A simple graph is a pair (V,E) where V is set (of nodes) and E is a set of 2-elementary subsets of V, which are called edges. General graphs are allowed to have multiple edges between a pair of nodes or loops, that is, edges from a node to itself.
The simple graphs are precisely the 1-dimensional abstract simplicial complexes.
A directed graph (also called digraph) has arcs instead of edges, where an arc is an ordered pair of nodes.
Typically, our graphs are finite.