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=Pebbling can be viewed as a model of resource transportation on 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 $t$-pebbling number of a graph $G$ is the minimum number of pebbles so that we can move $t$ pebbles on each vertex of $G$ regardless of the original distribution of pebbles. Let $\omega$ be a positive function on $V(G)$, the $\omega$-cover pebbling number of a graph $G$ is the minimum number of pebbles so that we can reach a distribution with at least $\omega(v)$ pebbles on $v$ for all $v\in V(G)$. In this paper, we give the $\omega$-cover pebbling number of trees for nonnegative function $\omega$, which generalized the $t$-pebbling number and the traditional weighted cover pebbling number of trees.