I would appreciate some feedback on my notation. I'm using Iverson brackets, which I'm sort of new to as a concept, but it seems simple enough.
The degree of a vertex $v_n$ is given by the sum of the edges containing the vertex: $$ \deg(v_n)=\sum_{i=1}^n \; [e_i=\{v_n,v_n'\}] $$
I also have the total number of edges of graph $G$ represented as: $$ \sum_{i=1}^m \; [e_i\in G] $$
With regard to using brackets in your notation as Iverson brackets, it's fine so long as you make it clear that that is what they are. The most important thing is that the meaning of the equations/expressions in your mathematics is clear to the reader, and even though you could use Iverson brackets to express something, there is often a clearer way without them. Using Iverson brackets could especially cause trouble since brackets are used quite often in mathematics with a variety of different meanings.
Now regarding your use of Iverson brackets, I think there is a problem with the bounds of your summation. The way you are expressing the degree of vertex $v_n$ in $G$, it looks your idea is to iterate over all the edges of $G$ and check and see if the edge is incident to $v_n$. I'm not sure what $n$ is suppose to be (or $v_n'$ for that matter), but you need to have this notion of "all the edges of $G$" in your expression. I would do it this way, with $E(G)$ representing the set of edges of $G$: $$ \deg(v_n) = \sum_{e \in E(G)} [v_n \in e] $$ Looking at your other example, having $E(G)$ gives us a much clearer way of representing the number of edges of $G$ than the notation using Iverson brackets. We can just use $|E(G)|$.