nLab weighted graph

Definition

Recall that a simple graph is a set $V$ of vertices equipped with

• a family of subsingletons $x \in V, y \in V \vdash E(x, y)$ whose elements are called edges,

• a family of bijections $x \in V \vdash i(x, y):E(x, x) \cong \emptyset$,

• a family of bijections $x \in V, y \in V \vdash s(x, y):E(x, y) \cong E(y, x)$.

A weighted graph is a simple graph equipped with a family of functions $x \in V, y \in V \vdash w(x, y):E(x, y) \to R$ from the edge subsingleton to a rig of numbers $R$. Usually $R$ is the natural numbers $\mathbb{N}$ or the real numbers $\mathbb{R}$.

References

• Wikipedia, Weighted graph

• Wikipedia, Graph (discrete mathematics)