On degree powers in the degenerate Tur\'an problem
Given a graph $G$ with degree sequence $d_{1},\ldots,d_{n}$ and a positive real number $p$, let $e_{p}(G)=\sum_{i=1}^{n} d_{i}^{p}$. For a fixed family of graphs $\mathcal F$, let $ex_{p}(n, \mathcal F)$ denote the maximum value of $e_{p}(G)$ over all $\mathcal F$-free graphs $G$ on $n$ vertices. In 2000, Caro and Yuster introduced the following Tur\'an-type problem: For a positive integer $p$ and a fixed graph $F$, determine $ex_{p}(n, F)$, and characterize the extremal graphs $G$ on $n$ vertices that attain $ex_p(n, F)$. Recently, Gao, Liu, Ma and Pikhurko proved that $ex_{p}(n, \mathcal F)=(\tau(\mathcal F)-1+o(1))n^p$ for real $p>\frac{1}{1-\alpha}$, where $\mathcal F$ is a degenerate family of graphs with classical Tur\'an number $ex(n, \mathcal F)=O(n^{1+\alpha})$ for some $\alpha\in[0,1)$, and $\tau(\mathcal F)$ is the minimum size of an independent vertex cover over all bipartite graphs $F\in\mathcal F$. Based on their method, we obtain a stability result for $ex_{p}(n, \mathcal F)$, and prove that all extremal graphs must contain the complete bipartite graph $K_{\tau(\mathcal F)-1,n-\tau(\mathcal F)+1}$ when $n$ is sufficiently large. Our results can be used to deduce all previously known results about $ex_{p}(n, F)$ when $F$ is a bipartite graph and $n$ is sufficiently large. We also obtain several new exact results for $ex_{p}(n, F)$, namely, when $F$ is an even cycle, a complete bipartite graph, a discrete hypercube, a caterpillar forest, and a spider forest.