What conditions on an undirected graph $G$ allow it to be uniquely determined by the spectrum of its adjacency matrix $A(G)$? Very simple examples show that one needs connectivity, and I imagine excluding self-loops and doubled edges will also help.
Is this enough? If not, is it enough for the spectrum of $A(G)$ to consist of distinct eigenvalues?
There is no useful set of conditions known. One cannot determine from the spectrum whether a graph is connected or not. Strongly regular graphs provide large classes of pairs of connected graphs that are not isomorphic, but are cospectral.
It is not enough to assume that the eigenvalues are all simple - there are cospectral trees on 10 vertices with all eigenvalues simple.