Consider a directed graph $G$ and its node-to-edge signed incidence matrix $B \in \mathbb{R}^{n \times l}$ where $n$ and $l$ are the number of nodes and links respectively.
I am interested in the so-called edge-based laplacian matrix: $$L_e = B^{\top}B \in \mathbb{R}^{l \times l}$$ Is there any useful interpretation of this matrix in terms of the graph structure? It seems that its main diagonal is always constant and equal to 2, I guess because each link has two neighbor nodes. I am also interested in its spectrum as it seems to be singular for any graph apart from very trivial ones like the one such that $B=[-1, 1]^{\top}$. In general, is there anything special about its eigenvalues?