In the following, the vertices and (indexed) edges for an arbitrary class
(called "graph" in the following) are defined and examined. The main
result of this section is to show that the set of vertices
of a graph is the first component of the graph if it is
represented by an ordered pair (see opvtxfv), or the base
set of the graph if it is represented as extensible
structure (see basvtxval), and that the set of indexed edges resp. the
edge function is the second component of the graph
if it is represented by an ordered pair (see
opiedgfv), or the component of the graph if it is
represented as extensible structure (see edgfiedgval). Finally, it is
shown that the set of edges of a graph is the range of its edge
function: , see edgval.

Usually, a graph is a set. If is a proper class, however, it
represents the null graph (without vertices and edges), because
and holds, see vtxvalprc and iedgvalprc.

Up to the end of this section, the edges need not be related to the vertices.
Once undirected hypergraphs are defined (see df-uhgr), the edges become
nonempty sets of vertices, and by this obtain their meaning as "connectors"
of vertices.