Definition:Loop-Graph
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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.
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:
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.
Sources
- Gary Chartrand: Introductory Graph Theory (1977): $\S 1.6$
