Remarks on random walks on graphs and the Floyd boundary

We show that for a uniformly irreducible random walk on graph, with bounded range, there is Floyd function which the converges to its corresponding boundary. Moreover if we add assumptions, $p^{(n)}(v,w)\leq C \rho^n$, where $\rho < 1$ spectral radius, then any $f$ satisfies $\sum_{n=1}^{\infty}nf(n)<\infty$, Dirichlet problem respect boundary solvable.