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Reviewer #1: In "Recognizing Distance-Count Matrices", the authors investigate so-called Distance Count Matrices (DCMs) associated to directed and undirected graphs. Such a matrix has at position (i,k) the number of nodes that sit at distance k to the node i, and thus (directly or indirectly) encodes many centrality properties of the graph.

In the first part of their paper, they show that deciding whether a given matrix is the DCM of an undirected graph is strongly NP-complete. This is a highly relevant result to the study of centrality, as it shows that it is not feasible to construct counterexamples to conjectures about DCMs, by first building a matrix with(out) required properties, and by then checking whether this matrix is the DCM of a graph. I believe this result about NP-completeness to have mathematical value in its own right, too.

In the second part of their work, they offer an alternative way of building DCMs, by showing that a great number of network products have DCMs that can be constructed from the DCMs of the constituents. By giving counter-examples, they prove that this is not the case for certain other products.

In my opinion, the paper is of very high quality. From what I can tell, there are (almost) no mistakes, the text is very complete and reads nicely, and above all it is highly original. Especially the idea for the proof of NP-completeness is very original, and frankly fun to read. I would only like to see most of the comments below addressed, but then I think it should be published.

Main:

1) Subsection 5.1 was at times very unclear to me. Definition 5.1 defines a TPP to be a sequence of numbers with certain properties, but then Theorem 5.3 introduces additional properties to these sequences. Often times throughout the text it is not clear if what is being referred to is sequences with some or with all of these properties. The proof of Theorem 5.3, at the start, just refers to "the problem" for instance.

2) In subsection 6.4, it's not always clear to me if we are considering directed or undirected graphs, especially around Proposition 6.30. Much more importantly though, I believe Proposition 6.30 to be wrong regardless. Consider the undirected chain graph with 8 nodes, G = P_8, consisting of the nodes 1 to 8, and the undirected 4-cycle, looking like A-B-C-D-, H = C4. A surjective graph homomorphism is given by

1->A, 2->B, 3->C, 4->D, 5->A, 6->B, 7->C, 8->D,

I believe.

Now, m_{G,2}(1) = 3 < m_{H,2}(1) = 4, if I am not mistaken. At any rate, I could not follow the proof of Proposition 6.30 very well. I believe the problem is that the restriction of the homomorphism from M_{G,k} to M_{H,k} may not be surjective itself, as the above counterexample shows. Clearly this part of the article is not the most important though.

Minor:

1) The authors might consider giving a very basic example of network centrality already in the introduction. The reader less familiar with them now starts off somewhat lost, and has to wait to later sections to catch up.

2) On page 1, "and called its distance-count matrix (DCM)" -> " and is called its distance-count matrix (DCM)"

3) On page 4, the paragraph starting with "Observe that" seems to be the only place where the subscript G is dropped from M_{G,k}(x) and N_{G,k}(x). I think this may confuse the reader.

4) On page 4, "one special instance of linear centralities are...", these are even strictly linear, right?

5) On page 5, the degree of P_g^i is at most |N_G|-1, isn't it? See also Thr. 6.14 (though no conclusions are wrong).

6) On page 5, "the standard reference is". I would put this more neutral; it may not be anymore in 10 years.

7) Proposition 4.7, I would briefly remind the reader that n = #G

8) Proposition 4.7,(i). Here m_{p}(n-1) should probably be m_{n-1}(i), correct?

9) The authors are not always very clear on if directed graphs can have self loops in this work or not. It seems mostly irrelevant for distance count issues, but Theorem 4.6 seems to me to only be true if self loops are allowed (because we allow d_1 = p), but Theorem 4.13 seems to only work (precisely as formulated) when we do not allow self loops. I think it's best to communicate to the reader what the convention will be early on.

10) On page 8, in the inequality under "Therefore", the j_k should probably just be a j, right?

11) On page 9, proof of Theorem 5.4, "the partition" should technically be "a partition", right? I see no reason to assume the solution, if it exists, is unique.

12) On page 10, "Finally, x_i has two vertices at distance three" --> "four" (?)

13) On page 11, "a_h + 1 \leq a_k + 2, that is a_h - a_k \leq 2" --> "that is a_h - a_k \leq 1"

14) In figure 6, aren't many arrows missing? Like from (4,B) to (3,C)?

15) On page 15, the Equation under "Recalling the definition of exponential-decay centrality, we have that", ((x,y)) ---> (x,y) (too many brackets)

16) Lemma 6.17, "Propositon 5.12 of ..." --> "Proposition"

17) In Theorem 6.18, in the case where n_{G,1}(x) = 0, why would n((x,y)) be given by n_{G,k}(x)|N_H| + n_{H,k}(y) for k>0, and not just by n_{H,k}(y)? after all, isn't it so that n_{G,1}(x) = 0 implies n_{G,k}(x) = 0 for all k>0? Obviously nothing is wrong, but it might confuse the reader. After all, the fact that n_{G,k}(x) = 0 for all k>0, also doesn't seem to be noticed in the proof.

18) On page 19, "the Kronecker product of two graphs is connected if and only if ...." this should be the Kronecker product of two connected graphs, I think.

19) On page 22, In the proof of theorem 6.27, in the second line of the paragraph starting with "Moving to node (x,y)..." we have "Additionally, in K_{left arrow} in k steps ... ", I believe the arrow has to be reversed here.

20) On page 23, the line above Proposition 6.30, the n_{H,3} and n_{G,3} should have an argument x, (it could be stated "for some x".)

21) On page 23, line 607, I believe n_{G,1}(3) = 1, not 0.

Reviewer #2: * Results are clear, but dependent on previously proved results, in new interpretations. Especially the results on distance degree sequences. For graph products also many results exist, the authors have proved them in different ways. Instead the authors are recommended to refer to the literature on graph products and distance degree sequences, to avoid repetitions. Some references that should have been checked first:

1. “Products of Distance Degree Regular and Distance Degree Injective graphs”, Medha Itagi Huilgol, Rajeshwari M., Syed AsifUlla S., Journal of Discrete Mathematical Sciences and Cryptography, Vol.15 (2012), No. 4&5, 303-314.

2. “New results on distance degree sequences of graphs”, Medha Itagi Huilgol, V. Sriram, Malaya Journal of Mathematik, 7(2) (2019), 345-352.

3. . “Tensor and Cartesian products for nanotori, nanotubes and zigzag polyhex nanotubes and their applications to 13C NMR Spectroscopy”, Medha Itagi Huilgol, V. Sriram, K. Balasubramanian, Molecular Physics, 118(16) (2020), 1-25.

DOI: 00.1080/00268976.1817594

4. “Distance degree vector and scalar sequences of corona and lexicographic products of graphs with applications to dynamic NMR and dynamics of nonrigid molecules and proteins”, Medha Itagi Huilgol, Divya B., K. Balasubramanian, Theoretical Chemistry Accounts, 140, 25 (2021). https://doi.org/10.1007/s00214-021-02719-y

* The authors are recommended to do a thorough literature survey and improve the results.

With corrections incorporated, the authors may resubmit.

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Reviewer #1: No

Reviewer #2: No

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Recognizing Distance-Count Matrices

PONE-D-26-09672R1

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