2025/01/06 by Dehqan, Juma Gul, Alikhani, Saeid, Khalafi, Ali Delavar +1
#05C30 #05C69 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2501.03004
Let G=(V,E) be a simple graph. A pebbling configuration on G is a function f:V→ ℕ∪ \0\ that assigns a non-negative integer number of pebbles to each vertex. The weight of a configuration f is w(f)=∑u∈ Vf(u), the total number of pebbles. A pebbling move consists of removing two pebbles from a vertex u and placing one pebble on an adjacent vertex v. A configuration f is a t-restricted pebbling configuration (tRPC) if no vertex has more than t pebbles. The t-restricted optimal pebbling number πt^*(G) is the minimum weight of a tRPC on G that allows any vertex to be reached by a sequence of pebbling moves. The distinguishing number D(G) is the minimum number of colors needed to label the vertices of G such that the only automorphism preserving the coloring is the trivial one (i.e., the identity map). In this paper, we investigate the 2-restricted optimal pebbling number of trees T with D(T)=2 and radius at most 2 and enumerate their 2-restricted optimal pebbling configurations. Also we study the 2-restricted optimal pebbling number of some graphs that are of importance in chemistry such as some alkanes.