What are loops in graph theory? Sometimes called self loops, a loop in a graph is an edge that connects a vertex to itself. These are not allowed in what are often called "simple graphs", which are the graphs we usually study when we begin studying graph theory.
In simple graphs, loop edges are not allowed. Also, multiple edges are not allowed to join the same pair of vertices in a simple graph. A pair of vertices is either joined by 0 edges or 1 edge.
Multigraph is a term used to refer to a graph where multiple edges are allowed, and loops may or may not be allowed. Pseudograph is a term usually used to refer to a graph where multiple edges and loops are allowed.
You might remember that with simple graphs, we defined an edge as being a 2-element subset of the vertex set. So an edge joining vertex a and vertex b could be written as { a, b }. This could cause some confusion with loops because the set { a, a } is just the set { a }, because repeated elements are not recognized in sets. There are two solutions I have seen to deal with this problem.
The first solution is to simply embrace the problem, and define a loop as being a one-element subset of the vertex set. So if an edge in the edge set is a one-element set, you would know that refers to a loop.
The other solution is to use multisets, which are like sets, except for the distinction that multiplicity is considered in multisets. That is, the multiset { a } is not the same as the multiset { a, a }. The multiset { a } contains the vertex a with multiplicity 1 (1 copy of a), the multiset { a, a } contains the vertex a but with multiplicity 2 (2 copies of a). So a loop can also be defined as a multiset with one vertex of multiplicity 2, representing an edge that goes from one vertex right back to the same vertex.
I hope you find this video helpful, and be sure to ask any questions down in the comments!
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