I have read [this paper][1]. 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 the guess is true, then how can we prove it? or some idea may go through.** 
 

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 There are plenty of cases of DRG that share the cyclotomic character value property, and a special one of them is called [Moore][2]. 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][3] 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][4] for a few days.


  [1]: https://pdf.sciencedirectassets.com/271586/1-s2.0-S0024379508X0014X/1-s2.0-S0024379507003631/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEGwaCXVzLWVhc3QtMSJGMEQCIDwpJohfCjrpnIr%2BOwh4g6onAx3wgUPRCMJGLLzRDSaWAiBhfqkFi%2FIbUPcmnjJSn2NqZnPld0F1qtj1iQ7mf2%2BMMyqyBQh0EAUaDDA1OTAwMzU0Njg2NSIMQ01QCo90Z04fAL0DKo8FMMyUUgysBFnky%2Fd3aSnpW3ZrGHUyjQBozhyYiSIyaX%2FRadXFa0Bg60%2FbN5tZ2oUit6pmqDxqou3oC4A4cMoK3%2FcrspG6lKOuTv%2BMEgsWwxw4Nu%2BAgYWfqaUX7HvKKC58e3cF5wajC0AEI4o3oHEA7hOUp7%2FpjpE1r6%2F3arnc15bO5V2gNw3eKtwrrmv2g91XKEwDCfxU1eA5JU0R4htfxdQ18688PK%2FfLwPzldfj9rDB7mCxqq9ULbB1md5%2BTjNizauAVi1GqjL2E00JSN9cgEsxVlQK%2BZHm2oT4ganZ0%2FWqu6P4chHr%2B1IIqDBDiH9b2JCZr%2BfZkJa2aKV2x15V03X8CA6nAj6PgX7nmEOZHEDCzwQp9b6dcFJEOSDBFPiI3ZvFLMORdhT8HVkcC4c5v60bwnhuSgPbUajoo%2BkBCf%2BJb5JoppTHttoP5gCBCPMhME5BON6mc6GmhcnZsyHmggh%2FPzXKuH4ru6mOeIBRS93iOHiQ4FWz1g9moV0Gl5QFLkn4Ir9OqZNfIzbPbya%2BzCgOmaRWx3%2FVZaBjTCzoLCLaMluYzyJnui%2FEB8RwpcRMFS5Id5WZDjiikBZTrLi8R5Uq6wDyT8tTBE4ywD%2BdfTtpvHmu2fNtCu7MmCQ%2Fxvc31ITgg5WFN6lbsCG2Lj6Ds04xPd6KuRGa%2BU6VZ5FvVyMMsH77xvBmHYlywC8AP5JV1pUGUwz8pVVHd1eMCKwOBI%2Fl1tGqscA456L8OKsTUxuTwG47io7nUCZIzvIs3hmLKJTi47Y1sG26k%2FH1UqEqNvqJqv3NbPvN%2FE5UuSy1jqoLRAYwjcQgLNibQSW8W7%2FKmBHHW8%2FTtN1Nq1tFyzMIR5D0Aj8ZajIlR%2B3oOQtUjjC8mZe2BjqyAW7dr%2F6okfiSvgIzeluVdmpBkhfcuJtzrcDHqZg0ZXKZtYdSthva0FZ%2BIPYcR5w1n1jsXHmIsRaH2pyuXgpt8HIGSbTcJvY4PeRNdmQIpcfqh4dcg1Ka%2Fx7uwpnpPLgyvGtOfIsQcMh9mmNTTDLHMVgvY6p7EyJeZd9s2GncsJF61Dp9SpTUlkFJ%2F0StRWTKhPLRraKiS5eSDJFqjPjECMwIVOdnwx9FOtgszj1xEf4Jeg0%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20240821T122614Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTY65P6BQHD%2F20240821%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=92fa5f84a203cdefab773273f98c85bcb87866efb922a8b9a81a05d7f63acece&hash=1338f2d574fc3874c554cf397c9064f1c1e34bec6bbd01930e927400ffa80760&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=S0024379507003631&tid=spdf-7a1dd128-a332-4801-9b0c-11901e15e082&sid=2df5481731e676478a88c9c53a8355f665afgxrqa&type=client&tsoh=d3d3LnNjaWVuY2VkaXJlY3QuY29t&ua=0e115d01510e59015d5d5d&rr=8b6a9c383fa8b312&cc=tw
  [2]: https://en.wikipedia.org/wiki/Moore_graph
  [3]: https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/on-moore-graphs/7493026A705CB99FF75E46AE47FB06AA
  [4]: https://math.stackexchange.com/questions/4961211/cyclotomic-character-value-property-for-distance-regular-graphs