Sum of squares of degrees in a graph

Let $\text{G}(v,e)$ be the set of all simple graphs with $v$ vertices and $e$ edges and let $P_2(G)=\sum d_i^2$ denote the sum of the squares of the degrees, $d_1,

…, d_v$,oftheverticesof$G$. It is known that the maximum value of $P_2(G)$ for $G\in \text{G}(v,e)$ occurs at one or both of two special graphs in $\text{G}(v,e)$–the \qs graph or the \qc graph. For each pair $(v,e)$, we determine which of these two graphs has the larger value of $P_2(G)$. We also determine all pairs $(v,e)$ for which the values of $P_2(G)$ are the same for the \qs and the \qc graph. In addition to the \qs and \qc graphs, we find all other graphs in $\text{G}(v,e)$ for which the maximum value of $P_2(G)$ is attained. Density questions posed by previous authors are examined.