On The T-pebbling Number Of Some Graphs

Given a distribution of pebbles on the vertices of a simple connected graph G, a pebbling move on G consists of taking two pebbles off one vertex and placing one on an adjacent vertex. The pebbling number f(G) of G is the smallest positive integer such that every distribution of f(G) pebbles on the vertices of G, we can move one pebble to any target vertex by a sequence of pebbling moves. Similarly, the t-pebbling number ft(G) of G is the smallest positive integer such that every distribution of ft(G) pebbles on the vertices of G, we can move t pebbles to any target vertex by a sequence of pebbling moves. Graham conjectured that for any connected graphs G and H, f(G×H)<f(G)f(H). Lourdusamy further conjectured that ft(G×H)≤ft(G)f(H) for any positive integer t. Herscovici further conjectured that fst(×H)≤fs(G)ft(H) for any positive integers s and t.In this thesis, we obtain the following results:1. Determine the pebbling number of the thorn graph of C5, and show that the thorn graph of C5has the2-pebbling property.2. Determine the t-pebbling number of the generalized friendship graph Ft1,t2...,tk;n, and show that Ft1,t2...,tk;n, has the2t-pebbling property and Lourdusamy’s conjecture is true when G is a generalized friendship graph and H is a graph having2t-pebbling property.3. Determine the t-pebbling number of the Kn-k, a graph obtained from Kn by deleting k in-dependent edges, and show that Kn-k has the2t-pebbling property and Lourdusamy’s conjecture is true when G=Kn-k and H is a graph having the2t-pebbling property.4. Determine the t-pebbling number of the thorn graph of Kn, and show that the thorn graph of Kn has the2t-pebbling property and Herscovici’s conjecture is true when G is a thorn graph of Kn and H is a graph having2t-pebbling property.