An upper bound on the sum of squares of degrees in a graph

Let G be a simple graph with n vertices, e edges and vertex degrees d1, d2, …, dn. It is proved that d12 + … + dn2 ⩽ e(2e/(n − 1) + n − 2) when n ⩾ 2. This bound does not generalize to all sequences of positive integers. A comparison is made to another upper bound on d12 + … + dn2, due to Székely et al. (1992). Our inequality follows from the positive semidefiniteness of a certain quadratic form in (n2) variables. We also apply the inequality to bounding the total number of triangles in a graph and its complement.