I'm trying to understand Feynman's theorem mentioned in this paper, Chapter 0.0.2.
In this paper, the Feynman amplitude of a graph $G$ is a number obtained as a result of the following process:
(graph is used in broadest sense, allowing self-loops and multiple edges)
(Assume the positive-definite bilinear form $B:V\otimes V \rightarrow \mathbb{R}$ and $m$-tensor $B_m:V^{\otimes m}\rightarrow \mathbb{R}$ for each $m\geq 0$ are fixed)
put the $m$-tensor $-B_m$ at each vertices of degree(valency) $m$.
For each edge $e$, take contraction of tensors at the two ends of $e$ using the bilinear form $B^{-1}:V^{\vee}\otimes V^{\vee}\rightarrow \mathbb{R}$. This will give a number $F_{G_i}$ for each connected component $G_i$ of $G$.
the Feynman amplitude is the product $F_{G}=\prod F_{G_i}$
Because I'm self-teaching myself and new to these topics(QFT), I'm not convincing I have understood in right way. So, please check my argument to calculate the value in case of a triangle is correct or not.
In the case of $G$ a triangle (having 3 vertices and 3 edges), each 3 vertices are given $2$-tensor $-B_2$.
Then, taking an edge and contracting give a $0$-tensor(i.e, a number) $c=-B_2(\bar{B}^{-1}\otimes \bar{B}^{-1} (-B_2))$, where $\bar{B}:V\rightarrow V^{\vee}$ is the linear isomorphism induced from $B$.
Taking another edge and contracting then give a $2$-tensor $-cB_2$.
Final contraction with the edge left gives a number $c^2$, which is the Feynman amplitude of this triangle.
Also, I'm not convincing this process is independent of the order of choice of edges and gives well-defined number in the end, especially because the contracted tensors generated on the way are at nowhere and do not satisfy valency condition(In this case, if we think of removing one edge on this triangle, the two ends of removed edges have degree 1. If we think of shrinking one edge to a vertex, then the resulted graph is a bigon. Hence, either of the two cases do not have a vertex of degree 0 that we can attach the $0$-tensor $c$).
Thus, is my calculation right? If right, then why this process is well-defined? And, there is plausible interpretation of numerous tensors obtained in the middle of the process?