SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

Let G = (V, E) be a simple graph with n vertices, e edges, and vertex degrees d1, d2, . . . , dn. Let d1, dn be the highest and the lowest degree of vertices of G and mi be the average of the degrees of the vertices adjacent to vi ∈ V . We prove that n ∑ i=1 di = e [ 2e n− 1 + n− 2 ] 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: n ∑ i=1 di ≤ e [ 2e n− 1 + (n− 2) ] − d1 [ 4e n− 1 − 2m1 − (n + 1) (n− 1)1 + (n− 1) ] , 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 ∑n i=1 d 2 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.