If $B\in\mathbb{R}^{n\times e}$ is the incidence matrix corresponding to a graph with $n$ vertices and $e$ edges, we know that $BB^T\in\mathbb{R}^{n\times n}$ is the graph Laplacian matrix.


I am curious that if there is any special meaning of the matrix $B^TB\in\mathbb{R}^{e\times e}$.



Incidence matrix in a graph, meaning of $B^TB$
Well, "special" is maybe an overstatement, because the meaning is really boring: If we consider edges of an undirected graph as two-element subsets of its vertex set, then
$$(B^T B)\_{e,e'} = \sum\_v B\_{v,e} B\_{v,e'} = \#\{v | v \text{ is incident to $e$ and $e'$}\}=|e\cap e'|$$
which is $2\cdot 1\_{n\times n}+$ the adjacency matrix of the line graph of $G$.


And and a slightly more complicated, but similarly boring formula can be derived directly from the definition if you consider directed graphs instead.


