# Definition:Loop-Graph

## Contents

## Informal Definition

A **loop-graph** is a graph which allows an edge to start and end at the same vertex:

Such an edge is called a loop.

### Incidence

Although a loop is incident to only one vertex, when measuring the degree of such a vertex, the loop is counted twice.

Thus, vertices $C$ and $D$ above have degree $5$, despite there being only four individual edges incident to those vertices.

## Formal Definition

A **loop-graph** $G$ is a non-empty set $V$ together with a symmetric relation $E$ on $G$.

Thus it can be seen that a loop-graph is a simple graph with the stipulation that the relation $E$ does not need to be antireflexive.

### Loop-Digraph

A **loop-digraph** is a directed graph which allows an arc to start and end at the same vertex:

### Loop-Network

A **loop-network** (directed or undirected) is a loop-graph together with a mapping which maps the edge set into the set $\R$ of real numbers.

That is, it is a **network** which may have loops.

### Loop-Multigraph

A **loop-multigraph** is a multigraph which is allowed to have loops:

## Also known as

Some presentations of this subject omit the hyphen and call this a **loop graph**.

**Loop-graphs** and **loop-multigraphs** are also often known as **pseudographs**.

However, the precise definition of the latter term varies in the literature, and its precise meaning can be misunderstood. Its use is therefore not recommended.

- Results about
**loop-graphs**can be found here.

## Sources

- 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): $\S 1.6$: Networks as Mathematical Models