## Network Theory and Discrete Calculus – Edge Algebra

This post is part of a series

In my last post, I noted that in following John Baez’ series, I’m finding the need to introduce operators that I haven’t previously used in any applications. In this post, I will introduce another. It turns out that we could get away without introducing this concept, but I think it helps motivate some things I will talk about later.

In all previous applications, the important algebra was a noncommutative graded differential algebra. The grading means that the degree of elements add when you multiply them together. For example, the product of two nodes (degree 0) is a node (degree 0+0), the product of a node (degree 0) and a directed edge (degree 1) is a directed edge (degree 0+1), and the product of a directed edge (degree 1) with another directed edge is a directed surface (degree 1+1).

Note the algebra of nodes is a commutative sub-algebra of the full noncommutative graded algebra.

There is another related commutative **edge algebra** with corresponding **edge product**.

The edge product is similar to the product of nodes in that it is a projection given by

It is a projection because for an arbitrary discrete 1-form

we have

and

The product of two discrete 1-forms is

I have not yet come across an application where the full edge algebra is needed. When the product does arise, one of the discrete 1-forms is usual the coboundary of a discrete 0-form, i.e.

When this is the case, the edge product can be expressed as a (graded) commutator in the noncommutative graded algebra, i.e.

An example of this will be seen when we examine electrical circuits.

[…] where is the edge product. […]

Network Theory and Discrete Calculus – Graph Divergence and Graph Laplacian « Phorgy PhynanceDecember 4, 2011 at 1:26 pm