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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.


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.


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



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.


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.