How to show this equality.
Let $L$ be the Laplacian matrix of a graph $G$ with eigenvalues $0,1,2..,n-1$, then this equality holds. $$\|{L}\|_{F}^{2} = \mbox{trace} \left( L^{2} \right) = \sum_{i=0}^{n-1} i^2$$
where $i=0,1,2,..,n$ are the eigenvalues of the Laplacian matrix of $G$.