**Journal of Graph Theory**

# Pebbling Graphs of Fixed Diameter

## Abstract

Given a configuration of indistinguishable pebbles on the vertices of a connected graph *G* on *n* vertices, a *pebbling move* is defined as the removal of two pebbles from some vertex, and the placement of one pebble on an adjacent vertex. The *m**-pebbling number* of a graph *G*, , is the smallest integer *k* such that for each vertex *v* and each configuration of *k* pebbles on *G* there is a sequence of pebbling moves that places at least *m* pebbles on *v*. When , it is simply called the pebbling number of a graph. We prove that if *G* is a graph of diameter *d* and are integers, then , where denotes the size of the smallest distance *k* dominating set, that is the smallest subset of vertices such that every vertex is at most distance *k* from the set, and, . This generalizes the work of Chan and Godbole (Discrete Math 208 (2008), 15–23) who proved this formula for . As a corollary, we prove that . Furthermore, we prove that if *d* is odd, then , which in the case of answers for odd *d*, up to a constant additive factor, a question of Bukh (J Graph Theory 52 (2006), 353–357) about the best possible bound on the pebbling number of a graph with respect to its diameter.