pebbling number of polymers

‎let $g=(v,e)$ be a simple graph‎. a function $f:vrightarrow mathbb{n}cup {0}$ is called a configuration of pebbles on the vertices of $g$ and the quantity $vert fvert=sum_{uin v}f(u)$‎ ‎is called the weight of $f$ which is just the total number of pebbles assigned to vertices‎. ‎a pebbling step from a vertex $u$ to one of its‎ neighbors $v$ reduces $f(u)$ by two and increases $f(v)$ by one‎. ‎a pebbling configuration $f$ is said to be solvable if for every vertex $ v $‎, ‎there exists a sequence (possibly empty) of pebbling moves that results in a pebble on $v$‎. ‎the pebbling number $ pi(g) $ equals the minimum number $ k $ such that every pebbling configuration $ f $ with $ vert fvert = k $ is solvable‎. let $ g $ be a connected graph constructed from pairwise disjoint connected graphs $ g_1,...,g_k $ by selecting a vertex of $ g_1 $‎, ‎a vertex of $ g_2 $‎, ‎and identifying these two vertices‎. ‎then continue in this manner inductively‎. ‎we say that $ g $ is a polymer graph‎, ‎obtained by point-attaching from monomer units $ g_1,...,g_k $‎. in this paper‎, ‎we study the pebbling number of some polymers‎. ‎