中文摘要：

图上的一个pebbling移动,是从图的一个顶点同时移除2个pebbles,并且在其某个邻点上放置1个pebble.图的优化t-pebbling数,记为f′t（G）,是指图G中所需要的pebbled的最小数目,使得存在该f′t（G）个pebbles在图上的一种分布,可以在经过一系列pebbling移动后,t个pebbles可以移动到任意一个给定的目标顶点上.f′（G）=f′1（G）称为图G的优化pebbling数.这里给出了路Pn和圈C5的优化t-pebbling数,证明了f′9t（P2×P3）=20t;f′9t＋1（P2×P3）=20t＋3;当2≤r≤8时,20t＋2r＋1≤f′9t＋r（P2×P3）≤20t＋2r＋2,其中,当5≤r≤8时,最后一个不等式取到等号.

英文摘要：

A pebbling move removes two pebbles from a vertex and places one pebble on one of its neighbours.For t≥1,the optimal t-pebbling number of a graph G,f′t（G）,is the minimum number of pebbles necessary so that from some initial distribution of them it is possible to move t pebbles to any target vertex by a sequence of pebbling moves.f′（G）=f′1（G）be the optimal pebbling number of G.Here the optimal t-pebbling numbers of the path Pnand the cycle C5 were given,respectively.In the final section,it was obtained that f′9t（P2×P3）=20t,f′9t＋1（P2×P3）=20t＋3,and 20t＋2r＋1≤f′9t＋r（P2×P3）≤20t＋2r＋2,for 2≤r≤8,the last equality holds for r=5,6,7,8.