TITLE: What makes Graph invariants so useful/important?
QUESTION [8 upvotes]: What makes Graph invariants so useful/important? If I were trying to create a useful graph invariant, what principles should I follow? 
My understanding is that they allow one to isolate and study specific properties of graphs algebraically or to classify graphs up to isomorphism (although, it seems to me that canonical labellings are the right tool for this). 
However, important graph invariants are constructed from counting proper colorings of a graph, for an appropriate definition of proper. A priori, why do we know that those graph invariants isolate and study specific properties or is there some other key motivation for graph invariants?

REPLY [5 votes]: Some graph invariants are useful to separate NP-complete problems from tractable problems. Bounded treewidth is such a parameter.
For example, the weighted max independent set problem is NP-complete but can be
solved in linear time for graphs of bounded treewidth.
Similarly, the NP-complete 3-coloring problem can be solved in linear time for graphs of bounded treewidth. The time complexities for these problems
are $O(2^w n)$ and $O(3^w n)$ respectively, for graphs of tree-width $w$.
Two classes of graphs with bounded treewidth are the Halin graphs and pseudoforests:

          



          

Images from Wikipedia: Halin graph, 
pseudoforest.