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).

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.

Maximum degree 2 would be such a class (which includes regular of degree $2$ as a subclass).

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).