Editing Graph (discrete mathematics) (section)

== Properties of graphs ==
{{see also|Glossary of graph theory|Graph property}}
Two edges of a graph are called ''adjacent'' if they share a common vertex. Two edges of a directed graph are called ''consecutive'' if the head of the first one is the tail of the second one. Similarly, two vertices are called ''adjacent'' if they share a common edge (''consecutive'' if the first one is the tail and the second one is the head of an edge), in which case the common edge is said to ''join'' the two vertices. An edge and a vertex on that edge are called ''incident''.

The graph with only one vertex and no edges is called the ''trivial graph''. A graph with only vertices and no edges is known as an ''edgeless graph''. The graph with no vertices and no edges is sometimes called the ''[[null graph]]'' or ''empty graph'', but the terminology is not consistent and not all mathematicians allow this object.

Normally, the vertices of a graph, by their nature as elements of a set, are distinguishable. This kind of graph may be called ''vertex-labeled''. However, for many questions it is better to treat vertices as indistinguishable. (Of course, the vertices may be still distinguishable by the properties of the graph itself, e.g., by the numbers of incident edges.) The same remarks apply to edges, so graphs with labeled edges are called ''edge-labeled''. Graphs with labels attached to edges or vertices are more generally designated as ''labeled''. Consequently, graphs in which vertices are indistinguishable and edges are indistinguishable are called ''unlabeled''. (In the literature, the term ''labeled'' may apply to other kinds of labeling, besides that which serves only to distinguish different vertices or edges.)

The [[category theory|category]] of directed multigraphs permitting loops is the [[comma category]] Set ↓ ''D'' where ''D'': Set → Set is the [[functor]] taking a set ''s'' to ''s'' × ''s''.