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

Given a graph G with degree sequence $d_1,\dots ,d_n$ and a positive integer r, let $e_r\left(G\right)={\sum }_{i=1}^n{d}_i^r$. We denote by ${\mathrm{ex}}_r(n,F)$ the largest value of $e_r\left(G\right)$ among n-vertex F-free graphs G, and by $\mathrm{ex}(n,S_r,F)$ the largest number of stars $S_r$ in n-vertex F-free graphs. The broom $B(\ell ,s)$ 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 ${\mathrm{ex}}_r(n,B(\ell ,s\left)\right)$ for $r\ge 2$, any $\ell ,s$ and sufficiently large n, proving a conjecture of Lan, Liu, Qin and Shi. We also determine $\mathrm{ex}(n,S_r,B(\ell ,s\left)\right)$ for $r\ge 2$, any $\ell ,s$ and sufficiently large n.