ANSWER TO REPORT 1

We thank the referee for his/her positive evaluation. We answer below to the requests for changes :

"1) Incomplete referencing "

We have added references to several reviews in the introduction, including the three mentionned by the referee: -C. Domb, Adv. Phys. 9 (1960) 149; -K.Y. Lin, Chinese J. Phys. 30 (1992) 287; -J. Strečka, M. Jaščur, Acta Phys. Slovaca 65 (2015) 235.

and in Table 1, we have added the year and reference where the critical temperature was first derived for each archimedean lattice, which led us to cite: -Alessandro Codello (2010), J. Phys. A: Math. Theor. 43, 385002; -I. Syozi (1972), in Phase Transitions and Critical Phenomena, Vol. 1, eds. Domb & Green; The reference 'J. Strečka (2006), Physics Letters A 349, 505–508' now appears in Sec.6, as it describe a star-triangle transformation between two lattices with different values of J on the links of two Archimedean lattices.

"2) In the early sections, the authors introduce several interrelated mathematical objects and symbols but do so purely textually. I recommend including a simple schematic diagram with a clear caption that visually illustrates these concepts. Such an illustration would greatly help readers navigate the formalism, especially those less familiar with this type of combinatorial approach, and would enhance the pedagogical clarity of the paper."

We have added a figure illustrating several notations (new Fig.2) used in sec. 2.1 and 2.2.

"3) Appendix references: Unless I overlooked something, the order in which appendices are cited in the main text appears inconsistent. I recommend checking and, if needed, adjusting the order or adding clarifying remarks."

We have reorganized the appendices so that they now appear in the same order as their references in the text.

"4) Ambiguous formulation regarding the sign of J: Below Equation (6), the text states “For F (resp. AF) interactions (J>0)...” This is ambiguous and potentially confusing. I suggest rewriting it more explicitly as: “ferromagnetic (J>0), antiferromagnetic (J<0).” 5) Figure 2 caption: The caption mentions “blue links,” but no blue edges are visible in the figure. Please check whether this is a leftover from an earlier version, or clarify what is meant. 6) Figure 5 caption: In the last sentence, the caption refers to lattice “A3334,” which does not appear elsewhere in the paper. This seems to be a typographical error; presumably it should read “A33344.” 7) Punctuation after equations: The manuscript contains several cases of inconsistent punctuation following displayed equations—sometimes a period or comma is missing, other times a full stop is used even though the sentence continues. I recommend reviewing the manuscript for consistent punctuation to improve clarity and readability. 8) Sentences beginning with variables: In many places, sentences begin with expressions such as f(T), c_v , etc., which visually appear to begin with a lowercase letter. While this is not technically incorrect in scientific writing, I suggest rephrasing such sentences (e.g., “The function f(T)...”) to ensure smoother readability and presentation."

Remarks 4 to 8 have been taken into account.

"9) Applicability to other lattices – open question: It would be valuable if the authors briefly commented in the conclusion on the potential applicability of their approach beyond Archimedean and Laves lattices. For example, can similar methods be applied to other periodic planar graphs with non-uniform vertex degree? Is constant coordination number a necessary condition? Even a short discussion on this point would benefit readers considering extensions of this work."

Similar calculations of the A and B quantities can be derived for planar graphs that are neither Archimedean nor Laves lattices, and that have several values of J on the links in a unit cell. We have precised all along the text the places where the results can easily be extended to more general lattices, and added in the conclusion: "The formulae in this article are directly applicable to any model on a planar lattice with a single type of link, for which the program given in Supp. Mat. [25] can be used. The formulae can be extended when several link types are present, as mentioned in Sec. 2.3 (examples of solutions in [13, 27])."

ANSWER TO REPORT 2

We thank the referee for his/her positive evaluation. We answer below to the requests for minor changes and to typos :

1- Minor changes

"a- References : while the main references are acknowledged and mentioned, a few references are still missing. As they are already mentioned in the other report, let me only add this one : - I.Syôzi, Statistics of Kagomé Lattice, Progress of Theoretical Physics, Volume 6, Issue 3, 1951, Pages 306–308"

This reference has been added, together with all the first references with the first mention of the critical temperature for the Archimedean lattices (in Tab. 1).

"b- Fig. 2 is missing some of the colors mentioned in its caption. I believe these would be extremely helpful to clarify the main text as the results are not completely trivial."

This has been corrected.

2- Typos etc.:

"a- It is unclear to me what u(\check{l}_f) means in Eq. (10). Is this correct ?"

