On the common neighborhood graphs

Let G be a simple graph with vertex set { v 1 , v 2 , … , v n } . The common neighborhood graph (congraph) of G, denoted by c o n ( G ) , is a graph with vertex set { v 1 , v 2 , … , v n } , in which two vertices are adjacent if and only if they have at least one common neighbor in the graph G. In this paper we compute the common neighborhood of some composite graphs. In continue we investigate the relation between hamiltonicity of graph G and c o n ( G ) . Also we obtain a lower bound for the clique number of c o n ( G ) in terms of clique number of graph G. Finally we state that the total chromatic number of G is bounded by chromatic number of c o n ( T ( G ) ) .