On degree powers and counting stars in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si25.svg"><mml:mi>F</mml:mi></mml:math>-free graphs

Given a positive integer r and a graph G with degree sequence d 1 , … , d n , we define e r ( G ) = ∑ i = 1 n d i r . We let 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 ex r ( n , F ) = e r ( G ) for a complete ( χ ( 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 ex r ( n , C 4 ) for r ≥ 3 . We also give simple proofs of multiple known results. Our key observation is the connection to 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 ex ( n , S r , F ) to prove our results. We also obtain several new results on ex ( n , S r , F ) .