Cloud Computing Network Empowered by Modern Topological Invariants

Round 1

Reviewer 1 Report

1. It is necessary to improve the first paragraph in the Introduction section with a few sentences about Rapid elasticity. It is one of the five fundamental aspects of cloud computing that explain scalable provisioning and the ability to provide scalable service in a cloud environment.

2. Please enter the full names related to the acronyms KBSO, CQ, and others where they first appear in the text.

3. In the Introduction section, you lack the problems you have addressed in your research. Also, It is important to list your key contributions in this section.

4. It would be suitable for the reader's sake if you separate the Background and Literature Review sections, that is, change the organization of the text.

5. The literature review must be improved with new references that would more closely indicate the subject of your research.

6. It is necessary to deepen the Research Methodology with more concrete details, which would point out to the readers the novelty that your approach carries.

7. Before The Experimental Results section, you must state what you want to perform by using graph theory. What are your main goals, and how did you organize the whole process?

8. The conclusion must be significantly rearranged. It is necessary to point out the solutions related to the problems you are dealing with. There are no benefits that would clearly indicate the need to implement your proposed solution in practice.

Author Response

Response 1: We have improved the Introduction section in following way “Cloud computing is the on-demand availability of workstation structure resources, especially data capacity limit (cloud limit) and computing power, without a direct unique organization by the client [1]. Gigantic clouds regularly have limits coursed over various regions, each region being a server ranch [2]. Cloud computing relies upon sharing of resources to achieve clarity and regularly uses a "pay-all-the-more just as expenses emerge" model which can help in diminishing capital expenses yet may similarly provoke frightening working expenses for clueless clients [3]. A model for conveying Web-based utility registering administrations, cloud computing has swiftly emerged. The rapid expansion of the cloud computing industry with its wide range of clients, from small businesses to major corporations, has made it difficult for cloud service providers to manage the vast amount of data and other resources in the cloud. Ineffective asset management can taint cloud computing's appearance. Therefore, resources should be distributed consistently to different partners without compromising the association's benefit or the satisfaction of clients. One of the most significant and constantly evolving cloud computing paradigms is a framework as a service (IaaS). Scalability, administration style, greatest utility, lower costs, increased throughput, decreased idle time, the particular environment, cost viability, and a softer connection point are several examples of the core cloud computing components. Additionally, modern information-focused organizations have noticed a rapid increase in the asset requirements of contemporary apps. Due to this, more cloud servers have been provisioned, raising concerns about manageability, scalability, and flexibility-related issues in part. Energy efficiency, heterogeneity, load balancing, task scheduling, resource management, quality of service, workload management, the enormous volume of data, the provision of affordable, simple, and flexible services, scalability, dynamic resource allocation, quality of service, optimum utility, decreased overheads, and higher throughput is just a few of the issues that need to be addressed, and reduce costs are among the problems and challenges with cloud computing, flexibility, capacity, scalability, and dependability [4] [5] [6] [7] [8] [9]. All of these attributes depend on the cloud's topology, which determines how different nodes, pathways, and clouds should be connected to maximize the previously described attributes or characteristics. The study only addresses the topology of the cloud network to achieve the best of all attributes and resolve all the issues and challenges to the possible extent with the help of Topological invariants.

The review is likewise vital because of the vigorous nature of the issues regarding cloud computing. Therefore, the study discusses and resolves the issues like scalability, higher throughput, reduced latency, efficient use, and adaptable cloud computing network to fulfill the desired outcomes in context of topology. For the formal reason, the study solves the existing cloud computing network with the help of freshly prepared certain topological invariants and gives the best fit topology for the existing cloud computing network and also gives the basis for the modeling of new scalable cloud computing networks with best feasible characteristics [10].

