  Given a configuration of pebbles on the vertices of a connected graph $G$, a
\emph{pebbling move} removes two pebbles from some vertex and places one pebble
on an adjacent vertex. The \emph{pebbling number} of a graph $G$ is the
smallest integer $k$ such that for each vertex $v$ and each configuration of
$k$ pebbles on $G$ there is a sequence of pebbling moves that places at least
one pebble on $v$.
  First, we improve on results of Hurlbert, who introduced a linear
optimization technique for graph pebbling. In particular, we use a different
set of weight functions, based on graphs more general than trees. We apply this
new idea to some graphs from Hurlbert's paper to give improved bounds on their
pebbling numbers.
  Second, we investigate the structure of Class 0 graphs with few edges. We
show that every $n$-vertex Class 0 graph has at least $\frac53n - \frac{11}3$
edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we
strengthen this lower bound to $2n - 5$, which is best possible. Further, we
characterize the graphs where the bound holds with equality and extend the
argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.
