A Note on Distance-Based Entropy of Dendrimers

Round 1

Reviewer 1 Report

This is a part of the authors' ongoing study of the mathematical description of chemical structures using the information-entropy-based structural descriptors. In the present paper, the authors provide the theorems on the properties of the graphs relating the dendrimeric (branched) molecules. I find the results novel and interesting to a narrow audience of mathematical chemists.

The manuscript is publishable after English grammar/style editing.

I were not able to select all the grammar mistakes through the text. But  even the abstract contains a lot of them.

For example:

'This paper introduces a variant entropy measures based on vertex eccentricity, and 2 applies it to all graphs representing isomers of octane' should be 'This paper introduces a variant OF entropy measures based on vertex eccentricity, and applies it to all graphs representing THE isomers of octane'.

Taking account of vertex degree as well 3 (degree-ecc-entropy), we find a high correlation with the acentric factor of octane isomers.

should be

Taking INTO account the vertex degree and degree-ecc-entropy, we find a good correlation with the acentric factor of the octane isomers.

and this list can be extended.

Author Response

I polished the  English grammar.

Reviewer 2 Report

The purpose of this paper under review is to introduce a degree-distance variant entropy and find a high correlation with the acentric factor of octane isomers. Also, the degree-ecc-entropy for three classes of dendrimer graphs are determined. The paper is interesting and it addresses an important topic. It contains some meaningful discussions. The derivations appear to be correct, and the organization of the paper is smooth and well planned. Based on the novel idea and interesting mathematical analysis, I recommend that the paper be accepted for publication by Axioms. The only suggestions I have are the following two points:

1. I can understand that you have to include your previous research works and studies in this manuscript. However, authors include a huge number of publications. It needs a clear justification (and maybe a reduction of self-cites). Self-citation is not the most proper procedure unless it is very well justified.

2. I suggest to cite the following recent published contributions in chemical graph theory:

R. Škrekovski, D. Dimitrov, J. M. Zhong, H. L. Wu, W. Gao, Remarks on multiplicative atom-bond connectivity index, IEEE Access, 2019, 7(1): 76806-76811.

A. Aslam, M. F. Nadeem, Z. Zahid, S. Zafar, W. Gao, Computing certain topological indices of the line graphs of subdivision graphs of some rooted product graphs, Mathematics, 2019, 7(5), 393; https://doi.org/10.3390/math7050393.

A. Aslam, S. Ahmad, M. A. Binyamin, W. Gao, Calculating topological indices of certain OTIS Interconnection networks, Open Chemistry, 2019, 17: 220-228.

Author Response

The following references added and I addressed them in the paper.

Škrekovski, D. Dimitrov, J. M. Zhong, H. L. Wu, W. Gao, Remarks on multiplicative atom-bond connectivity index, IEEE Access, 2019, 7(1): 76806-76811.Aslam, M. F. Nadeem, Z. Zahid, S. Zafar, W. Gao, Computing certain topological indices of the line graphs of subdivision graphs of some rooted product graphs, Mathematics, 2019, 7(5), 393; https://doi.org/10.3390/math7050393.Aslam, S. Ahmad, M. A. Binyamin, W. Gao, Calculating topological indices of certain OTIS Interconnection networks, Open Chemistry, 2019, 17: 220-228.

Reviewer 3 Report

The paper is lacking literature and connection to recent developments. For example, in this paper published in Phys Rev E (https://journals.aps.org/pre/abstract/10.1103/PhysRevE.96.012308) it has been shown how entropy of different graph properties can produce divergent values of entropy for exactly the same graph (see also https://www.hindawi.com/journals/complexity/2017/3250301/abs/). The authors should, therefore, include a section about limitations of measures such as distance measured by entropy. The current paper version seems to lack any self-criticism especially under the light of these new papers. Especially relevant papers that the authors neglect mention related to their approach is https://www.worldscientific.com/doi/10.1142/S0129626418500056 https://www.mdpi.com/1099-4300/20/7/534 and a recent review on algorithmic measures, in complementation to entropic ones, from the same group: https://www.mdpi.com/1099-4300/20/8/551 with several citations to Dehmer's papers and thus strongly connecting to this research.

Author Response

I did the following changes:

 

The english grammar is polished. The following references added and I addressed them in the paper. Škrekovski, D. Dimitrov, J. M. Zhong, H. L. Wu, W. Gao, Remarks on multiplicative atom-bond connectivity index, IEEE Access, 2019, 7(1): 76806-76811. Aslam, M. F. Nadeem, Z. Zahid, S. Zafar, W. Gao, Computing certain topological indices of the line graphs of subdivision graphs of some rooted product graphs, Mathematics, 2019, 7(5), 393; https://doi.org/10.3390/math7050393. Aslam, S. Ahmad, M. A. Binyamin, W. Gao, Calculating topological indices of certain OTIS Interconnection networks, Open Chemistry, 2019, 17: 220-228. Škrekovski, R.; Dimitrov, D.; Zhong, J. M.; Wu, H. L.; Gao, W. Remarks on multiplicative atom-bond

connectivity index. IEEE Access 2019, 7, 76806–76811.

Zenil, H.; Kiani , N. A.; Tegnér, J. Low-algorithmic-complexity entropy-deceiving graphs. Phys. Rev. E 2017,

96, 012308.

Zenil, H.; Kiani , N. A.; Tegnér, J. A Review of graph and network complexity from an algorithmic information

perspective. Entropy 2018, 20, 551; https://doi.org/10.3390/e20080551

I added the section Numerical results

Round 2

Reviewer 3 Report

I think the paper now reads better.