ISI-Equienergetic Graphs
Round 1
Reviewer 1 Report
The references should appear in order in the text
The novelty of the paper is to be clearly stated.
The construct pairs of connected, ISI-noncospectral, ISI-equienergetic graphs is to be described with more details.
The descriptions of the proofs are to be extended, for a better understanding for the readers.
What are the specific applications of the ISI-equienergetic graphs.
The term ‘’ vertices ’’ is to be defined.
The paper is to be checked against misprints.
Author Response
Dear reviewers,
Thank you very much for your kindness and we would like to
thank you for your useful comments and valuable suggestions. I
have carefully read your reports and revised the manuscript
according to your suggestions.
All problems mentioned in the reports are answered as follows.
1. The references should appear in order in the text.
Answer: I have reordered the references according your your helpful suggestion.
2. The novelty of the paper is to be clearly stated.
Answer: Thank you very much for your good suggestion. I have added some new content in the first paragraph to explain the novelty of this paper.
3. The construct pairs of connected, ISI-noncospectral, ISI-equienergetic graphs is to be described with more details.
Answer: I have added some new content at the end of the second section to describe the constructing of pairs of connected, ISI-noncospectral, ISI-equienergetic graphs.
4. The descriptions of the proofs are to be extended, for a better understanding for the readers..
Answer: I have extended the descriptions of the proofs for a better understanding for the readers according to your suggestion.
5. What are the specific applications of the ISI-equienergetic graphs.
Answer: $\varepsilon(G)$ and $\varepsilon_{isi}(G)$ of graphs are closely linked,
we can determine $\varepsilon(G)$ by means of $\varepsilon_{isi}(G)$. consequently, $\varepsilon_{isi}(G)$
study is not only of theoretical meaning, but also of realistic value. Our results of ISI-equienergetic graphs
can quickly get the ISI-energy of the ISI-noncospectral, ISI-equienergetic graphs, which can greatly reduce the workload of
calculating the ISI-energy of graphs.
6. The term ``vertices" is to be defined.
Answer: I have defined the term ``vertices" in the first paragraph.
7. The paper is to be checked against misprints.
Answer: I have checked against misprints according to your good suggestions.
8. The paper misses an illustration of the overall theory described in Section 2. It would have been useful to include as Section 3 a few examples illustrating the applicability of the proposed criterion for recognizing the two ISI-noncospectral ISI-equienergetic graphs.
Answer: I have added an illustration of the overall theory described in Section 2 according to your good suggestions. And I give some examples to illustrate the applicability of the proposed criterion for recognizing the two ISI-noncospectral ISI-equienergetic graphs.
9. References [2] and [14] are not cited in the manuscript. Please, insert them in the text.
Answer: Thank you very much for your good suggestions. I have cited References [2] and [14] in the manuscript.
Reviewer 2 Report
ISI-energy is one of the zillion currently studied graph energies based on matrices whose elements are functions of vertex degrees. The paper offers constructions of pairs of non-isomorphic graphs, having equal ISI-energy. I did not see any error in the mathematical considerations, but I also see no purpose of constructing ISI-equienergetic graphs. Since purposeless papers of such quality often appear in journals such as "Axioms", it would be unfair to refuse the publication of this particular article.
Author Response
Dear reviewers,
Thank you very much for your kindness and we would like to
thank you for your useful comments and valuable suggestions. I
have carefully read your reports and revised the manuscript
according to your suggestions.
All problems mentioned in the reports are answered as follows.
1. The references should appear in order in the text.
Answer: I have reordered the references according your your helpful suggestion.
2. The novelty of the paper is to be clearly stated.
Answer: Thank you very much for your good suggestion. I have added some new content in the first paragraph to explain the novelty of this paper.
3. The construct pairs of connected, ISI-noncospectral, ISI-equienergetic graphs is to be described with more details.
Answer: I have added some new content at the end of the second section to describe the constructing of pairs of connected, ISI-noncospectral, ISI-equienergetic graphs.
4. The descriptions of the proofs are to be extended, for a better understanding for the readers..
Answer: I have extended the descriptions of the proofs for a better understanding for the readers according to your suggestion.
5. What are the specific applications of the ISI-equienergetic graphs.
Answer: $\varepsilon(G)$ and $\varepsilon_{isi}(G)$ of graphs are closely linked,
we can determine $\varepsilon(G)$ by means of $\varepsilon_{isi}(G)$. consequently, $\varepsilon_{isi}(G)$
study is not only of theoretical meaning, but also of realistic value. Our results of ISI-equienergetic graphs
can quickly get the ISI-energy of the ISI-noncospectral, ISI-equienergetic graphs, which can greatly reduce the workload of
calculating the ISI-energy of graphs.
