The incidence matrix of a weighted graph How to correctly build the incidence matrix of a undirected weighted graph? May you show a little example?
 A: Although this question has already been resolved, I would like to add an additional comment since the current responses are, in my opinion, unsatisfactory.
The Unweighted Case
Following the notation of Biyikoglu et al., the incidence matrix $\nabla$ of an unweighted graph $G(V, E)$ is a $(|E| \times|V|)$ matrix defined by
$$
\nabla_{e, v}=\left\{\begin{array}{ll}
-1 & \text{if $x$ is the initial vertex of edge $e$}\\
+1 & \text{if $x$ is the terminal vertex of edge $e$} \\
0 & \text { otherwise }
\end{array}\right.
$$
Note that this is equivalent to 
$$
\nabla_{e, v}=\left\{\begin{array}{ll}
-\sqrt{A_e} & \text{if $x$ is the initial vertex of edge $e$}\\
+\sqrt{A_e} & \text{if $x$ is the terminal vertex of edge $e$} \\
0 & \text { otherwise }
\end{array}\right.
$$
where $A$ is the adjacency matrix of $G$. One crucial property of this matrix is that it satisfies the following relation
$$
\mathbf{L}=\mathbf{\nabla}^{\top} \mathbf{\nabla}
$$
where $L$ is the Laplacian matrix of $G$
$$
L_{x y}=\sum_{e \in E} \nabla_{e x} \nabla_{e y}=\left\{\begin{array}{ll}
-1 & \text { if } x y \in E \\
d(x) & \text { if } x=y \\
0 & \text { otherwise }
\end{array}\right.
$$
Equivalently, 
$$L = D - A$$
where $D$ is the degree matrix.
The Weighted Case
Now, if we consider a weighted graph $G(V, W)$, we would like similar properties. Since the Laplacian for the weighted case is defined as
$$L = D - W$$
(note that we replaced the adjacency matrix with the weight matrix)
We would like to find a $(|E| \times|V|)$ matrix $\nabla$ such that
$$
\mathbf{L}=\mathbf{\nabla}^{\top} \mathbf{\nabla}
$$
Fixing an arbitrary orientation, the matrix defined by
$$
\nabla_{e, v}=\left\{\begin{array}{ll}
-\sqrt{W_e} & \text{if $x$ is the initial vertex of edge $e$}\\
+\sqrt{W_e} & \text{if $x$ is the terminal vertex of edge $e$} \\
0 & \text { otherwise }
\end{array}\right.
$$
satisfies the desired relation.
Refs
Biyikoglu, T., Leydold, J., & Stadler, P. F. (2007). Laplacian eigenvectors of graphs: Perron-Frobenius and Faber-Krahn type theorems. Springer.
Kelner, J. (2007). An Algorithmist’s Toolkit: Lecture Notes. Lecture 2. MIT.
A: An incidence matrix $M$ is a matrix in $\mathbb{R}^{|V| \times |E|}$, where $M_{ij} = 1$ if vertex $i$ is incident to edge $j$, and $M_{ij} = 0$ otherwise.
You can replace the indicator value of $1$ with the edge weight instead.
A: I found a example in the following lecture (p.23):
https://www.fd.cvut.cz/personal/nagyivan/MatModEk/MME_Lectures.pdf
