The Effect of Arbitrary Connections on The Chromatic Number of A Graph

For graph operations we mainly have Join,Cartesian Product,Tensor Product,Strong Product,Lexicographic Product.For the join of two graphs, the chromatic number is the sum of the chromatic numbers of the two original graphs, but the connections between graphs are usually not complete. This paper focuses on the impact of arbitrary connections on the chromatic number of graphs; we can say it is the ``arbitrary connections'' of two graphs, especially complete graphs, and establishes some formal necessary and sufficient conditions regarding the chromatic number and the clique number.