Is it possible to enumerate all unlabeled, undirected graphs?

In other words, is there a bijective correspondence between the natural numbers and unlabeled, undirected (and connected) graphs?

Considering there is a finite number of connected graphs of such type for $$n$$ vertices (See sequence A001349 on the OEIS), this is trivially true. However, is there a "natural" way to enumerate such, where given the $$n^{th}$$ graph, one could deduce exactly how the vertices and edges connect?

This is perhaps analogous to the "natural" way of enumerating rooted identity trees, where given, say the $$5^{th}$$ rooted identity tree, the graph of which can be deduced by the binary expansion: Since $$5=2^2+2^0$$, then the $$5^{th}$$ rooted identity tree has 2 main branches, which by themselves are the $$2^{nd}$$ and $$0^{th}$$ rooteed identity trees.

There are at least two ways to enumerate the set of unlabeled graphs applying a suitable labeling algorithm. Suppose we have a graph $$G$$ with $$n$$ vertices. Then one of the following labelings can be used.
1. There are $$n!$$ ways to label the vertices using $$\{1,...,n\}$$. For all such labelings build the adjacency matrix choosing the lexicographically minimal matrix $$M$$. Choose the labeling corresponding to $$M$$.
2. Draw $$n$$ vertices $$\{1,...,n\}$$ adding each time the lexicographically minimal edge and checking if the obtained graph is a subgraph of $$G$$ (if not, return to the previous step and choose the next edge).