I am working in a field where the set of zero points of divergence-free 3D vector fields are investigated. These sets are comprised usually from a set of single (isolated) points or connected lines that may or may not branch. In addition closed curves (loops) are appearing. Traditionally these sets of zero points are called "stagnation graph".
I am wondering if such sets are really fitting into "conventional graph theory". For example the loops would be edges without vertices. Thus my question is there a term for an edge without vertex in graph theory? Can it be a "graph" at all?
According to (standandard) graph theory, a graph is an ordered pair $G=(E,V)$ of edges and vertices where the edges $E$ are defined by (ordered) pairs of vertices. So according to this definition such a vertex-free loop would not qualify as a graph. But I cannot exclude that a generalized definition or more general structure has been described and defined, somewhere.
In any case adjacency matrices for example would not exist, as far as I can see. Can anyone comment on this?
(This question can be seen as a kind of counter-part of this question.)
Best Answer
I think the pragmatic approach works best here. If you want the set of zero points to be a graph you just declare a single point on a loop to be a vertex and then you get a graph according to the classical graph theory definition. You may have to prove that the choice of the point you declare to be a vertex doesn't change anything important.