Optimal pebbling number of graphs with given minimum degree

Consider a distribution of pebbles on a connected graph $G$. A pebbling move removes two pebbles from a vertex and places one to an adjacent vertex. A vertex is reachable under a pebbling distribution if it has a pebble after the application of a sequence of pebbling moves. The optimal pebbling number ${\pi }^{\ast }(G)$ is the smallest number of pebbles which we can distribute in such a way that each vertex is reachable. It was known that the optimal pebbling number of any connected graph is at most $\frac{4n}{\delta +1}$, where $\delta$ is the minimum degree of the graph. We strengthen this bound by showing that equality cannot be attained and that the bound is sharp. If $\mathrm{diam}(G)\ge 3$ then we further improve the bound to ${\pi }^{\ast }(G)\le \frac{3.75n}{\delta +1}$. On the other hand, we show that for arbitrary large diameter and any $ϵ>0$ there are infinitely many graphs whose optimal pebbling number is bigger than $(\frac{8}{3}-ϵ)\frac{n}{(\delta +1)}$.