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Just wondering if any of you have encountered a term for a particular type of graph. It is like a graph that allows for loops, but for loops, instead of joining a vertex to itself, it joins a vertex to nothing. I just want to be consistent with existing terminology, if there are none, maybe you could make some recommendations (i.e. what not to use)

The idea is this, I'm looking at properties of a simple graph, in particular, edges that are connected to vertices of the same degree. So, I don't really care about any other vertices or edges, except that in the end, I'm still talking about a simple graph.

Essentially, what I'm looking at is a set of vertices, all of which have the same degree, and edges that are incident to those vertices. So, if two of those vertices are joined by an edge, there is no problem. But, I may have edges that are incident to only one vertex (as the other end of those edges may be connected to vertices that are not yet specified).

As of now, I'm calling such graphs semigraphs. A quick search didn't show that it was being used. Someone suggested calling it a pseudograph, and in the literal sense (i.e. pseudo as a prefix), I feel it is more appropriate, but unfortunately, it seems that pseudograph is sometimes used to mean multigraphs.

Hope someone can help. Thanks!