There are 3 formulas describing such polyhedra...
Euler's formula for polyhedra:
F = E-V+2
Another one saying that the number of Faces times the number of sides on each face equals twice the number of edges, because every edge connects two faces.
F*S = 2*E
And one more saying that the number of vertices times the number of edges connecting to it equals twice the number of edges, because every edge connects two vertices.
V*P = 2*E
These solve in terms of S and P as the formula for E, F, and V given.
For example if we want triangles so S=3 meeting 5 at each vertex:
It says there must be 30 edges, 20 faces and 12 vertices, which is the regular icosahedron.I find it interesting that the assumptions are quite general, Euler's formula for polyhedra can be proven even over topologies with holes in them like a donut, and the other two would certainly still be true. But it says you could never have say a 1000 vertex donut polyhedra where the whole surface is divided into triangles and 5 meet at every vertex, because the only solution is for there to be 12 vertices for that configuration of S and P.
I was thinking about how a toroid can be broken into almost any number of 4 sided figures with 4 edges meeting at each vertex.. there is a caveat that these formulas end up dividing by 0 for S=4 and P=4...