Laplace matrix for undirected graph

Question

So Laplacian matrix can be calculated by the following equations according to wiki:

$L=BB^{T}$

where $L$ is the Laplacian matrix and $B$ is the incidence matrix.

However, the incidence matrix for an undirected graph (e.g., see the following picture) is something like this:

$$ \begin{pmatrix} 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ \end{pmatrix} $$

I can't get the correct Laplacian matrix from this incidence matrix. According to the Laplacian matrix wiki, the incidence matrix for this undirected graph should be:

$$ \begin{pmatrix} 1 & 1 & 1 & 0 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 1 \\ 0 & 0 & -1 & -1 \\ \end{pmatrix} $$

My question is, when calculating the Laplacian matrix for undirected graph, should I treat it as directed graph for the incidence matrix.

Answer 1

This is a simple graph. If you need the Laplacian matrix and the method is not important, you can use $L=D-A,$ where $D$ is the degree matrix and $A$ the adjacency matrix. Here

\begin{pmatrix} 3 & -1 & -1 & -1 \\ -1 & 1 & 0 & 0 \\ -1 & 0 & 2 & -1 \\ -1 & 0 & -1 & 2 \\ \end{pmatrix}

Answer 2

Citing Wikipedia:

Even though the edges in this definition are technically directed, their directions can be arbitrary, still resulting in the same symmetric Laplacian.

We can see that any change of orientation of an edge has no result on the Laplacian. Changing the orientation of an edge is equivalent to multiplying the corresponding column by $-1$.

In $BB^T$, the coefficients of this column will only be multiplied with coefficients of this same column (but displayed as a row in $B^T$). As every coefficient of this column changes its sign, the signs will cancel in the product, and $BB^T$ is unchanged.