On the other hand, a network's dynamic development is intimately related to its arrangement of internal connections [11]. The lack of periodic patterns in the great majority of computer networks, particularly recurrent neural networks, makes it difficult to demonstrate this link from a theoretical and formal standpoint, leaving only descriptive statistical characteristics to explain network dynamics [12]. Algebraic topology gives us in-variants, which are a great tool for understanding the structure of abstract spaces and can also be defined for graphs. The study proposes that these in-variants be used in network science. A topological index is a quantity derived from a graph that reflects relevant structural properties of the underlying network. It is, in reality, a numerical number associated with the network used to correlate computer structures with specific physical qualities. A topological index is created by converting a computer network into a numerical value. It is a numeric number associated with a computer structure (graph) that characterizes the structure's topology and is invariant under a structure-preserving mapping. The study uses K-Banhatti Sombor (KBSO), Contraharmonic-quadratic invariants (CQI), and Dharwad invariants for the solution of cloud computing networks [13].

Gutman in 2021, define the possibility of the sombor indices. Another vertex degree-based invariant graph named the sombor index is utilized to catch the sharp lower and upper limits of the associated network and the attributes of the network arriving at the limits. There are two variations of KBSO indices, the first is the KBSO index and the second is its diminished adaptation means reduced form [14]. A KBSO Index is a topological index that is a number related to a network graph that catches the balance of the network structure and gives a logical language to foresee the qualities of the network [15].

In 2021 V. R Kulli presented some topological degree-based indices following Gutman's sombor indices. These indices are called Dharwad indices. It has a couple of different structures like diminished Dharwad, decreased Dharwad remarkable, and δ-Dharwad index which is utilized to tackle the geography of sweet-smelling compounds [16].”

Response 2: We have added the full names as “The study computed newly prepared topological invariants, K-banhatti sombor (KBSO) invariants, Dharwad Invariants, Quadratic-contraharmonic invariants (QCI), and their reduced forms with other forms of cloud computing networks. These are used to explore and enhanced their characteristics such as scalability, efficiency, higher throughput, reduced latency, and best-fit topology.” 

Response 3: We have improved the Introduction section, according to our objectives, added 1.1 Research Motivation, 1.2 Research questions about deals with and added  Expected Contributions as “Cloud computing is the on-demand availability of workstation structure resources, especially data capacity limit (cloud limit) and computing power, without a direct unique organization by the client [1]. Gigantic clouds regularly have limits coursed over various regions, each region being a server ranch [2]. Cloud computing relies upon sharing of resources to achieve clarity and regularly uses a "pay-all-the-more just as expenses emerge" model which can help in diminishing capital expenses yet may similarly provoke frightening working expenses for clueless clients [3]. A model for conveying Web-based utility registering administrations, cloud computing has swiftly emerged. The rapid expansion of the cloud computing industry with its wide range of clients, from small businesses to major corporations, has made it difficult for cloud service providers to manage the vast amount of data and other resources in the cloud. Ineffective asset management can taint cloud computing's appearance. Therefore, resources should be distributed consistently to different partners without compromising the association's benefit or the satisfaction of clients. One of the most significant and constantly evolving cloud computing paradigms is a framework as a service (IaaS). Scalability, administration style, greatest utility, lower costs, increased throughput, decreased idle time, the particular environment, cost viability, and a softer connection point are several examples of the core cloud computing components. Additionally, modern information-focused organizations have noticed a rapid increase in the asset requirements of contemporary apps. Due to this, more cloud servers have been provisioned, raising concerns about manageability, scalability, and flexibility-related issues in part. Energy efficiency, heterogeneity, load balancing, task scheduling, resource management, quality of service, workload management, the enormous volume of data, the provision of affordable, simple, and flexible services, scalability, dynamic resource allocation, quality of service, optimum utility, decreased overheads, and higher throughput is just a few of the issues that need to be addressed, and reduce costs are among the problems and challenges with cloud computing, flexibility, capacity, scalability, and dependability [4] [5] [6] [7] [8] [9]. All of these attributes depend on the cloud's topology, which determines how different nodes, pathways, and clouds should be connected to maximize the previously described attributes or characteristics. The study only addresses the topology of the cloud network to achieve the best of all attributes and resolve all the issues and challenges to the possible extent with the help of Topological invariants.”,

“1.1 Research Motivation:

The investigation of cloud computing networks' topological invariants is the primary goal of this study. The study determines the seriousness and intensity of topological indices in particular cloud networks. The paper demonstrates the advantages of some topological invariants, such as KBSO, CQI, and Dharwad, as well as their reduced forms. Its main goal is to provide formulas that can be used to evaluate the topology and performance of certain cloud networks both before they are manufactured and without doing experiments. The research yielded mathematical conclusions that are used to the modelling of specific cloud networks.

