The Crossing Number Of The Join Product Of Small Order Graph With NK₁ Path And Cycle

The crossing number of a graph is a significant notion, originated from Modern Graph Theory, which starts at the fifties of the 19th century, and it is difficulty in the forefront of Topological Graph Theory. The crossing number of a graph mainly studies the minimum crossing number of the graph’s best drawing on he plane or curved surface, it’s a characterization of a graph of non-plane of an important parameter. Due to the use of the design of electronic circuits and the graphical representation of DNA in biology engineer and so on, many professors and scholars home and aborad have engaged in the investigation in this field. But Garey and Johnson have proved that in general the problem to obtain the crossing number of a graphs is NP-complete. Therefore, few achievements of the studies about the crossing number of graphs we’ve had for now. But there are quite some conclusions of the crossing number of some special graphs and single graphs we have obtained.In this paper, we have got the crossing number of the join graph of a disconnected graph on six vertices with path and cycle.and the crossing number of the join graph of a special graph on six vertices with path and cycle.One. We introduce the origins and backgrounds of the crossing number, and the structure of this paper.Two. Introduce some basic concepts and some lemma and properties, which associated with the graph.Three. On the crossing number of the join graph of a disconnected graph on six vertices with path and cycle.Four. On the crossing number of the join graph of a special graph on six vertices with path and cycle.Five. Summarize our researches and look forward to the work in future.