On degree sums of a triangle-free graph

For a simple triangle-free kk-chromatic graph GG with k≥2k≥2 the upper bound m(n−f(k−2))m(n−f(k−2)) on the sum Σ2(G)=∑x∈V(G)d2(x)Σ2(G)=∑x∈V(G)d2(x) of the squares of the degrees of GG is proved, where nn, mm, and f(l)f(l) are the order of GG, the size of GG, and the minimum order of a triangle-free ll-chromatic graph, respectively. Consequences of this bound are discussed.Moreover, we generalize the upper bound on Σp(G)=∑x∈V(G)dp(x)Σp(G)=∑x∈V(G)dp(x) for p=2p=2 to p≥3p≥3.