A graph pebbling algorithm on weighted graphs

TL;DR

This paper introduces an algorithm to compute the pebbling number of weighted graphs, extending pebbling theory to weighted cases and applying it to small connected graphs.

Contribution

The paper presents a novel algorithm for calculating pebbling numbers in weighted graphs and applies it to all connected graphs with up to nine vertices.

Findings

Algorithm successfully computes pebbling numbers for weighted graphs.

Determined regular pebbling numbers for all small connected graphs.

Provides a method to simplify complex pebbling problems.

Abstract

A pebbling move on a weighted graph removes some pebbles at a vertex and adds one pebble at an adjacent vertex. The number of pebbles removed is the weight of the edge connecting the vertices. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The pebbling number of a weighted graph is the smallest number $m$ needed to guarantee that any vertex is reachable from any pebble distribution of $m$ pebbles. Regular pebbling problems on unweighted graphs are special cases when the weight on every edge is 2. A regular pebbling problem often simplifies to a pebbling problem on a simpler weighted graph. We present an algorithm to find the pebbling number of weighted graphs. We use this algorithm together with graph simplifications to find the regular pebbling number of all connected graphs with at most nine vertices.