proposition if is graph with edgesthen justificationan edge ( ,vis counted twice in the summation aboveonce by its endpoint and once by its endpoint thusthe total contribution of the edges to the degrees of the vertices is twice the number of edges proposition if is directed graph with edgesthen justificationin directed graphan edge ( ,vcontributes one unit to the outdegree of its origin and one unit to the in-degree of its destination thusthe total contribution of the edges to the out-degrees of the vertices is equal to the number of edgesand similarly for the out-degrees we next show that simple graph with vertices has ( edges proposition let be simple graph with vertices and edges if is undirectedthen < ( )/ and if is directedthen < (