  In this short note, we prove some consequences of Burnside's vanishing
property \cite{b-pa}. It is known that Harada's identity concerning the product
of all conjugacy classes of a finite group is a consequence of Burnside's
vanishing property of characters. We prove a similar formula for any
weakly-integral fusion category. In the second part, we prove some structural
results concerning nilpotent and modular fusion categories. As an application
we show that any fusion category of dimension $p^2q^2r^2d$ with $p<q<r$ prime
numbers and $d$ a square-free integer is weakly-group theoretical.