Due to its incremental and quick character, it is also uncovering new and significant formulas or solutions for modelling and creating specific cloud networks, for which no acceptable solution has yet been identified.

The idea is to create new, highly effective cloud networks with the best features while also enhancing the ones that already exist. This is because vendors and manufacturers require products that are reliable and effective. The study gives the ability to create the strongest, most reliable, and error-free specific networks.”

“1.2 Research Questions:

Our research questions deal with better cloud networks used in interconnection networks, parallel processing, power generation networks, integrated circuits, bioinformatics, chemical compound development, and robotics. The study focuses to provide mathematical results for modeling purposes before the manufacturing of the above-mentioned products by avoiding compromised cloud networks.

The following questions are arising from the said topic:

• How does the study solve the topology of the cloud networks involved in interconnection networks mathematically by graph theory?
• How modeled the interconnection networks with the help of deduced mathematical results.

How did the study enhance the existing interconnection networks, reduced their irregularities, and found error-free, failure-free, and efficient advanced cloud networks as compared to existing networks?” and “Expected Contributions:

The following are the expected contributions of said research:

• The expected contribution of this research is to analyze how existing cloud networks can be improved by optimizing their adaptability.
• During the said research certain cloud networks were modeled through deduced results by topological invariants. These results will be developed over the solution of networks graphically by freshly prepared topological indices.
• Existing networks will be studied for topological perspectives and QSPR and QSAR models will be developed and analyzed.
• The relation between the lower bounds and upper bounds of the network or graph will be discovered. Further, these relationships will be defined through optimization.
• Cloud networks and other certain computer networks are solved and evaluated with the help of topological invariants.

The outcomes of the research will provide designing guidelines for advanced cloud networks and their applications in interconnection networks, power generation interconnection networks, chemical compound interconnection networks, and robotics.”

Response 4: We have improved the section, divided into Background and Literature Review also. We have added the Authors Contributions and Scope of the Study in the manuscript as

“Expected Contributions:

The following are the expected contributions of said research:

• The expected contribution of this research is to analyze how existing cloud networks can be improved by optimizing their adaptability.
• During the said research certain cloud networks were modeled through deduced results by topological invariants. These results will be developed over the solution of networks graphically by freshly prepared topological indices.
• Existing networks will be studied for topological perspectives and QSPR and QSAR models will be developed and analyzed.
• The relation between the lower bounds and upper bounds of the network or graph will be discovered. Further, these relationships will be defined through optimization.
• Cloud networks and other certain computer networks are solved and evaluated with the help of topological invariants.

The outcomes of the research will provide designing guidelines for advanced cloud networks and their applications in interconnection networks, power generation interconnection networks, chemical compound interconnection networks, and robotics.” And

 “Scope:

The research work concentrates on the topological properties and solutions of cloud networks for interconnection networks, power generation networks, chemical compounds, robotics, etc through topological invariants. The topological properties include lower bounds, upper bounds, and prediction qualities of deduced mathematical results. Though cloud networks are modeled through these solved results. So, engineers and manufacturers foresee concerning products before manufacturing or developing them.”

Response 5: We have removed little bit irrelevant references like references [30][33][34]

Response 6: We have added methodology flow diagram and explanation also as “This systematic study will take an existing cloud computing network associate it with a graph and solve the topology of the graph with the help of KBSO indices, QCI, Dharwad index, and their reduced forms. The concerning results in the form of formulas will compare with existing results. These deduced results will be used for the modeling and development of a best-fit network having the best feasible characteristics. This model is very concerning as it solved the topology of cloud computing networks in numeric and graphical form and gives accurate results. After analysis, a simulation tool maple is used for the verification and validation of results [32]. A ‘C_{r, s}’ is an existing cloud network which is under investigation, the study find vertices and edges of the given network,  define certain modern topological invariants KBSO, CQI, Dharwad and their redusced forms, then convert the cloud network into graph after mapping, afterward solve the mapped network graph through given topological invariants. At the end validation and optimization has been done by ML-based mathematical tool.