There was some ambiguity in the notations, as \check{l_f} could refer to the check applied to the site l_f or the link l. We thank the referee for seeing it. To avoid this, we do no more use this notation for sites, and only use it for links. Moreover, a picture has been added illustrating the meaning of this notation (fig.2).

"b- The reference to Eq. (19) at the beginning of paragraph (2.3) seems a bit strange (albeit perhaps correct) since it is only presented much later on in the paper."

This has been corrected, and now the reference is to Eq. (14).

"c- Above Eq. (18), I suspect P_0(1) should either have an overline or should be set equal to -2^{2 \tilde{m}}. d- In Eq. (18) I have a doubt: should it not be n_{\nu} ?"

This two typos have been corrected.

"e- There is a typo in the residual entropy for A3636 in sec. 5.2 (0.0518 should be 0.5018), as stated correctly in the related table."

Thanks to the referee for seeing this typo. We have corrected it.

"- I believe "insure" should be "ensure" in both cases. - bottom of P. 6 "we fall is the situation"-> "we fall in the situation". "

This has been corrected.

4- Optional changes / questions for the authors' consideration:

"a- The angle \alpha is introduced quite early in the text but mostly used around Eq. (9). It could be helpful to the reader to only introduce it there and to explicitly mention it in Appendix A."

The definition of alpha has been moved just before eq. 9 (now Eq. 10) and a new figure now illustrates it (Fig 2).

"b- The appendices may be easier to navigate if presented in the same order as the text, or alternatively if the choice of their structure is explained in the introduction or at the very beginning of the appendices."

We have reorganized the appendices so that they now appear in the same order as their references in the text.

"c- The various geometrical quantities introduced in the first sections can be hard to follow. A figure summarizing their different roles might be helpful, or an "intuition" behind their impact on the calculations e.g. above Eq. 9 and e.g. in the very beginning of sec. 2.4 when mentioning Eq. (70)."

We have added a figure illustrating several notations (new Fig.2) used in sec. 2.1 and 2.2. In the beginning of Sec. 2.4, the reference to Eq. (70) of the appendix has been removed, and we now precise in this section what is needed for the following, refering to the appendix only for more details. The dual lattices are now defined here, with the associated figure moved here (Fig. 7 is now Fig. 3).

"c2 - The square torus is used to discuss periodic boundary conditions. In numerics, people sometimes use other constructions for tori to respect the lattice symmetries (e.g. in the triangular lattice case). May the author comment on whether this could affect their results ?"

We misused "square torus" and "square flat torus" instead of "flat torus". We fixed it. Actually our tori are always flat, but may be square or not.

"d- The authors comment in the conclusion about the appearance of geometric characteristic playing an important role in the statistical mechanics of the Ising model. I may have missed it : I am curious whether their systematic analysis (see Table 1 & 3) provides further insight on possible systematic correlations between e.g. the value of n_nu and A, B."

We have removed this remark as we didn't found any simple correlations between geometric characteristics and the values of A and B. This remark was more related to the polynomials properties, as for example the multiplicity of the factor (1+v), or n_v appearing in the ground state energy. But we added two remarks in the main text: we recall the already known correlation between the coordination number and the critical temperature: "On this figure, we recover the strong correlation (quasi-linear dependancy) between the coordination number $z$ and the critical temperature $T_c$, which is not specific to the Ising model" (p.12) and emphasize the lack of simple correlations between geometric characteristics and the values of A and B: " Despite strong variations in the parameters z, m, nl , n v (see Tab. 1), the coefficients A and B of the singularity for the F models show no simple correlations with them." (p.16)

e- The authors may consider summarizing in the conclusions the directions they already pointed out where their results may or may not be generalizable, possible references where this might have been considered, and perhaps comment on a few other possible directions. For instance, is there a way to use the combinatorial method to charaterize the magnetization or the correlations (despite the absence of a solution in the presence of a magnetic field), and for instance identify in the AF model the presence of a zero-temperature critical point? Would the approach be applicable to spin glasses (e.g. J.F. Valdés, W. Lebrecht, E.E. Vogel, Physica A: Statistical Mechanics and its Applications, Volume 391, (2012))? Would it be applicable to amorphous lattices ? These last few questions are mostly out of curiosity inspired by this work.

We have added further details in the text on the validity of the formulas for non-Archimedean lattices, and clarified them in the conclusion: "The formulae in this article are directly applicable to any model on a planar lattice with a single type of link, for which the program given in Supp. Mat. [25] can be used. The formulae can be extended when several link types are present, as mentioned in Sec. 2.3 (examples of solutions in [13, 27]). However, extension to magnetization calculations, or to disordered lattices are left for future work."