On the Laplacian, the Kirchhoff Index, and the Number of Spanning Trees of the Linear Pentagonal Derivation Chain

Round 1

Reviewer 1 Report

At the end of March 2022, this paper was submitted to Entropy (entropy-1670379).

After negative comments of reviewers, the paper was rejected by the academic editor.

The authors have not changed anything and submit the same paper to Axioms (axioms-1730592).

I do not recommend this paper for publication.

Author Response

Dear Editor and Referees,

We are very pleased to learn from your letter about revision for our manuscript entitled ``On Laplacian, Kirchhoff index, and spanning tree number of the linear
pentagonal derivation chain''. Thank you for your helpful comments and advice concerning our manuscript. Those comments are all valuable and very helpful for revising and improving our manuscript, as well as the important guiding significance to our researches. We have studied comments carefully and have made correction which we hope meet with approval. Revised portions are marked in red in the manuscript.

Please see the attached pdf for a point-by-point response to the reviewer’s comments.

Author Response File: Author Response.pdf

Reviewer 2 Report

In the present article, the authors deal with the linear pentagonal chain QP_n. First, they explicitly obtain the Laplacian spectrum of QP_n. Then, by using the decomposition theorem for the Laplacian characteristic polynomial, they present the explicit closed- form formulae for the Kirchhoff index and the number of spanning tree of QP_n. Finally, it is shown that the ratio of the Kirchhoff index and the Wiener index of a linear pentagonal derivation chains QP_n approaches 1/2 as n tends to infinity. This is an interesting observation. The paper is well-written. 

In page 7, line 1, the equation number appears as (??). The TeX file should be fixed for avoiding this.

Author Response

Dear Editor and Referees,

We are very pleased to learn from your letter about revision for our manuscript entitled ``On Laplacian, Kirchhoff index, and spanning tree number of the linear
pentagonal derivation chain''. Thank you for your helpful comments and advice concerning our manuscript. Those comments are all valuable and very helpful for revising and improving our manuscript, as well as the important guiding significance to our researches. We have studied comments carefully and have made correction which we hope meet with approval. Revised portions are marked in red in the manuscript.

Please see the attached pdf for a point-by-point response to the reviewer’s comments.

Author Response File: Author Response.pdf

Reviewer 3 Report

    In this paper the authors investigate the Kirchhoff index and spanning tree number of the linear pentagonal derivation chain. They prove that the Kirchhoff index is almost one half of the Wiener index of a linear pentagonal derivation chain QPn. The results appear to be connected. Before it is ready for publication there are some comments that are listed below that should be addressed.
    
    Comments:
         The paper includes a reference to Vieta's theorem, but it is not clear which theorem they are referring to. This should be clarifed with a reference included.
    Tables 1 and 2 provide numerical values, but it is not clear what this adds to the paper. The necessity including these tables should be addressed.
        
Minor Comments / Proper use of English / Typographical errors
         - Abstract, second to last line: "one half to Wiener index" -> "one half of the Wiener index"
    - Abstract, last line: "chains" -> "chain"
    - Page 2, line 5: "called Wiener index" -> "called the Wiener index"
    - Page 2, second to last line: "The study on the hexagonal systems have aroused widespread concern" -> "The study of hexagonal systems have attracted interest"
    - Page 3, last sentence of Section 1: "Finally, it is wonderful to find " -> "Finally, it is inteesting to find"
    - Page 4, statement of Lemma 2.3: "M1 and M4 invertible" -> "M1 and M4 being invertible"
    - Page 4, second sentence of the statement of Lemma 2.3: "is called" -> "are called"
    - Page 6, Statement of Theorem 3.1: "chains with lenth n" -> "chain with length n".
    - Page 7, line 1: There is a missing equation reference
    - Page 9, Two lines after the matrix. There appears to be an extra space after the word "respectively".
    - Page 14: There should be a period at the end of the last equation.
    - Page 16, line 5: I am not sure what is meant by "Therefore, be Claims 2,3."
    - Page 16, Statement of Theorem 3.3: "chains" -> "chain"
    - Page 17, first sentence: "then added all together" -> "then add them all together"
    

Author Response

Dear Editor and Referees,

We are very pleased to learn from your letter about revision for our manuscript entitled ``On Laplacian, Kirchhoff index, and spanning tree number of the linear
pentagonal derivation chain''. Thank you for your helpful comments and advice concerning our manuscript. Those comments are all valuable and very helpful for revising and improving our manuscript, as well as the important guiding significance to our researches. We have studied comments carefully and have made correction which we hope meet with approval. Revised portions are marked in red in the manuscript.

Please see the attached pdf for a point-by-point response to the reviewer’s comments.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The authors have improved their paper. The paper can be publiched.