Common Neighborhood Graph

Let G be a simple graph with vertex set {v1, v2, … , vn}. The common neighborhood graph of G, denoted by con(G), is a graph with vertex set {v1, v2, … , vn}, 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 con(G). Also, we obtain a lower bound for the clique number of con(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 con(T(G)).