AUTHOR=Xia Zheng-Jiang , Hong Zhen-Mu 

TITLE=Generalization of the Cover Pebbling Number for Networks

JOURNAL=Frontiers in Physics

VOLUME=Volume 8 - 2020

YEAR=2020

URL=https://www.frontiersin.org/journals/physics/articles/10.3389/fphy.2020.00197

DOI=10.3389/fphy.2020.00197

ISSN=2296-424X

ABSTRACT=<p>Pebbling can be viewed as a model of resource transportation for networks. We use a graph to denote the network. A pebbling move on a graph consists of the removal of two pebbles from a vertex and the placement of one pebble on an adjacent vertex. The <italic>t</italic>-pebbling number of a graph <italic>G</italic> is the minimum number of pebbles so that we can move <italic>t</italic> pebbles on each vertex of <italic>G</italic> regardless of the original distribution of pebbles. Let ω be a positive function on <italic>V</italic>(<italic>G</italic>); the ω-cover pebbling number of a graph <italic>G</italic> is the minimum number of pebbles so that we can reach a distribution with at least ω(<italic>v</italic>) pebbles on <italic>v</italic> for all <italic>v</italic> ∈ <italic>V</italic>(<italic>G</italic>). In this paper, we give the ω-cover pebbling number of trees for nonnegative function ω, which generalized the <italic>t</italic>-pebbling number and the traditional weighted cover pebbling number of trees.</p>