# The t-pebbling number and 2t-pebbling property of rooted product of some graphs

Published: 11 June 2024| Version 1 | DOI: 10.17632/9ww4zjjd26.1
Contributors:
, Joice Punitha M, DHIVVIYANANDAM I

## 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.

## Categories

Mathematics, Discrete Mathematics