  Let $G$ be a graph on $n$ vertices with degree sequence $(d_1,d_2......d_n)$.
For a real $p \geq 1$, let $D_p(G)=\sum_{i=1}^nd_i^p$. A Tur\'an-type problem
of degree power sum was initiated by Caro and Yuster \cite{caro2000degpower}:
determining the function $D_p(n,H) :=\max \{D_p(G): \text{$G$ is an $n$-vertex
$H$-free graph}\}$. They obtained some exact values for certain graphs $H$. For
a path $P_k$, they mentioned that ``a close examination of the proof of Theorem
1.2 shows that the value of $n_0(k)$ in the statement of the theorem is
$O(k^2)$", namely, they could show the $n$-vertex $P_k$-free graph with maximum
degree power sum is $W_{n,k-1,\lfloor \frac{k}{2} \rfloor -1} = K_{\lfloor
\frac{k}{2} \rfloor -1} \vee \left((n - \lceil \frac{k}{2} \rceil)K_1 \cup
K_{1+k-2\lfloor \frac{k}{2} \rfloor} \right)$ when $n \geq c k^2$ for some
constant $c$. In this note, we improve their result to a linear size of $k$ by
a different approach. The bound is tight up to a constant factor.
