How many countable graphs are there? Even though there are uncountably many subsets of $\mathbb{N}$ there are only countably many isomorphism classes of countably infinite - or countable, for short - models of the empty theory (with no axioms) over one unary relation.

How many isomorphism classes of
  countable models of the empty theory
  over one binary relation (a.k.a.
  graph theory) are there? I.e.: How many countable unlabeled graphs are there?

A handwaving argument might be: Since the number of unlabeled graphs with $n$ nodes grows (faster than) exponentially (as opposed to growing linearly in the case of a unary relation), there must be uncountably many countable unlabeled graphs. (Analogously to the case of subsets: the number of subsets of finite sets grows exponentially, thus (?) there are uncountably many subsets of a countably infinite set.) 
How is this argument to be made rigorous?

Another very simple way to show that they are uncountable is with well-orderings: By definition the well-orderings of a countable set are $\aleph_1$.
EDIT: Here's another simple way to show that they are $2^{\aleph_0}$ many. Take all but one (let's call it $p$) of the countable nodes and order them with the standard order of the natural numbers. This essentially assigns to every of these nodes a natural number. Now use the edgeless node $p$ that we kept to describe characteristic functions: For every subset $X$ of the natural numbers add an edge from the node that denotes $n$ to $p$ if and only if $n\in X$. It's trivial to see that all these graphs are not isomorphic: If two of them were then two different sets of natural numbers would coincide.
