# Notation for set of links in a graph

Let $\mathcal{G} = (\mathcal{N}, \mathcal{L})$ represent a directed graph, where $\mathcal{N}$ is the set of nodes.

What would be the correct way to represent the set of links between them?

$$\mathcal{L} = \{(i,j) \mid i,j \in \mathcal{S}, i \neq j \}\rightarrow$$ Is this correct?

Does this imply that $\mathcal{L}$ contains all such $(i,j)$ pairs? Because it would be wrong to assume that the graph is a complete graph.

You can represent the set of links as $\mathcal{L} \subset \mathcal{N} \times \mathcal{N}$. The crossproduct is all the pairs and subset indicates that not all pairs are present. Also typically directional graphs don’t prohibit nodes that have an edge to themselves.