Sharp bounds, stability, and arithmetic fluctuations for induced pentagons

We prove that every graph with $m>0$ edges contains at most $\frac{3}{2\sqrt{2}}(\frac{m}{5})^{5/2}$ induced pentagons. Equality holds precisely for balanced blow-ups of the Clebsch graph, together with isolated vertices. The bound and equality classification extend to arbitrary nonnegative vertex w