On the bounds of feedback numbers of (n,k)-star graphs

The feedback number of a graph G is the minimum number of vertices whose removal from G results in an acyclic subgraph. We use $f(n,k)$ to denote the feedback number of the $(n,k)$-star graph $S_{n,k}$ and $p(n,k)$ the number of k-permutations of an n-element set. This paper proves that$p(n,k)?2(k?1)!\left(\begin{array}{c}\hfill n\hfill \\ \hfill k?1\hfill \end{array}\right)?f(n,k)?p(n,k)?2(k?1)!\sum _{i=1}^{\theta }\left(\begin{array}{c}\hfill n?2i+1\hfill \\ \hfill k?i\hfill \end{array}\right),$ where $\theta =\min \{k?1,n?k+1\}$.