Corresponding to each group Γ, a mixed graph G = (Γ,E,E′) called C-graph is assigned, such that the vertex set of G is the group itself. Two types of adjacency relations, that is, one way and two way communication is defined for vertices, to get a clear idea of the underlying group structure. An effort to answer the question, ‘Is there any relation between the order of an element in the group and degrees of the corresponding vertex in the C-graph’, by proposing a mathematical formula connecting them is made. Established an upper bound for the total number of edges in a C-graph G. For a vertex z in G, the concept Connector Edge CEz is defined, which convey some structural properties of the group Γ. The Connector Edge Set is defined for both a vertex z and the whole C-graph G, and is denoted as C E z and C E G respectively. Proposed the result, C E G = E if and only if |Γ| = 2n, n ∈ N. Finally, the properties of G, which the Connector Edge Set C E G carry out are discussed.