The Nature Of Pebbling Number Of Graphs Under Reidemeister Moves

This paper combines knot theory with graph theory to employ the nature of pebbling number of the graph of a knot by the theory of the graph of knots and the relevant knowledge of the pebbling number of graphs.This paper tries to find out how the pebbling number of the graph of a knot changes under Reidemeister moves.The first part of the article highlights some preliminary knowledge related to the content of this article,including the definitions of the universe、graph、path、tree、circle、 complete graph 、Reidemeister moves of knots and graphs、pebbling move、pebbling number of the graph、chromatic polynomial and Tutte polynomial.The second part shows some known knot invariants: Alexander polynomial、Conway polynomial、Jones polynomial and Kauffman polynomial.Then some known conclusions in the pebbling number of the graph are given,such as the relation between pebbling number f（G）of the graph G and the number of vertices （?）is f（G）（?）;The paper gives pebbling numbers of some special graph,including path、 tree with n vertices、circle and complete graph.The third section discusses the nature of the pebbling number of the graph of a knot underR₁-move.The paper concludes the pebbling number of graph is not always equal to the number of vertices and then gives how the pebbling numbers of complete graph、path、tree with n vertices and circle change underR₁-move.Finally,we conclude the pebbling number of the graph of knot is not an invariant by giving some specific examples of knots and links.The fourth section discusses the nature of the pebbling number of the graph of a knot underR₂-move.The paper focuses on tree and circle to find out how the pebbling numbers of these two kinds of graphs change underR₂-move.