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

Let G(v, e) be the set of all simple graphs with v vertices and e edges and let P2(G) = ∑ di denote the sum of the squares of the degrees, d1, . . . , dv, of the vertices of G. It is known that the maximum value of P2(G) for G ∈ G(v, e) occurs at one or both of two special graphs in G(v, e)—the quasi-star graph or the quasi-complete graph. For each pair (v, e), we determine which of these two graphs has the larger value of P2(G). We also determine all pairs (v, e) for which the values of P2(G) are the same for the quasi-star and the quasi-complete graph. In addition to the quasi-star and quasi-complete graphs, we find all other graphs in G(v, e) for which the maximum value of P2(G) is attained. Density questions posed by previous authors are examined. AMS Subject Classification: 05C07, 05C35