Finite weighted directed multigraphs, weighted walks and their transfer matrices

Let $R$ be a commutative ring (Commutative ring). A finite weighted directed multigraph over $R$ consists of a finite vertex set $p=\{0,\dots ,p-1\}$, a finite edge set $E$, source and target maps $s,t:E\to p$, and a weight map $w:E\to R$. Parallel edges and loops are allowed.

A walk of length $n$ from $u$ to $v$ is a sequence $(e_0,\dots ,e_{n-1})$ of edges with $s(e_0)=u$, $t(e_{n-1})=v$, and $t(e_i)=s(e_{i+1})$ whenever $i+1<n$. Its weight is ${\prod }_{i<n}w(e_i)$. At length zero there is one empty walk from $u$ to $u$, of weight $1_R$, and no empty walk between distinct vertices.

The transfer matrix or weighted adjacency matrix is $A\in M_p(R)$ (Finite rectangular matrices over a commutative ring, their entries, rows and columns) defined by

$A_{uv}:={\sum }_{\begin{array}{c}e\in E\\ s(e)=u,t(e)=v\end{array}}w(e)\mathrm{.}$

An empty edge sum is $0_R$. Rows record sources and columns record targets.