# nLab empty graph

Contents

### Context

#### Graph theory

graph theory

graph

category of simple graphs

# Contents

## Idea

The empty graph is the graph with empty set of vertices and empty set of edges.

## Properties

The empty graph is the initial object in any reasonable category of graphs (such as a category of simple graphs).

The free category on the empty graph is the empty category.

## Some notions of non-empty graphs

For concreteness let us consider the topos of quivers. The inhabited graphs, i.e. those satisfying $\top\vdash (\exists x)\top$ are precisely the graphs $G$ having an edge since the formula on the right is interpreted as the image of the composite $G\overset{id}{\hookrightarrow} G\to 1$ but $1$ being the loop graph and in order to validate the sequent, its loop must be contained in the image requiring an edge in the source. This notion of being inhabited is stronger than being non-empty $\neq\empty$ but weaker as having a (global) element $1\to G$ which corresponds to graphs containing a loop. A uninhabited graph satisfying $(\exists x)\top\vdash \bot$ is necessarily empty whence edgeless graphs with a node are internally neither inhabited nor uninhabited.

Last revised on May 24, 2020 at 13:41:42. See the history of this page for a list of all contributions to it.