On Divided-Type Connectivity of Graphs

Round 1

Reviewer 1 Report

This paper is interesting and mostly well-written. I only have the following minor comments:

(1) It is better to add a table to describe clearly the mathematical notations used in this paper;

(2) Authors are suggested to add some comments/remarks to indicate the practical usage of their obtained theorems.

Author Response

C1-1. It is better to add a table to describe clearly the mathematical notations used in this paper.

Answer C1-1. We have summarized the mathematical notation used in the paper and added a table, detailed as shown in  Table 1.

C1-2. Authors are suggested to add some comments/remarks to indicate the practical usage of their obtained theorems.

Answer C1-2. We have added several remarks to illustrate the practical application, please refer to remark 3 remark 4.

Author Response File: Author Response.docx

Reviewer 2 Report

I find the paper very interesting. It discusses graphs - systems whose application in various fields of science is constantly increasing. Many of the properties of these systems, particularly those relating to topology, are used by scientists to study a wide range of phenomena occurring in nature, such as chaotic quantum systems, and yet they remain largely unexplored. The paper is written correctly and coherently, with no substantive issues. The only thing missing for me is a stronger emphasis on the role that boundary conditions play at the vertices in the divided operation. I would like this to be elaborated on in more detail, as I fear that without further explanation certain aspects may not be comprehensible to the reader (for example, the edge splitting operation illustrated in Figure 1 c and d). It would also be advisable to emphasize the role of boundary conditions at the vertices of the graph: vertex s has a value of two. If this vertex has standard edge conditions (von Neumann's), then from a mathematical point of view it is not a vertex, i.e. splitting does not occur. For such a vertex, the probability of a particle bouncing, a wave traveling along the arm is 0 and the transition is 1, thus there are no scattering conditions – there is no vertex. I can recommend the paper for publication; however, I am of the opinion that it is worth expanding certain sections to clarify the subject matter for the reader.

It may be worth reading and citing publications:
Entropy 2022, 24(3), 387; https://doi.org/10.3390/e24030387 Mathematics 2022, 10(20), 3785; https://doi.org/10.3390/math10203785   both describe experimental implementations of graph splitting at vertices and edge cutting.

Author Response

Answer C2-1. We have referred two references in this paper.

Author Response File: Author Response.docx