The Maximum Number of Stars in a Graph Without Linear Forest

For two graphs J and H , the generalized Turán number, denoted by ex ( n , J , H ), is the maximum number of copies of J in an H -free graph of order n . A linear forest F is the disjoint union of paths. In this paper, we determine e x ( n , S r , F ) when n is large enough, which generalizes the results on e x ( n , S r , P k ) and e x ( n , ( k + 1 ) P 2 ) . Finally, we prose a problem related to the number of graph copies in an F -free graph under shifting operations.