Minimizing the number of matchings of fixed size in a \(K_s\)-saturated graph

For a fixed graph \(F\), a graph \(G\) is said to be \(F\)-saturated if \(G\) does not contain a subgraph isomorphic to \(F\) but does contain \(F\) after the addition of any new edge. Let \(M_k\) be a matching consisting of \(k\) edges and \(S_{n,k}\) be the join graph of a complete graph \(K_k\) and an empty graph \(\overline{K_{n-k}}\). In this paper, we prove that for \(s \geq3\) and \(k\geq 2\), \(S_{n,s-2}\) contains the minimum number of \(M_k\) among all \(n\)-vertex \(K_s\)-saturated graphs for sufficiently large \(n\), and when \(k \leq s-2\), it is the unique extremal graph. In addition, we also show that \(S_{n,1}\) is the unique extremal graph when \(k=2\) and \(s=3\).