$2$-Restricted Optimal Pebbling Number of Some Graphs

TL;DR

This paper studies the minimum number of pebbles needed to reach any vertex in certain graphs with restrictions on pebbles per vertex, focusing on trees with specific symmetry and radius, and some chemical graphs.

Contribution

It characterizes the 2-restricted optimal pebbling number for trees with automorphism number 2 and radius at most 2, and explores pebbling configurations in chemical graphs.

Findings

Determined the 2-restricted optimal pebbling number for specific trees.

Enumerated all optimal pebbling configurations for these trees.

Analyzed pebbling numbers for chemical graphs like alkanes.

Abstract

Let $G=(V,E)$ be a simple graph. A pebbling configuration on $G$ is a function $f:V\rightarrow \mathbb{N}\cup \{0\}$ that assigns a non-negative integer number of pebbles to each vertex. The weight of a configuration $f$ is $w(f)=\sum_{u\in V}f(u)$, 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 ($t$RPC) if no vertex has more than $t$ pebbles. The $t$-restricted optimal pebbling number $\pi_t^*(G)$ is the minimum weight of a $t$RPC 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.