The t-pebbling number and 2t-pebbling property of rooted product of some graphs
Description
Graph pebbling is a network optimization model for the transportation of resources during the transit. Given a distribution of pebbles on the vertices of a connected graph $G$, the pebbling move consists of removal of two pebbles from a vertex and placing one pebble on one of its adjacent vertices, discarding the other pebble. The loss of one pebble in a pebbling move can be related to the packet loss in message transmission between the computers. The generalization of packet loss in this process is defined as $t$-pebbling. The $t$-pebbling number $f_t(G)$ is the smallest positive integer such that, for every distribution of $f_t(G)$ pebbles, $t$ pebbles can be moved to a vertex $v$, for every $v$ in $G$ by a sequence of pebbling moves. A graph $G$ is said to satisfy a $2t$-pebbling property if $2t$ pebbles can be moved to $v$ when the total starting number of pebbles is $2f_t(G)-q+1$, where $q$ is the number of vertices with at least one pebble. In this paper a lower bound for rooted product of two graphs G and H is determined and the bound is proved to be sharp when one of the graphs in the rooted product is a path, complete graph or a star. Furthermore, this paper explores some new results on pebbling in triangle free graphs. In particular, $2t$- pebbling property is verified for rooted product whenever one of the graph is a Triangle free graph.