Root Cover Pebbling on Graphs

A graph is a mathematical object that can be described as a set of vertices and a set of edges. An edge joins one vertex to another. The existence or absence of an edge between two vertices can represent a relationship or absence of a relationship between two objects. Two vertices are said to be adjacent if there is an edge that joins them. Imagine placing pebbles on the vertices of a graph. We can move a pebble from one vertex to an adjacent vertex using certain pebbling moves in which two pebbles are removed from a vertex and one is placed on an adjacent vertex while one is removed from the graph entirely. We have defined a concept called root cover pebbling, a variation on cover pebbling which is a well documented concept. In root cover pebbling we begin with all pebbles on one vertex of a certain graph and attempt to place at least one pebble on every vertex of the graph by using pebbling moves. Thus the root cover pebbling number is the least number of pebbles needed to achieve a configuration with at least one pebble on each vertex of a graph when starting from a configuration with all pebbles on the root vertex. We construct an algorithm for calculating root cover pebbling numbers for certain graphs. We also examine graphs with a root vertex and paths attached to it. With these graphs, we explore the relationships between the number of paths in the graph and the root cover pebbling number of the graph.https://ecommons.udayton.edu/stander_posters/1589/thumbnail.jp