The Crossing Numbers Of The Joins Of Special Graph With N Isolated Vertices,Path And Cycle

The crossing number is gradually developed in the modern graph theory.It is a subject which mainly study how to draw the graph in a plane when the crossing number is at the least.Pure mathematical methods are often used.However,to determine the crossing numbers of graphs is proved to be an NP-complete problem.Therefore,there are few results of the crossing number except for some special and simple graphs.In recent years,more and more scholars begin to do research on the crossing number of the join graph of special graph with n isolated vertices,path and cycle.This paper used combination method,inductive inference,exclusive method and proof by contradiction to determine the crossing numbers of join of the spe-cial graph on six vertices with n isolated vertices,path and cycle,along with studying the crossing numbers of join of a five-order unconnected graph with path and cycle.In chapter one.We provide detailed information on research backgrounds,the significance,the research tendency of the crossing number,and the structure of this paper.In chapter two.Preliminary Knowledge,including some basic conceptions,properties and some lemmas which associated with the graph.In chapter three.We obtain the crossing numbers of join of a special graph on six vertices with n isolated vertices?path and cycle,respectively.In chapter four.The crossing numbers of join of a special five-order un-connected graph with path and cycle are determined.In chapter five.Summarize our researches and look into the future.