Research On The Pebbling Number Of Graphs

In recent years, the graph pebbling has become a hot topic in the graph theory. Within the past twenty years, it has been attracting the mathematicians deeply. It is closely related to number theory and it can deal with some unresolved problems on number theory.The pebbling number of a graph G , f(G), is the least m such that, nomatter how m pebbles are placed on the vertices of G, one pebble can be moved to any vertex by a sequence of pebbling moves, each move taking two pebbles off one vertex and placing one on an adjacent vertex. Graham conjectured that for all connected graphsGand H,f(G×H)≤ f(G)f(H).This paper studies the pebbling of graphs. The research background of the graph pebbling is introduced; the development of the pebbling number of the graph and the 2-pebbling property of graph are presented. Based on the previous studies, this paper mainly focuses on the pebbling number of thorn graphs and middle graphs. The main investigation is the pebbling number of the thorn graph of complete and the middle graph of star. In this paper, we computer the pebbling number of these graphs and show that Graham's conjecture holds when G is a thorn graph of complete or a middle graph of star and H satifies 2-pebbling property. At last, we also obtain the bound on the pebbling number of a Cartesian product of thorn graphs of complete and number of middle graphs of star.