Bounds on sum of squares of node degrees in undirected graphs
It's $\Theta(m^2)$ for multigraphs (consider a graph with $m$ edges between the same pair of nodes and all other nodes isolated) and $\Theta(mn)$ for simple graphs (the sum of squared degrees equals the number of directed two-edge walks; each edge can be continued to form a two-edge walk in $O(n)$ ways; consider $K_{m/n,n}$).