Definition:Loop Graph

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



Loop
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 pseudograph or loop-graph $$P$$ is a non-empty set $$V$$ together with a symmetric relation $$E$$ on $$P$$.

Thus it can be seen that a pseudograph 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.

A loop-multigraph is also known as a pseudograph.