Weighted pebbling numbers on graphs
  We expand the theory of pebbling to graphs with weighted edges. In a weighted pebbling game, one player distributes a set amount of weight on the edges of a graph and his opponent chooses a target vertex and places a configuration of pebbles on the vertices. Player one wins if, through a series of pebbling moves, he can move at least one pebble to the target. A pebbling move of p pebbles across an edge with weight w leaves the floor of pw pebbles on the next vertex. We find the weighted pebbling numbers of stars, graphs with at least 2|V|-1 edges, and trees with given targets. We give an explicit formula for the minimum total weight required on the edges of a length-2 path, solvable with p pebbles and exhibit a graph which requires an edge with weight 1/3 in order to achieve its weighted pebbling number. 
Stephanie Jones, Joshua D. Laison, Cameron McLeman, and Kathryn Nyman
math.CO