Some stability and exact results in generalized Tur\'an problems

Given graphs $H$ and $F$, the generalized Tur'an number $\mathrm{e}\mathrm{x}(n,H,F)$ is the largest number of copies of $H$ in $n$-vertex $F$-free graphs. Stability refers to the usual phenomenon that if an $n$-vertex $F$-free graph $G$ contains almost $\mathrm{e}\mathrm{x}(n,H,F)$ copies of $H$, than $G$ is in some sense similar to some extremal graph. We obtain new stability results for generalized Tur'an problems and derive several new exact results.