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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 the guess is true, then how can we prove it? or some idea may 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.

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.

I have read this paper. So, I am just thinking about if 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 statement is true, then how can we prove it? or some idea may 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.

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 the guess is true, then how can we prove it? or some idea may 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.

I have read this paper. So, I am just thinking about if 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 $D\geqslant 5$.

Is there any reference or proof for the question? Or some idea may 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.

I have read this paper. So, I am just thinking about if 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 statement is true, then how can we prove it? or some idea may 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.