The Incidence Matrix: A

The incidence matrix will be denoted as 'A' throughout the rest of this project. The importance of this matrix is that is is the combinatorial gradient operator and its transpose is the combinatorial divergence operator. These operators will be defined in detail in a future post. 

Now, the incidence matrix is defined as:

where eij is the edge adjacent to nodes i and j; and vk is the node k. 

So, by defining the columns as eij and the rows as vk we end up with a matrix which includes a diagonal of 1s since for every k=i, the value is +1. From there, for every node j adjacent to k=i, the value in the matrix is -1. The orientation for the edges here is assigned by taking i=k for node k (always) and then the adjacent node would just be j.

To demonstrate, I have made the incidence matrix for the graph we have been using throughout the blog.

The incidence matrix is always symmetrical.

What I don't understand though is what new information this provides over the adjacency matrix. I realize that it is represented differently by having a diagonal row of 1s, but don't both matrices just show the adjacent nodes of every node?