Upper bounds for Steklov eigenvalues of subgraphs of polynomial growth Cayley graphs

We study the Steklov problem on a subgraph with boundary $(\Omega,B)$ of polynomial growth Cayley graph $\Gamma$. prove that for each $k \in \mathbb{N}$, $k^{\mbox{th}}$ eigenvalue tends to $0$ proportionally $1/|B|^{\frac{1}{d-1}}$, where $d$ represents rate The method consists in associating manifold $M$ $\Gamma$ and bounded domain $N \subset M$ $(\Omega, B)$ find upper bounds spectrum $N$ transfer these by discretizing using comparison Theorems.