TITLE: Invariants that might determine graph up to isomorphism
QUESTION [10 upvotes]: Are there any graph invariants which have a reasonable chance of capturing the graph up to isomorphism? In other words, some candidates for a function $f$ such that $f(G)=f(H)$ if and only if $G$ and $H$ are isomorphic.
For instance, in the case of trees, weighted graph polynomial ($U$-polynomial) of Welsh/Noble 1999 is a candidate because no counter-example has been found. Are there such candidates for general graphs?

Clarification: I'm interested in examples of functions which
  capture some graph invariant, are practical to compute, and are not
  yet proven to assign the same value to a pair of non-isomorphic graphs

REPLY [7 votes]: In some ways, provably, no (assuming the graphs are infinite).  See MR1011177 (91f:03062) Friedman, H; Stanley, L; "A Borel reducibility theory for classes of countable structures." J. Symbolic Logic 54 (1989), no. 3, 894–914.
This paper shows (although the argument is terse, and at least some is older folklore) that any Borel (in an appropriate sense) function f mapping graphs to any thing else with an equivalence relation E in such a way that G is isomorphic to H iff f(g) E f(H) must be at least as complicated as the graphs themselves.
For a similar result on finite graphs, see MR2135387 (2006e:03049) Calvert, Cummins, Knight, and Miller, Comparing classes of finite structures. (Russian) Algebra Logika 43 (2004), no. 6, 666--701, 759; translation in Algebra Logic 43 (2004), no. 6, 374–392.