  Let $\widetilde{X}$ be a locally finite complete Gromov hyperbolic metric
graph with the geometric boundary consisting of infinitely many points. Suppose
that there is a discrete subgroup of the isometry group $Iso(\widetilde{X})$
acting geometrically on $\widetilde{X}$. The $\lambda$-Martin boundary is the
boundary of the image of an embedding from $\widetilde{X}$ to the space of
$\lambda$-superharmonic functions.
  We show that the $\lambda$-Martin boundary coincides with the geometric
boundary for any $\lambda \in [0, \lambda_0],$ in particular at the bottom of
the spectrum $\lambda_0$.
