in the propositions that followwe explore few important properties of graphs proposition if is graph with edges and vertex set then deg( in justificationan edge (uvis 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 edges and vertex set then indeg(voutdeg(vm in in justificationin directed graphan edge (uvcontributes one unit to the out-degree 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 in-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 < ( justificationsuppose that is undirected since no two edges can have the same endpoints and there are no self-loopsthe maximum degree of vertex in is in this case thusby proposition < ( now suppose that is directed since no two edges can have the same origin and destinationand there are no self-loopsthe maximum in-degree of vertex in is in this case thusby proposition < ( there are number of simple properties of treesforestsand connected graphs proposition let be an undirected graph with vertices and edges if is connectedthen > if is treethen if is forestthen <