I have recently been considering modelling graph walks of length 3 using a rank 3 tensor as opposed to modelling walks of length 2 with an adjacency matrix. I am interested to see what happens if you square this tensor but currently I don't know the operations for how to do that.
Is there a 3D equivalent to squaring a matrix?
edit:
Given a graph $G$ with $|V|\geq3$, model walks $x$-$y$-$z$ as $1$ in position $A_{xyz}$ of an "adjacency" tensor and $0$ otherwise.
- When squaring an adjacency matrix of some graph $G$ the output $A^2$ can be interpreted as the number of walks that can be made in 2 steps.
I'm hoping to gain some insight similar to this from squaring an adjacency tensor. I don't know what relationship would come out as that's what I am curious to see. Maybe the trace of the tensor might contain the information whether or not $K_3$ are present for each vertex.