$2$-Restricted Optimal Pebbling Number of Some Graphs

TL;DR

The paper addresses the computation of the $2$-restricted optimal pebbling number $\pi_2^*(G)$ for graphs of interest, focusing on trees with distinguishing number $D(T)=2$ and radius at most $2$, and on alkane graphs modeled as molecule trees. It formalizes the $2$-restricted pebbling framework and enumerates $2$-restricted optimal pebbling configurations ($P_2^*$) for a structured family of trees, establishing that $2\le\pi_2^*(T)\le 6$ and $D(T)\le\pi_2^*(T)$ in this class, while cataloging the number of optimal configurations across many instances. The study extends to chemical graph models, computing exact $\pi_2^*$ and $P_2^*$ for methane ($CH_4$), ethane ($C_2H_6$), propane ($C_3H_8$), butane ($C_4H_{10}$), and several isomers of pentane ($C_5H_{12}$), illustrating how pebbling requirements scale with molecular size and topology. The results connect graph pebbling concepts with chemical graph theory and raise open questions about broader molecule classes and how $\pi_2^*(G)$ can be bounded via the distinguishing number $D(G)$.

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 $π_t^*(G)$ 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.

Figures (11)

• Figure 1: 2-restricted optimal pebbling number.
• Figure 2: The $2$-restricted optimal pebbling configurations of some trees $T\in \mathcal{T}$.
• Figure 3: The unique $2$-restricted optimal pebbling configuration of graph $G$
• Figure 4: Unique $2$-restricted optimal pebbling configuration of $CH_4$
• Figure 5: Two 2-restricted optimal pebbling configurations of $C_2H_6$

Theorems & Definitions (21)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Corollary 2.4
• Theorem 2.5
• Corollary 2.6
• Theorem 2.7