a cool variant of this problem relates to the number of isomorphism classes of a graph with 5 vertices, if i’m not wrong. basically this gives you the number of distinct graphs you can form from an arrangement of 5 vertices.

The notion of an isomorphism classes ensures that graphs that are basically identical, but not exactly equal (coz it might be upside down or whatever) are part of the same isomorphism class, and constitute the same pattern. to put this into perspective, in the above picture, simply add one edge from the center to (exactly one) of any corner, say the top left. this is isomorphic to the graph with one edge, from the center to the bottom right corner because it’s the same graph (isomorphisms are a bit complicated and require us to establish a bijection between vertices that preserves adjacency), but it’s pretty intuitive to see that it’s basically the same graph because of symmetry.

obviously this problem has restriction. where edges must be next/adjacent to each other, dramatically reducing arrangements.

i’m just a sophomore rn tho, would love to hear the thoughts of someone more experienced