# nLab vertex

### Context

#### Graph theory

graph theory

graph

category of simple graphs

# Contents

## Idea

A vertex is a point in a graph or simplicial set or similar. In the context of simplicial sets it is a 0-dimensional simplex.

## Terminological remarks

Vertices are also sometimes called points (somewhat old-fashioned, e.g. Harary, Chapter 2) or nodes (more frequently in computer science). In specialized contexts other terms are used. For instance, the two classes of vertices of a Petri net are called “places” or “species” and “transitions” or “reactions”, respectively; in an automaton the vertices are called “states”; and for quasicategories (particular simplicial sets) the vertices are also called objects.

Historically, the use of “vertex” for a point of an abstract graph probably derives from the natural connections between graph theory and the theory of polyhedra, which have “vertices”. One such connection is Steinitz's 1916 theorem? on the one-dimensional skeleta of polyhedra in three-dimensional Euclidean space.

• Frank Harary: Graph Theory. Addison-Wesley. 1969