I am trying to enumerate the number of distinct polyhedra that can be formed from a given number of vertices. So far, i have managed to finish the sets for 4, 5 and 6 vertices and is now a third of the way through the set for 7 vertices.
I am proceeding through this exercise without using Schlegel diagrams because i am more concerned with how they look in 3-space. The method i use is this: starting with the set for a given number of vertices, say, 5, then adding a vertex, then joining that new vertex to other vertices (whenever possible), to obtain the set of polyhedra with 6 vertices. The task is not easy (for me that is) and i realise that having names that i can refer to for each polyhedron will make the enumeration a bit easier.
I am aware that in chemistry, there is a systematic way of naming chemical compounds: the IUPAC Nomenclature system, which, is essence is mathematical in nature because it concerns permutations, graphs, and 3D spatial relationships of the constituent atoms.
Is there a similar system of nomenclature for polyhedra? for example, what are the names of the seven distinct polyhedra with seven vertices?
There is no wholly systematic naming convention for polyhedra.
In general, authors choose descriptive names based on a more well-known figure from which the new one can be most elegantly derived. These names can stick, as with the Johnson solids, or somebody might come along in a hundred years or so and try to change them, as Conway did with Cayley's names for some of the regular star polyhedra.
I know of no attempt to name those with seven vertices. They are the duals of heptahedra (having seven faces), so one approach you could use would be to call each one the "dual [relevant heptahedron]".
In passing, I wonder whether you are including nonconvex polyhedra in your count, such as the Császár polyhedron.