  We study both function theoretic and spectral properties of the weighted
Laplacian $\Delta_f$ on complete smooth metric measure space $(M,g,e^{-f}dv)$
with its Bakry-\'{E}mery curvature $Ric_f$ bounded from below by a constant. In
particular, we establish a gradient estimate for positive $f-$harmonic
functions and a sharp upper bound of the bottom spectrum of $\Delta_f$ in terms
of the lower bound of $Ric_{f}$ and the linear growth rate of $f.$ We also
address the rigidity issue when the bottom spectrum achieves its optimal upper
bound under a slightly stronger assumption that the gradient of $f$ is bounded.
Applications to the study of the geometry and topology of gradient Ricci
solitons are also considered. Among other things, it is shown that the volume
of a noncompact shrinking Ricci soliton must be of at least linear growth. It
is also shown that a nontrivial expanding Ricci soliton must be connected at
infinity provided its scalar curvature satisfies a suitable lower bound.
