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Abstract: We investigate generalizations of pebbling numbers and of Graham’s pebbling conjecture that , where is the pebbling number of the graph . We develop new machinery to attack the conjecture, which is now twenty years old. We show that certain conjectures imply others that initially appear stronger. We also find counterexamples that shows that Sjöstrand’s theorem on cover pebbling does not apply if we allow the cost of transferring a pebble from one vertex to an adjacent vertex to depend on the weight of the edge and we describe an alternate pebbling number for which Graham’s conjecture is demonstrably false. [Copyright &y& Elsevier]

Abstract: A pebbling move on a connected graph consists of removing two pebbles from some vertex and adding one pebble to an adjacent vertex. We define as the smallest number such that whenever pebbles are on , we can move pebbles to any specified, but arbitrary vertex. Graham conjectured that for any connected and . We define the -pebbling number and prove that when none of the cycles is , and satisfies one more criterion. We also apply this result with by showing that satisfies Chung’s two-pebbling property, and establishing bounds for . [Copyright &y& Elsevier]