2024/01/16 by Aghaei, Fatemeh, Alikhani, Saeid
#05C99 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2401.08528
Let G=(V,E) be a simple graph. A function f:V→ ℕ∪ \0\ is called a configuration of pebbles on the vertices of G and the quantity \vert f\vert=∑u∈ Vf(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 π(G) equals the minimum number k such that every pebbling configuration f with \vert f\vert = k is solvable. Let G be a connected graph constructed from pairwise disjoint connected graphs G1,...,Gk by selecting a vertex of G1 , a vertex of G2 , 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 G1,...,Gk . In this paper, we study the pebbling number of some polymers.