# What is a Euclidean Graph? Can edges be negative in a Euclidean Graph?

To my understanding a graph is Euclidean if each edges connecting two vertices represents the distance between those two vertices, where the vertices are points in a plane.

Is this really the only thing that makes a graph Euclidean? I mean, I feel like this is a skimpy description. It doesn't seem like a rigorous definition. I am wondering things like:

Are they referring to the "planes" we know from geometry? Or some other type of coordinate system?

If that's the case, then all edges must be nonnegative, right?

A Euclidean graph takes a graph object and embeds it into $\mathbb{R}^{n}$. The vertices are points in Euclidean space, and we use the standard Euclidean metric to denote edge distance. That is, in $\mathbb{R}^{2}$ (for example), $d( (x_{1}, y_{1}) , (x_{2}, y_{2})) = \sqrt{ (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} }$.
Metrics, by definition, are non-negative. Essentially what we are doing is treating our graph as a continuous object. So if we are halfway along an edge between two vertices, we know exactly where in $\mathbb{R}^{n}$ we are. This makes a lot of sense with the cities and roads analogy commonly used to describe graphs.