Is there already an established name for properties or subgraphs of finite, symmetric graphs, that are invariant under a change of vertex weights?

Some of those invariants are

• the extremal Hamilton cycles
• the relative order of the weights of perfect matchings in complete subgraphs with 4 vertices
• the relative order of the weights of detours over a common vertex $ c$ , i.e. whether $ w\left(e_{ac}\right)+w\left(e_{cb}\right)-w\left(e_{ab}\right)\lt w\left(e_{a’c}\right)+w\left(e_{cb’}\right)-w\left(e_{a’b’}\right)$

as it seems likely that there are more of those invariants and, that there will be interesting properties related to them, I would like to have a name for the mathematical property, that is invariant.

A good examples of what I am looking for, is “topological” invariants that are defined as properties, that do not change under continuous deformations.
So, for the name I am looking for, the change of vertex weights is analogous to the continuous deformations, and I would like to know, what the name of the “object” analogous to “topology” is.