  Graph pebbling is the study of moving discrete pebbles from certain initial
distributions on the vertices of a graph to various target distributions via
pebbling moves. A pebbling move removes two pebbles from a vertex and places
one pebble on one of its neighbors (losing the other as a toll). For t >= 1 the
t-pebbling number of a graph is the minimum number of pebbles necessary so that
from any initial distribution of them it is possible to move t pebbles to any
vertex. We provide the best possible upper bound on the t-pebbling number of a
diameter two graph, proving a conjecture of Curtis, et al., in the process. We
also give a linear time (in the number of edges) algorithm to t-pebble such
graphs, as well as a quartic time (in the number of vertices) algorithm to
compute the pebbling number of such graphs, improving the best known result of
Bekmetjev and Cusack. Furthermore, we show that, for complete graphs, cycles,
trees, and cubes, we can allow the target to be any distribution of t pebbles
without increasing the corresponding t-pebbling numbers; we conjecture that
this behavior holds for all graphs. Finally, we explore fractional and optimal
fractional versions of pebbling, proving the fractional pebbling number
conjecture of Hurlbert and using linear optimization to reveal results on the
optimal fractional pebbling number of vertex-transitive graphs.
