Pebbling Number of Polymers

Let G = (V, E) be a simple graph. A function f : V -> N U {0} is called a configuration of pebbles on the vertices of G and the quantity |f| = Sigma(u is an element of 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 |f| = k is solvable. Let G be a connected graph constructed from pairwise disjoint connected graphs G1, ..., 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. (c) 2025 University of Kashan Press. All rights reserved.