An edge that connects 3 vertices Are there any practical situations that can be represented using a modified definition of an edge - where each edge connects three vertices instead of two?
 A: You're talking about $3$-uniform hypergraphs.  They will arise in a range of contexts, some of which are:


*

*Steiner triple systems are $3$-uniform hypergraphs in which each pair of vertices belongs to exactly one hyperedge.

*Triangulations of surfaces (e.g. the plane).

*The problem of completing a partial triangulation of the complete tripartite graph $K_{n,n,n}$ has been shown to be NP-complete.  Charlie Colbourn gave a polynomial time reduction of this problem to the problem of completing partial Latin squares, thereby showing that this problem is also NP-complete (ref.).
These contexts are more areas of pure mathematics, and can be rephrased to not be $3$-uniform hypergraphs (however, I'm guessing all $3$-uniform hypergraph problems can be thus rephrased).
Combinatorial designs, such as Steiner triple systems, have applications in software testing.  Hypergraphs in general arise in the study of complex networks, but I haven't seen the particular case of $3$-uniform hypergraphs arise.
However, we don't need applications for mathematics to be interesting; the following quote applies to mathematics as well as physics.

Physics is like sex. Sure you can get some interesting results, but that's not why we do it.  -- Richard Feynman (disputed).

