  Given a configuration of pebbles on the vertices of a graph $G$, a pebbling
move removes two pebbles from a vertex and puts one pebble on an adjacent
vertex. The pebbling number of a graph $G$ is the smallest number of pebbles
required such that, given an arbitrary initial configuration of pebbles, one
pebble can be moved to any vertex of $G$ through some sequence of pebbling
moves. Through constructing a non-tree weight function for $Q_4$, we improve
the weight function technique, introduced by Hurlbert and extended by Cranston
et al., that gives an upper bound for the pebbling number of graphs. Then, we
propose a conjecture on weight functions for the $n$-dimensional cube. We also
construct a set of valid weight functions for variations of lollipop graphs,
extending previously known constructions.
