The generalized Turán number $ex(n,K_s,H)$ is defined to be the maximum number of copies of a complete graph $K_s$ in any $H$-free graph on $n$ vertices. Let $S_{\ell }$ denote the star on $\ell +1$ vertices, and let $k{S}_{\ell }$ denote the disjoint union of $k$ copies of $S_{\ell }$. Gan et al. and Chase determined $ex(n,K_s,S_{\ell })$ for all integers $s\ge 3$, $\ell \ge 1$ and $n\ge 1$. In this paper, we determine $ex(n,K_s,2S_{\ell })$ for all integers $s\ge 4$, $\ell \ge 1$ and $n\ge 1$.

In the report, based on the important branch of extremal graph theory, the Turán-type problem for the degree powers of graphs without 2K_p+1 was studied. Based on the degree sequences of two 2K_p+1-free graphs, some degree sequence properties of extremal graph were obtained. Then the degrees of the extremal graph were categorized and discussed. With the help of relevant lemmas and the Erdös-Gallai theorem, when 2p+2≤n≤2p+4, the maximum values of the q-power sum of degrees of 2K_p+1-free graphs were proved, in which n is the number of vertices, p≥2 and q≥2.