Definition:Graph of Mapping

Let $$f: S \to T$$ be a mapping.

Then the relation $$\mathcal{R} \subseteq S \times T$$ defined as $$\mathcal{R} = \left\{{\left({x, f \left({x}\right)}\right): x \in S}\right\}$$ is called the graph of $$f$$.

Alternatively, this can be expressed:
 * $$G_f = \left\{{\left({s, t}\right) \in S \times T: f \left({s}\right) = t}\right\}$$

where $$G_f$$ is the graph of $$f$$.

The word is usually used in the context of a diagram:


 * GraphOfFunction.png

The defining nature of a mapping means that each vertical line through a point in $$A$$ intersects the graph at one and only one place, corresponding to a single point in $$B$$.

Graph of a Relation
The concept can still be applied when $$f$$ is a relation, but in this case a vertical line through a point in the graph is not guaranteed to intersect the graph at one and only one place.

Note
Not to be confused with a graph theoretic graph.