5 Conclusions

In this paper we are computed the common neighborhood of some product graphs such as Cartesian product, join, composition and corona product. Also we obtained the common neighborhood graph of the splice and link of two graphs according to their common neighborhood graphs. In continue computed the common neighborhood graph of subdivision graph, total graph and two extra subdivision-related graphs that named R(G) and Q(G). Next the relation between hamiltonicity of graph G and con(G) has been investigated. Also we gave a lower bound for the clique number of con(G) in terms of clique number of graph G. Finally it is stated that total chromatic number of graph G is bounded by chromatic number of con(T(G)).