares of the Dirichlet $l$-order poly-Laplace eigenvalues.