A digraph is defined as $G=(V,E,\phi)$ with
- a set of nodes $V$
- a set of edges $E$
- a mapping $\phi : E \rightarrow V \times V$
A weighting $\mathcal{W}$ for a directed graph $G=(V,E,\phi)$ is a mapping $E \rightarrow R$ from $E$ to a ring $R$. This ring is considered to be the noncommutative polynomial ring $R=\mathbb{C}\left<\Sigma\right>$ over an alphabet $\Sigma$.
Let $\Sigma$ be an alphabet. The addition and multiplication on the set $\mathbb{C}\langle \Sigma \rangle := \left\{ \sum_{w \in \Sigma^*} a_w w ~|~\begin{array}{c} a_w \in \mathbb{C} \\ a_w = 0/; \mbox{for almost all } w \in \Sigma^* \end{array} \right\}$ are defined by $(\sum_{w \in \Sigma^*} a_w w) + (\sum_{w \in \Sigma^*} b_w w) := \sum_{w \in \Sigma^*} (a_w + b_w) w.$
$(\sum_{w \in \Sigma^*} a_w w) \cdot (\sum_{w \in \Sigma^*} b_w w) := \sum_{w \in \Sigma^*} (\sum_{uv = w} a_ub_v) w.$
The set $\mathbb{C}\left<\Sigma\right>$ describes in combination with the given definitions for addition and multiplication a ring with 1.
Hi!
I do understand what a directed graph is, and I generally understand what a weighted direct graph is as well. But I have only used integers for the weightings before.
Could you please explain to me what this ring is (could you please provide a simple example) and how I might understand the last sentence?
Thank you in advance!