T-pebbling number and 2t-pebbling property for book graphs

A pebbling step takes two pebbles from one vertex and places one pebble at a neighbouring vertex if certain pebbles are placed on the vertices of a connected graph G. The pebbling number, f(G), of a graph G is the least positive integer m such that we can transport a pebble to the destination using a series of pebbling motions, regardless of how many pebbles are dispersed on the vertices of G. The t-pebbling number, ƒ_t(G), is the smallest positive integer such that from each placement of ƒ_t(G) pebbles, t pebbles can be moved to any target vertex by a sequence of pebbling moves. If 2t pebbles may be moved to a designated vertex when the total initial number of pebbles is 2ƒ_t(G) – q+1, where q is the number of vertices with at least one pebble, then the graph G satisfies the 2t-pebbling property. We determine ƒ(G), ƒ_t(G) for Book graphs in this article, and show that the graph B_m satisfies the 2-pebbling and 2t-pebbling conditions.