The Complexity of the Super Subdivision of Cycle-Related Graphs Using Block Matrices

number

Before

After

1

0" from intro was supposed to be 1

has been modified

2

The list of references is short.

The connectivity of your research with other studies is low.

We update the introduction part to ensure that our research is well connected to previous studies, and we add some related and recent references to the list. 

3

Figures 7 are more or less trivial. Please consider compacting them in only one or removing them entirely. The same for Fig. 8, Fig. 9 and Fig. 10.

What is the program that is suitable to give comparisons in

Figures 8, 9 and 10.

We improved the last section  “  Diagrammatic comparison of the obtained graphs complexities “  by delete the individual figures and   merging the important  parts ( comparison figures )   of Figures 7, 8, 9, and 10 into a single, significant figure , Figures 7.   We  use  Matlab  program to  make the comparisons  of these  figures .

4

In abstract line 3 "The best and most dependable network is the

one with the most spanning trees" will be "The best and most dependable network is one of the most spanning trees". 

has been modified in line 3

5

In abstract line 10, the sentence " 3D graphs will be utilized to

illustrate our deductions." Is not suitable, authors must state

their deductions in the abstract.

has been modified

6

The introduction must not begin with a sentence " In this paper,

" and the sentence " undirected and connected graph " can be

written before defining the graphs    Γ = (V(Γ), E(Γ)).

has been modified

7

In introduction, line 2, the reference [1] must be stated after the definition of spanning trees.

has been modified in line  2

8

The definition of "the degree and adjacency matrices" must be  presented before Lemma 1 in introduction and its citation.

  We add the definitions and its citation.

9

In Lemma 2, "the complement of Γ." Must be "where  ̅ is the

complement of   ".

                  has been modified  

10

There are many typos in Definition 2 such that "Let Γ be a graph," must be "Let Γ be a graph." . "the super subdivision of  Γ ," is a new sentence.

has been modified

11

In the proof of Lemma 4, authors used "Using the properties of  determinants and matrix row and column operations". These are the properties of matrices. It is also better to state properties

(C1+C2, C2, C2+C3) and so on.

   In Lemma 4  and  Lemma 5 we added  the properties of matrices like adding C_{2} to   C_{1} and  so  on .

To clarify how we obtain the next matrix from the preceding one, the properties of matrices were added to Lemma 6.  Similar procedures can be carried out in other Lemmas.

12

Theorems 2, 3 and 4 are a general calculation for Theorem 1 to calculate the number of spanning trees for C4, C5 and C6. It is better to write it as corollaries without proof. 

We reduce the procedures in the proof of Theorems 2, 3, and 4 to avoid repetition.

13

Please provide the reason for different colors in Figure 1.

We modified all the figures (Figure 1,…,Figure 6)  to make  every color represent one of the steps of super subdivision operation.

14

"The complexity (number of spanning trees) of communication networks (graphs)" - complexity and the number of spanning trees is not the same. Nor graphs and comm. networks.

In our paper by   the complexity  we  mean  number of spanning trees  and    graphs  by networks 

number

Before

After

1

0" from intro was supposed to be 1

has been modified

2

The list of references is short.

The connectivity of your research with other studies is low.

We update the introduction part to ensure that our research is well connected to previous studies, and we add some related and recent references to the list. 

3

Figures 7 are more or less trivial. Please consider compacting them in only one or removing them entirely. The same for Fig. 8, Fig. 9 and Fig. 10.

What is the program that is suitable to give comparisons in

Figures 8, 9 and 10.

We improved the last section  “  Diagrammatic comparison of the obtained graphs complexities “  by delete the individual figures and   merging the important  parts ( comparison figures )   of Figures 7, 8, 9, and 10 into a single, significant figure , Figures 7.   We  use  Matlab  program to  make the comparisons  of these  figures .

4

In abstract line 3 "The best and most dependable network is the

one with the most spanning trees" will be "The best and most dependable network is one of the most spanning trees". 

has been modified in line 3

5

In abstract line 10, the sentence " 3D graphs will be utilized to

illustrate our deductions." Is not suitable, authors must state

their deductions in the abstract.

has been modified

6

The introduction must not begin with a sentence " In this paper,

" and the sentence " undirected and connected graph " can be

written before defining the graphs    Γ = (V(Γ), E(Γ)).

has been modified

7

In introduction, line 2, the reference [1] must be stated after the definition of spanning trees.

has been modified in line  2

8

The definition of "the degree and adjacency matrices" must be  presented before Lemma 1 in introduction and its citation.

  We add the definitions and its citation.

9

In Lemma 2, "the complement of Γ." Must be "where  ̅ is the

complement of   ".

                  has been modified  

10

There are many typos in Definition 2 such that "Let Γ be a graph," must be "Let Γ be a graph." . "the super subdivision of  Γ ," is a new sentence.

has been modified

11

In the proof of Lemma 4, authors used "Using the properties of  determinants and matrix row and column operations". These are the properties of matrices. It is also better to state properties

(C1+C2, C2, C2+C3) and so on.

   In Lemma 4  and  Lemma 5 we added  the properties of matrices like adding C_{2} to   C_{1} and  so  on .

To clarify how we obtain the next matrix from the preceding one, the properties of matrices were added to Lemma 6.  Similar procedures can be carried out in other Lemmas.

12

Theorems 2, 3 and 4 are a general calculation for Theorem 1 to calculate the number of spanning trees for C4, C5 and C6. It is better to write it as corollaries without proof. 

We reduce the procedures in the proof of Theorems 2, 3, and 4 to avoid repetition.

13

Please provide the reason for different colors in Figure 1.

We modified all the figures (Figure 1,…,Figure 6)  to make  every color represent one of the steps of super subdivision operation.

14

"The complexity (number of spanning trees) of communication networks (graphs)" - complexity and the number of spanning trees is not the same. Nor graphs and comm. networks.

In our paper by   the complexity  we  mean  number of spanning trees  and    graphs  by networks 

number

Before

After

1

The connectivity of the study with other studies is still low.

 We  modified  the introduction part  and  make a connection  with the other studies 

2

The literature survey and references list are still not covering enough the topic.

We update the references part to ensure that our research is well connected to previous studies, by  adding  8 references related to our works

3

Applications part must be more elaborated

We improved the application part  to be related with  chemistry, topological complexity for selected molecular graphs, and networks   reliability .