For a graph $G$ whose degree sequence is $d_{ 1},..., d_{ n}$, and for a positive integer $p$, let $e_{ p}(G)=\sum_{ i=1}^{ n}d_{ i}^{ p}$. For a fixed graph $H$, let $t_{ p}(n,H)$ denote the maximum value of $e_{ p}(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $t_{ 1}(n,H)$ is twice the Tur\'{ a}n number of $H$. In this paper we consider the case $p>1$. For some graphs $H$ we obtain exact results, for some others we can obtain asymptotically tight upper and lower bounds, and many interesting cases remain open.
Categories: math.CO
