Check it out

GraphMatrix

Imagine graph G with vertexes A, B, C (sketch this if that's your thing). G has the following directed edges:
A->B
B->C
C->A

Basically this is a triangle with a loop. Now, let's say we model this with a binary matrix. Edge A->B means that G[A][B]=1. Similarly: G[B][C], G[C][A] = 1, 1. Think about this 2D array as a matrix G1:
 ABC
A010
B001
C100

Pretty slick. But hey, what if it was an undirected graph? So that any connection A->B implies B->A? B<->A, if you will. Then you'd get G2:
 ABC
A010
B101
C010

Note that this is a symmetric graph (across the NW->SE axes). This is special, because the Transpose of the graph is the same graph. A somewhat intuitive theorem is:
All directed graphs, when represented as matricies as above, correspond to a symmetric matrix.

Furthermore, I feel like there's something special about the relation of G1 and G2. I'm doing more testing and researching now.

For some reason this reminds me of the Linear Algebra class I never went to in college. That prof was a baller...

Oh, and think about if it wasn't a binary matrix, but instead a value between 0-1, even irrational expressions. These could represent weights on the graph. Sick. Let's go make some neural net matrix calculations in JavaScript!