Vertex capacity in a graph structure

In a graph structure G = (V, E1, E2… Ek), the capacity of a vertex c(v) is the number of different Ei edges incident at v. In this paper, we have made an attempt to answer the problem of examining the graph structures for which sum of vertex capacities is equal to sum of number of relations and the number of edges in the structure for some families of graphs viewed as graph structures.