Toroidal Spectral Drawing
Round 1
Reviewer 1 Report
Dear Authors,
Below please find my specific comments to your paper.
Page 1, line 6 (the 1st line of the Introduction) “… on the euclidean space …” + 28 more occurrences of the word “space” in the text: I recommend that you use the word “space” without any article. Space (in mathematics) functions like a proper noun and is usually written without any article, so say just “space” throughout the text but not “the space”.
Page 2, line 28: … can be found in many literature sources, …
Page 2, line 38 and Page 10, line 212: … fullerene …
Page 2, line 42: “In this section, we recall the spectral drawing algorithm …” Recall? A reference is necessary. My understanding that all Section 2 is a review of known results; if yes, give references; if no, state explicitly that such and such a theorem is new.
Titles of Sections 2 and 3: I think Section 2 should be entitled “Spectral Drawing” while Section 3 should remain to be “Spectral Drawing Revisited”.
Page 3, Theorem 1: Shouldn’t it be “U_k [instead of E(λ)] is ρ_σ invariant for any automorphism σ of X”?
Page 4, Example 1: I recommend that you mention around here or around Example 7 that a symmetric drawing of the graph in Example 7 (page 9) has been obtained in 4-space in the following two publications (check [L, Theorem 1]):
[L] Lawrencenko, S. Geometric realization of toroidal quadrangulations without hidden symmetries. Geombinatorics 24 (2014), no. 1, 11--20. URL: https://arxiv.org/pdf/1307.1054
[LS] Lavrenchenko, S. A.; Shchikanov, A. Yu. Euclidean realization of the product of cycles without hidden symmetries. (Russian) Sib. Èlektron. Mat. Izv. 12 (2015), 777--783.
URL: http://dx.doi.org/10.17377/semi.2015.12.063
Page 6, lines 117 and 140: … partially symmetric …
Page 6, line 146: “Fix a symmetry σ of X” Do you mean “Fix an automorphism σ of X”?
Page 6: The drawing algorithm, as described in Subsection 4.1, finds explicit coordinates of the vertices, but no coordinates appear in Examples 3-8.
Page 7, lines 159-160: Remark. Combining with the standard parametric equation of the torus in R^3, one can also obtain a drawing on R^3.
Page 7, lines 163-164: Example 3 says:
“… let σ = (123 · · · n) be an automorphism of K_n …”
but there are no labels of the vertices in Figures 3-5.
Page 7, Example 3: It ought to be explained how the planes W_1 and W_2 are used to obtain the pictures in Figures 3-5.
Page 8, Example 5: The notations “F_2” and “PGL_2 (F_2)” only appear in Example 5 and were not introduced before.
Page 9, Example 7: The notation “1 with a bar” is not introduced in the text.
Sincerely,
Reviewer
Author Response
Dear reviewer,
Thanks for your valuable comments. The paper has been modified according to the comments. The following is the point-by-point response.
Best regard.
--------------
Page 1, line 6 (the 1st line of the Introduction) “… on the euclidean space …” + 28 more occurrences of the word “space” in the text: I recommend that you use the word “space” without any article. Space (in mathematics) functions like a proper noun and is usually written without any article, so say just “space” throughout the text but not “the space”.
Response: All of "the euclidean space" in the paper has been changed to "Euclidean space". Other terminologies like "the eigenspace", "the subspace" would be kept.
Page 2, line 28: … can be found in many literature sources, …
Response: the typo has been revised.
Page 2, line 38 and Page 10, line 212: … fullerene …
Response: the typo has been revised.
Page 2, line 42: “In this section, we recall the spectral drawing algorithm …” Recall? A reference is necessary. My understanding that all Section 2 is a review of known results; if yes, give references; if no, state explicitly that such and such a theorem is new.
Response: the reference is added.
Titles of Sections 2 and 3: I think Section 2 should be entitled “Spectral Drawing” while Section 3 should remain to be “Spectral Drawing Revisited”.
Response: the sections have been renamed.
Page 3, Theorem 1: Shouldn’t it be “U_k [instead of E(λ)] is ρ_σ invariant for any automorphism σ of X”?
Response: the typo has been revised.
Page 4, Example 1: I recommend that you mention around here or around Example 7 that a symmetric drawing of the graph in Example 7 (page 9) has been obtained in 4-space in the following two publications (check [L, Theorem 1]):
[L] Lawrencenko, S. Geometric realization of toroidal quadrangulations without hidden symmetries. Geombinatorics 24 (2014), no. 1, 11--20. URL: https://arxiv.org/pdf/1307.1054
[LS] Lavrenchenko, S. A.; Shchikanov, A. Yu. Euclidean realization of the product of cycles without hidden symmetries. (Russian) Sib. Èlektron. Mat. Izv. 12 (2015), 777--783.
