Generalization of the Cover Pebbling Number for Networks - Xia, Zheng-Jiang; Hong, Zhen-Mu (Frontiers in Physics, 2020)
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 $ω$ be a positive function on $V(G)$, the $ω$-cover pebbling number of a graph $G$ is the minimum number of pebbles so that we can reach a distribution with at least $ω(v)$ pebbles on $v$ for all $v\in V(G)$. In this paper, we give the $ω$-cover pebbling number of trees for nonnegative function $ω$, which generalized the $t$-pebbling number and the traditional weighted cover pebbling number of trees.