Research On Pebbling Number Of The Graph D_n,G

Graph theory originated in the eighteenth Century, is a relatively old subject. It has been rapidly developed and widely applied since the twentieth Century. Graph theory is a branch of a mathematical involving various fields such as chemistry, physics, network theory, computer science and control theory and so on. Pebbling problem of a graph was first introduced in article by Chung in 1989. In the past twenty years, the pebbling number has been as popular problems in graph theory attracting many attentions of mathematicians. Simultaneously, the pebbling problem has also been applied to the fields of transportation resources, Network optimization configuration etc.A pebbling move on a simple connected graph refers to:Given a distribution D of pebbles on the vertices of a graph G, a pebbling move on G consists of taking two pebbles off from a given vertex and placing one of them onto an adjacent vertex (the other one is discarded). The pebbling number of a graph, denoted by f(G), is the minimal integer such that any distribution of f(G) pebbles on G allows one pebble to be moved to any specified vertex by a sequence of pebbling moves. Pebbling move can be understood as a transport model. The discarded pebble can be seen as resource depletion or the cost of transportation, the pebble be add to the adjacent vertex can be regarded as the goods need to be transported.This paper mainly studies on the pebbling number of graph Dn,G. The graph Dn,G refers to the union of graph G with only one common vertex. The main results are as follows:1) The background, basic concepts and conclusions of the graph pebbling are introduced. Based on the pigeonhole principle, we obtain a sufficient condition of pebbling move on the graph Dn,Cm.2) we calculate the pebbling number of the odd-cycle-union graph DnC2m+1; generalize and proof a lemma derived from the pigeonhole principle3) The t-pebbling number of the graph Dn,C2m is calculated. The q-t-pebbling number of even cycle is verified, in order to show that the graph Dn,C2 has 2t-pebbling property.