There are essentially two orthogonal problems: how to handle the lifetime of the graph and how to handle it's mutability.
The first problem essentially boils down to what kind of pointer to use to point to other nodes in the graph. Since graph-like data structures are recursive (the types are recursive, even if the data is not) we are forced to use pointers of some kind rather than have a totally value-based structure. Since graphs can be cyclic, and ownership in Rust cannot be cyclic, we cannot use
Box<Node>as our pointer type (as we might do for tree-like data structures or linked lists).
No graph is truly immutable. Because there may be cycles, the graph cannot be created in a single statement. Thus, at the very least, the graph must be mutable during its initialisation phase. The usual invariant in Rust is that all pointers must either be unique or immutable. Graph edges must be mutable (at least during initialisation) and there can be more than one edge into any node, thus no edges are guaranteed to be unique. So we're going to have to do something a little bit advanced to handle mutability.
Read the full tutorial, with examples. There's also some discussion of potential language improvements in Rust to make dealing with graphs easier.