There seems to be a slight difference in terminology among a couple of different traditions within theoretical computer science.
- To have a quantum computer simulate the Hamiltonian evolution of $\exp(-iAt)$ of some $2^n\times 2^n$ hermitian matrix $A$ giving the adjacency matrix of a sparse undirected graph on $2^n$ vertices, it suffices that we are also provided oracle access to query different vertices to have adjacent vertices returned in order, so that we can partition the adjacency matrix $A$ with edge-colors and subsequently Trotterize the partition;
- In the theory of expanders, as explained by O'Donnell here it's a rather trivial result that most (Erdős–Rényi) sparse random graphs have good expansion properties; what applications generally require is an explicit family of graphs. But, when we are also provided oracular access to adjacent vertices, this has been referred to as a strongly explicit family of expanders; while
- Wikipedia defines an implicit graph representation as is a graph whose vertices or edges are not represented as explicit objects in a computer's memory, but rather are determined algorithmically.
It's these last two terms that I'm wondering about - is it correct to say that an implicitly defined graph could be a member of a strongly explicit family of expanders (and, if so, could it be amenable to Hamiltonian simulation)?