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