Given two graphs H and F, let ex(n, H, F) denote the maximum number of copies of H in an F-free graph on n vertices. Let \(F_{k,r}\) be a graph with \((r-1)k+1\) vertices consisting of k cliques each with r vertices, which intersect in exactly one common vertex. Liu and Wang [16] determined \(ex(n,K_r,F_{2,r})\) for n sufficiently large. Zhu [23] et al. determined \(ex(n,K_3, F_{k,3})\) for \(k\ge 3\) and \(n\ge 4k^3\) . Gerbner [9] determined \(ex(n,K_s,F_{2,r})\) for \(3\le s<r\) and n sufficiently large. In the paper, we determine \(ex(n,K_s,F_{k,r})\) for \(2\le k<r,3\le s<r\) and n sufficiently large.
