On degree powers and counting stars in $F$-free graphs

Given a positive integer $r$ and a graph $G$ with degree sequence $d_1,\dots,d_n$, we define $e_r(G)=\sum_{i=1}^n d_i^r$. We let $\mathrm{ex}_r(n,F)$ be the largest value of $e_r(G)$ if $G$ is an $n$-vertex $F$-free graph. We show that if $F$ has a color-critical edge, then $\mathrm{ex}_r(n,F)=e_r(G)$ for a complete $(\chi(F)-1)$-partite graph $G$ (this was known for cliques and $C_5$). We obtain exact results for several other non-bipartite graphs and also determine $\mathrm{ex}_r(n,C_4)$ for $r\ge 3$. We also give simple proofs of multiple known results. Our key observation is the connection to $\mathrm{ex}(n,S_r,F)$, which is the largest number of copies of $S_r$ in $n$-vertex $F$-free graphs, where $S_r$ is the star with $r$ leaves. We explore this connection and apply methods from the study of $\mathrm{ex}(n,S_r,F)$ to prove our results. We also obtain several new results on $\mathrm{ex}(n,S_r,F)$.