”

Response 7: We have added and improved the experimentation process by adding methodology workflow diagram and explaining it as “This systematic study will take an existing cloud computing network associate it with a graph and solve the topology of the graph with the help of KBSO indices, QCI, Dharwad index, and their reduced forms. The concerning results in the form of formulas will compare with existing results. These deduced results will be used for the modeling and development of a best-fit network having the best feasible characteristics. This model is very concerning as it solved the topology of cloud computing networks in numeric and graphical form and gives accurate results. After analysis, a simulation tool maple is used for the verification and validation of results [32]. A ‘C_{r, s}’ is an existing cloud network which is under investigation, the study find vertices and edges of the given network,  define certain modern topological invariants KBSO, CQI, Dharwad and their redusced forms, then convert the cloud network into graph after mapping, afterward solve the mapped network graph through given topological invariants. At the end validation and optimization has been done by ML-based mathematical tool.

” and also by adding in the Results section as “The ‘C_{r, s}’ cloud network mapped, converted into graph, solved, validated and optimized the results according to the steps and methodology discussed in methodology section with the help of topological invariants mentioned in Eq. (1) to (6)

”

Response 8: We have added and improved the conclusion as “TIs have lots of uses and implementations in many fields of computer science, chemistry, biology, informatics, arithmetic, material sciences, and many more, especially in cloud networks and other network architectures. But the utmost significant application is in the non-exact QSPR and QSAR. TIs are associated with the structure of cloud networks used in cloud computing. The study discusses the KBSO invariants, CQIs, and Dharwad invariants and their reduced forms which are freshly presented and have numerous prediction qualities for different variants of cloud computing networks for improvements in context with scalability, efficiency, higher throughput, best-fit topology, and latency in context to the topology. The study achieves improvements in all mentioned characteristics through the best-fit topology of the cloud network. For this purpose solve the existing network by converting it into a graph through topological invariants and get the solution in mathematical and graphical form. Graphical results show the irregularities in cloud networks as mentioned by the KBSO, CQI, and its reduced forms. Future work is to deal with these irregularities. Mathematically deduced results from Eq. 3 to Eq. 5 will be used for the modeling and improvements of cloud networks used in cloud computing as well as in different chemical structure development with the best characteristics indeed.”

Author Response File: Author Response.docx

Reviewer 2 Report

The idea of using topological invariants to improve cloud computing network performance sounds interesting. I have a few critical points to improve the article.

1. Please explain the term topological invariant first, next, explain  KBSO, QCI, and Dharwad invariants. It will be hard for readers without basic knowledge of topological invariants who use cloud networks.

2. In the introduction section, the characteristics of cloud computing are explained well but missed to explain about basics of the topological invariants and how solving them helps to solve cloud computing challenges. I noticed that the paper mentioned that topological invariants help to solve many cloud computing challenges but did not explain how.

3. Explaining how topological invariant helps to solve a specific issue such as reducing the latency or improving the scalability will add strength to the paper. Showing experiments with results and graphs or mathematical proofs will make the proposed work more interesting.

4. In line number 44,  the term "pay-all-the-more just as expenses emerge" needs to be explained.

5. In the background section 2.1, the cited paper [19] is not relevant to the explanation given.

6. In general please avoid using "in this work" while referring to other literature. It is misleading to the readers that you are saying something related to the current paper. example, line numbers 145, 154-155, 175, 208, 244, 266, and 276. Please use a third-person point of view while writing the literature review.

7. "The research work concentrates on the topological properties and solutions of cloud networks for interconnection networks, power generation networks, chemical compounds, robotics, etc through topological invariants". The above statement is repeated multiple times in the paper.

