Lambda the Ultimate

A graph isn't necessarily reflexive and if I remember right some kinds of graph are precluded from having self loops.

Graphs where every node has a self loop attached are the same as graphs without self loops, just like graphs that are drawn with a blue pen are the same as graphs drawn with a red pen. The properties of these two structures are essentially the same.

A graph is a set of vertices and a collection of edges (which may be a set or a multi-set). Saying that the edge (multi)set always includes self loops is a different structure from saying that it never does or even sometimes does.

See Simple Graph for a kind of graph that explicitly precludes self loops.

"In a simple graph the edges of the graph form a set (rather than a multiset) and each edge is a pair of distinct vertices."

Sure, they are not exactly the same, but there is no interesting difference between them. I'm saying that if you want to know more about this structure, graph theory is exactly the place to look. A structure that *sometimes* has self loops *is* different, however.

Suppose that we take the natural numbers, and give each natural number a property whizzbang, which always has the value flitzpop. Do we now have a different thing than natural numbers? Sure. Is the difference interesing? No.

Okay, I see what you mean, there's an isomorphism between a totally non-reflexive graph and a totally reflexive graph.

Yes, exactly. So the OP is lucky because everything there is hope to learn about his structure is very likely already known in the context of graph theory.