Degree powers and number of stars in graphs with a forbidden broom

Given a graph G with degree sequence d 1 , … , d n and a positive integer r , let e r ( G ) = ∑ i = 1 n d i r . We denote by ex r ( n , F ) the largest value of e r ( G ) among n -vertex F -free graphs G , and by ex ( n , S r , F ) the largest number of stars S r in n -vertex F -free graphs. The broom B ( ℓ , s ) is the graph obtained from an ℓ -vertex path by adding s new leaves connected to a penultimate vertex v of the path. We determine ex r ( n , B ( ℓ , s ) ) for r ≥ 2 , any ℓ , s and sufficiently large n , proving a conjecture of Lan, Liu, Qin and Shi. We also determine ex ( n , S r , B ( ℓ , s ) ) for r ≥ 2 , any ℓ , s and sufficiently large n .