The spectrum of a graph is the (multi)set of eigenvalues of its adjacency matrix (or Laplacian, depending on what you're interested in). In general, two non-isomorphic graphs might have the same spectrum. 


Prompted in part by [this discussion on reverse engineering a graph from its spectrum](https://cstheory.stackexchange.com/questions/3742/reverse-graph-spectra-problem), I was wondering: 



> 
> Are there interesting classes of
>  graphs for which isospectrality
>  implies isomorphism ?
> 
> 
> 



Classes of graphs for which isospectrum implies isomorphism?
Maximum degree 2 would be such a class (which includes regular of degree $2$ as a subclass). Transitive graphs (by which I mean that the relation of being connected by an edge is transitive) are another example (there is a less obscure description of that class of graphs but I wanted it to sound mysterious for a few moments).


I assume that you are asking for a class $\mathcal{C}$ of graphs such that $G,H \in \mathcal{C}$ and $G,H$ cospectral implies isomorphism. If you mean classes of graphs $\mathcal{C}$ such that $G \in \mathcal{C}$ and $G,H$ cospectral implies isomorphism, then <http://mathworld.wolfram.com/DeterminedbySpectrum.html> might be worth a look.


