The adjacency algebra of a graph is the algebra consisting of all polynomials in the adjacency matrix of the graph. An association scheme is a commutative matrix algebra containing the identity and all ones matrices and which is closed under entrywise product and transposition. An association scheme is symmetric if it contains only symmetric matrices. It is not difficult to see that since an association scheme is closed under entrywise product, it must have an orthogonal (with respect to the Hilbert-Schmidt inner product) basis of 01 matrices. These matrices can be viewed as the adjacency matrices of some (di)graphs and these are often called the **classes** of the scheme.

From here it is easy to see that the adjacency algebra of a graph is a (necessarily symmetric) association scheme if and only if it is closed under entrywise product and contains the all ones matrix (the latter of which is equivalent to the graph being connected and regular).

I am interested mostly in sufficient conditions on a graph for its adjacency algebra to be a symmetric association scheme. This is a little open ended, so maybe I should ask something more specific like the following:

If $G$ is a connected graph that is a **single** class in an association scheme, then is it true that the adjacency algebra of $G$ is an association scheme?

Two obvious necessary conditions on $G$ are that it must be connected and be the **union** of classes in a symmetric association scheme, but those are not sufficient. On the other hand it is sufficient for $G$ to be distance regular, but I was hoping for something a bit more general than that.