Yep, that all sounds right. In fact, a directed graph can be called transitive if… well, take a guess. And k-uniform hypergraphs (edit: not k-regular, that’s different) correspond to k-ary relations.
Here’s another thought for you: Adjacency matrices. There’s a one-to-one correspondence between matrices and edge-weighted directed graphs. So large chunks of graph theory could, in principle, be described using matrices alone. We only choose not to do that out of pragmatism.
(I’ve also heard of something even more general called matroid theory. Sadly, I never took the time to learn about it.)
Yep, that all sounds right. In fact, a directed graph can be called transitive if… well, take a guess. And k-uniform hypergraphs (edit: not k-regular, that’s different) correspond to k-ary relations.
Here’s another thought for you: Adjacency matrices. There’s a one-to-one correspondence between matrices and edge-weighted directed graphs. So large chunks of graph theory could, in principle, be described using matrices alone. We only choose not to do that out of pragmatism.
(I’ve also heard of something even more general called matroid theory. Sadly, I never took the time to learn about it.)
And then when we do that, its called spectral graph theory, and its the origin of many clustering algorithms among other things.