Graph

A graph $G = (V, E)$ is a pair consisting of a set $V$ of vertices and a set $E$ of edges, where each edge connects two vertices. In an undirected graph, an edge is an unordered pair $\{u, v\}$; in a directed graph, an edge is an ordered pair $(u, v)$.

Graphs formalize relations between discrete objects. A symmetric binary relation on a set corresponds exactly to an undirected graph; a general binary relation corresponds to a directed graph.

In category theory, a directed graph (or quiver) is a diagram in the category of sets — a pair of functions $s, t : E \to V$ assigning source and target to each edge. A category can be seen as a graph with composition and identities.