URL: http://dx.doi.org/10.17377/semi.2015.12.063
Response: Two suggested citations have been added in the end of Example 7.
Page 6, lines 117 and 140: … partially symmetric …
Response: the typo has been revised.
Page 6, line 146: “Fix a symmetry σ of X” Do you mean “Fix an automorphism σ of X”?
Response: the typo has been revised.
Page 6: The drawing algorithm, as described in Subsection 4.1, finds explicit coordinates of the vertices, but no coordinates appear in Examples 3-8.
Response: Instead of giving explicit coordinates, we think it is more clear to draw these vertices directly on (R/Z)^2
Page 7, lines 159-160: Remark. Combining with the standard parametric equation of the torus in R^3, one can also obtain a drawing on R^3.
Response: the typo has been revised.
Page 7, lines 163-164: Example 3 says:
“… let σ = (123 · · · n) be an automorphism of K_n …”
but there are no labels of the vertices in Figures 3-5.
Response: the labels are added to Figures 3-5.
Page 7, Example 3: It ought to be explained how the planes W_1 and W_2 are used to obtain the pictures in Figures 3-5.
Response: It is explained in the algorithm how the planes W_1 and W_2 are used.
Page 8, Example 5: The notations “F_2” and “PGL_2 (F_2)” only appear in Example 5 and were not introduced before.
Response: the explanation of notations F_2“and PGL_2 (F_2)” have been added.
Page 9, Example 7: The notation “1 with a bar” is not introduced in the text.
Response: We have changed the notation 1 with a bar by 1.
Reviewer 2 Report
The spectral method is used to draw a graph onto a torus, and we present a deterministic drawing algorithm for doing so. In contrast to the conventional spectral drawing algorithm, which is effective only for planar graphs, our method successfully derives a good embedding for the majority of well-known toroidal graphs onto the torus.
The following concern must be addressed in the revised manuscript:
- What is the rationale behind considering the proposed approach? The context and applications are not clear to me.
- The coherence in the paper is missing. Authors are advised to read the paper very carefully.
- Write the main contribution of this paper in bullet format.
- The research gap is not clearly stated in the paper. Thus it is very difficult to understand,
Author Response
Dear reviwer,
Thanks for your valuable comments. we have revised the introduction according to your comment. Especially, we highlight the contribution of the paper as follows.
"There are several studies of drawing graphs on a torus \cite{KNS, KNS2,CDF}. However, these approaches do not give explicit drawings directly and they also require the extra structure of graphs, namely the choice of the set of "faces" (or so-called rotation systems).
The main contribution of the paper is to give a deterministic drawing algorithm for highly symmetric toroidal graphs without using any extra structure."
Best regard
Round 2
Reviewer 1 Report
Dear Authors,
Some final corrections (listed below) still have to be done.
Page 8, lines 179-180: I believe this sentence should be as follows: "The automorphism group of X contains the group PGL_2(F_2) as an index two subgroup of order 168."
Page 10, line 209: I believe this should be as follows: "Set σ(x) = x + (1, 2)."
Page 12, reference [4]: The surname of the second author should be "Shchikanov".
Page 11, lines 212-213: Replace "in 4-sphere" with either "in 4-space" or "in the 3-sphere".
Regards,
Editor
Author Response
Dear Reviewers,
Thanks for your careful reading and valuable comments. We have revised the paper according to your suggestions.
Best regard
------------------------------------
Page 8, lines 179-180: I believe this sentence should be as follows: "The automorphism group of X contains the group PGL_2(F_2) as an index two subgroup of order 168."
Respons: The typo is fixed.
Page 10, line 209: I believe this should be as follows: "Set σ(x) = x + (1, 2)."
Respons: The typo is fixed.
Page 12, reference [4]: The surname of the second author should be "Shchikanov".
Respons: The typo is fixed.
Page 11, lines 212-213: Replace "in 4-sphere" with either "in 4-space" or "in the 3-sphere".
Respons: The typo is fixed.
Reviewer 2 Report
Thank you for improving the revised manuscript as per the instruction. However, I am suggesting authors to --
include reproducibility of the work and must be mentioned how to improve the work.
The clarity is missing in the paper. Authors are requested to read the paper very carefully and try to improve it further.
Author Response
Dear Reviewers,
Thanks for your valuable comments. For the concern of reproducibility of the work, our proposed algorithm is very simple and explicit and we also applied the algorithm in detail in many examples. Therefore, it should be easy to obtain the result of all examples by readers. On the other hand, we add a section of future works to say a few words about how to improve the work. Finally, we feel sorry that the whole paper is not clear enough to you. We have done our best.
Best regard.
