The Pebbling Number And2-pebbling Property Of Some Graphs

In recent years, the pebbling number of graphs have become a newhot topic in the graph theory. They have deeply attracted great interest of manyscholars. Their theoretical results can be widely used in goods distribution, computer,communication networks and other related fields, which with broad prospects. Thepebbling number of graph G, denoted by f(G), is the least positive integer n suchthat any distribution of n pebbles on G allows one pebble to be moved to any specifiedvertex by a sequence of pebbling moves. A pebbling move is taking two pebbles ofone vertex and then placing one on an adjacent vertex. Given a pebbling of G, let pbe the number of pebbles, q be the number of vertices with at least one pebble. Wesay that G satisfies the two-pebbling property, if it is possible to move two pebbles toany specified target vertex whenever p and q satisfy the inequality p+q>2f(G).Graham conjectured that for any connected graphs G and H, f(G×H)≤f(G)f(H).This paper mainly studies the pebbling number and the2-pebbling property ofgraphs. Firstly, the background and the development of the graph pebbling, andthe research contents of this paper are introduced simply. Then we introduce thepebbling number and2-pebbling property of graph. At last, the pebbling numberand2-pebbling property of graph G Pkare presented. The main achievements arelisted as follows:(1)The pebbling number and2-pebbling property of multi-fan graph,and Graham’s conjecture holds when G and H are multi-fan graphs.(2)The pebblingnumber and2-pebbling property of graph Fn Pk.(3)The pebbling number of graphWn Pkand double-wheel graph.