The graph Laplacian

In our project, in order to generate a Laplacian graph, we start with generateing a adjacency matrix. I use the Matlab to do the following generation.

For example, we can randomly generate a 3 by 3 adjacency matrix with the following code:

The sample result is:

We continue with this matrix to generate a Laplacian matrix according to the formula:

where D is a degree matrix, and A is an adjacency matrix.

Since we already have the adjacency matrix, the only thing we need to do is to generate the degree matrix and plug them into the formula above:

The result is:

Then we can use this matrix to generate the Laplacian graph directly:

We can test it with higher dimension, for example, we can set dimension n=10, then we can randomly generate a laplacian matrix:

The graph of this adjacency matrix is:

Incidence matrix

We want to test and verify this property. First, we need generate the matrix according to the property of incidence matrix above, here for convenience, we set the n=3, for example, we give the Laplacian matrix:

Then we use the code in Matlab to gain the incidence matrix:

After running the code, the incidence matrix is:

which is the same as our Laplacian matrix.

Supersymmetric GQM

From the previous process, we can get the incidence matrix from any Laplacian matrix, so the process of generating the super Laplacian is very straightforward. For example, assume we already have the incidence matrix:

With the help of the code, we can generate the super Laplacian based on the incidence matrix:

Here, S(u,w) is the super Laplacian matrix: