For those of you who don't know, a graph is a set of points (commonly referred to as nodes) connected by lines (commonly referred to as edges).
For the graphs we will be considering in this problem, it will follow these criteria
- It has finitely many points.
- Each point is colored the same color.
- Each edge is colored the same color.
- No edges have any directionality to them.
- There may be no more than 1 edge per pair of points.
- Edges cannot connect points to themselves.
We consider 2 graphs to be the same if you can continuously distort one into another. In other words, the only things you aren't allowed to do is to remove or add edges or nodes.
Intuitively, you'd think that the answer must be 2^(n choose 2). For the first few values this formula works. But once you consider 3 or more nodes, the formula begins to overestimate. However, this function can still be useful to this problem. We can use it as an upper bound of the true function.
Let's call the function we're looking for Graph(x). Here are the first few inputs and their outputs of Graph(x)
- Graph(1) = 1
- Graph(2) = 2
- Graph(3) = 4
- Graph(4) = 10
It begins to get hard to tell whether or not a graph is the same as another once you reach 5 or more nodes.
