Definition:Loop-Graph

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


 * Pseudograph.png

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


 * LoopMultigraph.png

Pseudograph
A loop-graph and loop-multigraph is also often known as a pseudograph. 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.