Abstract. Let G=(V,E) be a simple graph with n vertices, e edges, and vertex degrees d₁,d₂,¼,d_n. Let d₁, d_n be the highest and the lowest degree of vertices of G and m_i be the average of the degrees of the vertices adjacent to v_i Î V. We prove that

if and only if G is a star graph or a complete graph or a complete graph with one isolated vertex. We establish the following upper bound for the sum of the squares of the degrees of a graph G:

with equality if and only if G is a star graph or a complete graph or a graph of isolated vertices. Moreover, we present several upper and lower bounds for åⁿ_i=1d²_i and determine the extremal graphs which achieve the bounds and apply the inequalities to obtain bounds on the total number of triangles in a graph and its complement.