6. The term ``vertices" is to be defined.
Answer: I have defined the term ``vertices" in the first paragraph.
7. The paper is to be checked against misprints.
Answer: I have checked against misprints according to your good suggestions.
8. The paper misses an illustration of the overall theory described in Section 2. It would have been useful to include as Section 3 a few examples illustrating the applicability of the proposed criterion for recognizing the two ISI-noncospectral ISI-equienergetic graphs.
Answer: I have added an illustration of the overall theory described in Section 2 according to your good suggestions. And I give some examples to illustrate the applicability of the proposed criterion for recognizing the two ISI-noncospectral ISI-equienergetic graphs.
9. References [2] and [14] are not cited in the manuscript. Please, insert them in the text.
Answer: Thank you very much for your good suggestions. I have cited References [2] and [14] in the manuscript.
Reviewer 3 Report
1. The manuscript treats an important and actual problem for obtaining ISI-noncospectral graphs on same number of vertices having equal ISI-energy, because the ISI-energy for each pair cospectral graphs on same number of vertices is equal.
2. Title and abstract: The title is appropriate for the content of the manuscript. The abstract summarizes the essential information of the paper precisely and clearly.
3. The problem considered and the basic theory about it are defined in the introduction.
4. In Section 2 a theoretical base of graph theory and matrices analysis (especially about eigenvalues) for finding ISI-noncospectral ISI-equienergetic graphs is described in detail. Seven theorems and six Lemmas and some Corollaries are written. Some of these already exist in the literature and others have been formulated by the authors. As a result, the end of this section makes clear the requirements under which two ISI-noncospectral graphs has the same ISI-energy.
5. I notice no plagiarism in the article. All material is the personal work of the authors. References used are appropriately cited.
6. Conclusion is appropriate formulated.
Some remarks and recommendations:
1. The paper misses an illustration of the overall theory described in Section 2. It would have been useful to include as Section 3 a few examples illustrating the applicability of the proposed criterion for recognizing the two ISI-noncospectral ISI-equienergetic graphs.
2. References [2] and [14] are not cited in the manuscript. Please, insert them in the text!
Author Response
Dear reviewers,
Thank you very much for your kindness and we would like to
thank you for your useful comments and valuable suggestions. I
have carefully read your reports and revised the manuscript
according to your suggestions.
All problems mentioned in the reports are answered as follows.
1. The references should appear in order in the text.
Answer: I have reordered the references according your your helpful suggestion.
2. The novelty of the paper is to be clearly stated.
Answer: Thank you very much for your good suggestion. I have added some new content in the first paragraph to explain the novelty of this paper.
3. The construct pairs of connected, ISI-noncospectral, ISI-equienergetic graphs is to be described with more details.
Answer: I have added some new content at the end of the second section to describe the constructing of pairs of connected, ISI-noncospectral, ISI-equienergetic graphs.
4. The descriptions of the proofs are to be extended, for a better understanding for the readers..
Answer: I have extended the descriptions of the proofs for a better understanding for the readers according to your suggestion.
5. What are the specific applications of the ISI-equienergetic graphs.
Answer: $\varepsilon(G)$ and $\varepsilon_{isi}(G)$ of graphs are closely linked,
we can determine $\varepsilon(G)$ by means of $\varepsilon_{isi}(G)$. consequently, $\varepsilon_{isi}(G)$
study is not only of theoretical meaning, but also of realistic value. Our results of ISI-equienergetic graphs
can quickly get the ISI-energy of the ISI-noncospectral, ISI-equienergetic graphs, which can greatly reduce the workload of
calculating the ISI-energy of graphs.
6. The term ``vertices" is to be defined.
Answer: I have defined the term ``vertices" in the first paragraph.
7. The paper is to be checked against misprints.
Answer: I have checked against misprints according to your good suggestions.
8. The paper misses an illustration of the overall theory described in Section 2. It would have been useful to include as Section 3 a few examples illustrating the applicability of the proposed criterion for recognizing the two ISI-noncospectral ISI-equienergetic graphs.
Answer: I have added an illustration of the overall theory described in Section 2 according to your good suggestions. And I give some examples to illustrate the applicability of the proposed criterion for recognizing the two ISI-noncospectral ISI-equienergetic graphs.
9. References [2] and [14] are not cited in the manuscript. Please, insert them in the text.
Answer: Thank you very much for your good suggestions. I have cited References [2] and [14] in the manuscript.
