(Solved): [10 points] The incidence matrix of a directed graph \( G=(V, E) \) with no self-loops (no edge st ...

[10 points] The incidence matrix of a directed graph \( G=(V, E) \) with no self-loops (no edge starts and ends at the same vertex) is a \( |V| \times|E| \) matrix \( B=\left(b_{i j}\right) \) such that \[ b_{i j}=\left\{\begin{array}{ll} -1 & \text { if edge } j \text { leaves vertex } i \\ 1 & \text { if edge } j \text { enters vertex } i \\ 0 & \text { otherwise } \end{array}\right. \] Describe what the entries of the matrix product \( B B^{\mathrm{T}} \) represent, where \( B^{\mathrm{T}} \) is the transpose of \( B \).