Maximizing 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,...,d_n. Also, let d₁, d _n be, respectively, the highest degree and the lowest degree of G and m_i be the average of the degrees of the vertices adjacent to vertex v_i∈V. It is proved thatmax{d_j+m_j:v _j∈V}≤2en-1+n-2with equality if and only if G is an S _n graph (K_1,n-1⊆S_n⊆K_n) or a complete graph of order n-1 with one isolated vertex. Using the above result we establish the following upper bound for the sum of the squares of the degrees of a graph G:∑i=1nd_i²≤e2en-1+n-2n-1d ₁+(d₁-d_n)1-d₁n-1with equality if and only if G is a star graph or a regular graph or a complete graph K _d1+1 with n-d₁-1 isolated vertices. A comparison is made to another upper bound on ∑_i=1ⁿd_i ², due to de Caen (Discrete Math. 185 (1998) 245). We also present several upper bounds for ∑_i=1ⁿd_i ² and determine the extremal graphs which achieve the bounds.