I have read this paper. So, I am just thinking about if the following guess is true:
GUESS: Any Distance-regular graph (DRG) has cyclotomic character value property (which means the eigenvalues of a commutative association scheme have to be cyclotomic algebraic integers) over rational number field $\mathbb Q$ for sufficiently large diameter $D$, at least $D\geqslant 5$.
If my guess is true, then how can we prove it? or some idea might go through.
There are plenty of cases of DRG that share the cyclotomic character value property, and a special one of them is called Moore. Here is the proof:
First, if the maximal degree $\Delta=1$ then the diameter $D=1$. So, we just focus on the maximal degree $\Delta\geqslant 2$. There is no Moore graph of maximal degree $\Delta\geqslant 3$ and diameter $D\geqslant 3.$ This was shown by Damerell on an application of Distance-regular graphs to the classification of Moore graph. For $D\geqslant 3$ and $\Delta = 2$, Moore graphs are just cycle and their eigenvalues are all of the form $2\cos\frac{2k\pi}{n}$ and, of course, these eigenvalues are all in some cyclotomic field $\mathbb{Q}(e^{\frac{2\pi i}{m}})$ for some integer $m$. Moreover, it is not hard to see that any Distance-regular graph of diameter $D\leqslant 2$ has cyclotomic character value property. So, our conclusion now follows.
In Theorem $7.11$ of the book “Algebraic Combinatorics I Association Schemes,” written by Bannai and Ito published in $1984,$ they proved in particular that any ($P$ and $Q$)-polynomial association scheme also shares the cyclotomic character value property when its diameter is at least $34$. So, this implies that if we assume DRG is a ($P$ and $Q$)- polynomial association scheme, then everything is fine if its diameter is at least $34$. Actually, I have asked the question in Math. Stack Exchange for a few days.
