# Full-rank factorization of the graph Laplacian

Is there a combinatorially meaningful full-rank factorization of the Laplacian matrix of a graph?

The usual factorization $L=BB^{T}$, where $B$ is an oriented incidence matrix, is full-rank if and only if the graph is a tree (because then $B$ has $n-1$ columns and rank $n-1$). Is there a nice way to to factorize a connected graph with cycles?