Response to comments of the referee:

We appreciate the comments made by the referee, and have revised our paper accordingly.
Detailed responses to all comments are given below (after each comment).

Author: EDWIN R. VAN DAM, JACK H. KOOLEN, AND HAJIME TANAKA
Title: DISTANCE-REGULAR GRAPHS

General Comments

1. Since the classification of distance-regular graphs with unbounded diameter or large
diameter is the main goal, it is convenient for the reader to have their list.
>We have extended Section 3 and now explicitly mention all known families with unbounded diameter.

2. Since geometric distance-regular graphs seem to be as important as Q-polynomial
distance-regular graphs, please explain it somewhere.
>We moved the definition of geometric distance-regular graphs from section 4.5 to a new section (Section 2.8) in the
introduction.

3. As for the reference [77], sometimes it is cited
as ‘BCN’ [77]. Is there a policy?
>Usually we refer to [77] without 'BCN'. We use 'BCN' sometimes to emphasize the status of 'BCN'.

Typos and Suggestions In the following this reviewer lists misprints and comments.
Some of them may be a matter of taste.

1. p.3 `.8 from the bottom: ‘Johnson graphs’ −! ‘the Johnson graphs’, or other
construction.
>DONE

2. p.5 `.1 from the bottom: Add ‘if v > 1’.
>DONE

3. p.15 `.13: ‘isomorphic to "2).’ −! ‘isomorphic to "2, a Johnson graph).’
>DONE

4. p.21 `.16: Is a complete graph Kn strongly regular? This definition of strongly
regular graphs includes Kn and hence ‘dimameter two’ is not valid.
>DONE

5. p.21 `.14 and 13 from the bottom: When there is a triangle and the local graph is
connected, the definition of geometric girth should be more carefully written, if you
still want to define geometric girth of such a graph.
>The geometric girth for such a graph is defined in the next
sentence. We reformulated these sentences slightly for clarity.

6. p.22 `.11 from the bottom: The definition of the i-thin condition
is slightly di↵erent from the one defined by Terwilliger,
especially when i is large.
>We added a footnote where we mention this.

7. p.24 `.4 from the bottom: It is better to have a comma before ‘where’.
>DONE

8. p.27 `.25: Add ‘a kite of length 2’.
>DONE

9. p.28 `.14 from the bottom: Is it correct to say ‘the maximum integer t for which Y
is a t-design? The indices i with Ei" = 0 may have a leap.
>DONE

10. p.32 `.2 from the bottom: Delete one period.
>DONE

11. p.43 `.5: Collins did not discuss the numerical girth, the result on the geometric
girth is in [578].
>Indeed, but the result on the numerical girth follows easily.

12. p.43 `.9: ‘a regular near 2D-gon’ −! ‘a regular near polygon’. The paper [580]
involves an error.
>DONE

13. p.54 `.13–14: This is a part of Corollary 7.7.
>We added: see also Corollary 7.7.

14. p.44 `.20–23: This part is a repetition of the statements in p.15.
>(It is p.54 instead of p.44 (proof of 9.9), and p.51 instead of p.15 (prop.
9.1))
>We added a sentence: Using Proposition 9.1 we now obtain the
following result.

15. p.56 `.1: ‘intersection parameters’ −!
‘intersection numbers’ or ‘intersection array’.
>DONE
>We also changed "parameters" in Section 3.2.3 to "intersection array".
>Similar in some problems and a footnote

16. p.59: The essential part of spectral excess
theorems is presented as the thin condition of modules of the
Terwilliger algebra by Terwilliger in [615]. See also [579].
>We added footnotes to indicate this.

17. p.75 `.6 from the bottom: Something is wrong. When " has a Delsarte
clique, then the multiplicity of the smallest eigenvalue would be
s.
>We changed the proposition to characterize the case of equality.

18. p.89, Problem 5: The problem setting is similar to Problem 4, express in a similar
way. ‘Complete this classification’ should be changed.
>DONE

19. p.90, Problem 13 (ii): ‘Determine all j such that’ −! ‘Determine j 6= 0, i such
that’.
>DONE

20. p.94, Problem 54: Do you want a condition besides the tight condition of the vector
in E⇤1W?
>We removed this problem

21. p.96, Problem 66: ‘For s = 0’ −! ‘For s = 1’, and ‘For s = 1’ −! ‘For s = 2’.
>DONE. In addition, Yamazaki's result is for $s \geq 3$.

22. p.96, Problems 71, 73, 74: Should these be included as a list of problems?
>We reorganized the problem section a little, so now the mentioned problems fit
better.