Is there a branch of mathematics that studies and classifies (infinite) graphs where nodes are natural numbers and edges represent some relationship between them?
A few examples I found:
"...vertices are the natural numbers, with an edge between j and k if j∣k." in a question about definition of graph.
"...two vertices being connected if the sum of the numbers on the two vertices add up to a prime number." in Proving that a "prime graph" is connected.
If the Collatz conjecture is true, that relationship also organises natural numbers into a graph: https://www.jasondavies.com/collatz-graph/