8. In the research methodology, Please explain C, r, s in 'C_r,s '. Similarly, explain what is KBSO(G) and similar terms. Also, explain how did you obtain equation (1) and other equations. Readers without background knowledge may not have any clue about how the equations are obtained.

9. Figure 2 cloud network graph is repeated in Figure 3 as well.

10. Figures 4, 5, and 6 need to be explained.

11. All the equations need some brief explanations.

11. Please add a section to explain how the equations help to solve at least any one of the cloud computing challenges or in the discussion section. We have no clue how the equations will be used to solve the mentioned cloud networks.

12. Are all the contributions listed in section 2.3 explained or proved?

13. In line number 314, it is mentioned as validation and optimization have been done by Ml-based tools. I don't see any optimization or ML tools being used. Please clarify.

14. Please check the formatting on line number 311.

15. While referring to the table, please use Table instead of Tab. Also, a brief explanation about edge partition will be helpful.

16. The conclusion should be mapped to your research questions and show that your work has found answers to those questions. 

Author Response

Response to Comments

Manuscript ID: applsci- 2061428

Title: Cloud Computing Network Empowered IoT by Modern Topological Invariants

Reviewer 2

Dear Reviewer,

Thank you very much for giving us the opportunity to revise the manuscript. We would like to thank the editor and all the reviewers for their valuable comments and suggestions. Based on the feedback, we have revised our manuscript. We marked the revisions in the manuscript as red color. For clarity, we have marked our responses in blue. We also highlight the revised content in the draft to facilitate the reviewer and editor.

Comment 1. Please explain the term topological invariant first, next, explain  KBSO, QCI, and Dharwad invariants. It will be hard for readers without basic knowledge of topological invariants who use cloud networks.

Response to Comment: We have explained as “A KBSO Index is a topological index that is a number related to a network graph that catches the symmetry of the network structure and gives a logical language to foresee the qualities of the network to enhance the cloud network topologically[15].

In 2021 V. R Kulli presented some topological degree-based indices following Gutman's sombor indices. These indices are called Dharwad indices. It has a couple of different structures like diminished Dharwad, decreased Dharwad remarkable, and δ-Dharwad index which is utilized to tackle the geography of sweet-smelling compounds called aromatic compounds. Dharwad indices can be used to solve the topology of cloud networks as these are already used for aromatic compounds [1]. These indices are used to solve the topology of a cloud networks very effectively and efficiently and find the lower bounds and upper bounds of cloud network or graph. [16-17].

Quantitatively predicting hidden aspects of various network constructions, such as cloud networks, bridge networks, Sierpinski networks, and chemical component networks used in the production of various networks utilized in different products development, is made possible by CQIs. It can also be used for the verification, improvements, and exploration of irregularities from the network under discussion [18].”

Comment 2. In the introduction section, the characteristics of cloud computing are explained well but missed to explain about basics of the topological invariants and how solving them helps to solve cloud computing challenges. I noticed that the paper mentioned that topological invariants help to solve many cloud computing challenges but did not explain how.

Response to Comment: We have explained as “All of these attributes depend on the cloud's topology, which determines how different nodes, pathways, and clouds should be connected to maximize the previously described attributes or characteristics. The study only addresses the topology of the cloud network to achieve the best of all attributes and resolve all the issues and challenges to the possible extent with the help of Topological invariants. The performance, efficiency, quality of service, security of cloud networks, flexibility and cost effectiveness are dependent on topology of the cloud network to some extent. That’s the reason, this study solve the cloud network topologically by the help of graph theory and cheminformatics which is a combination of computer, chemistry and mathematics. The deduced results will be used for the scalability, modeling, capacity nhancement and other challenges discusses above for the cloud computing networks.”

Comment 3. Explaining how topological invariant helps to solve a specific issue such as reducing the latency or improving the scalability will add strength to the paper. Showing experiments with results and graphs or mathematical proofs will make the proposed work more interesting.

Response to Comment: We have given proved results as

“KBSO (G) =

                             (7)    

KBSO red (G) =

                                    (8)                                               

Eq. (7) and Eq. (8) represent the proven results of the graph of C r, s of the cloud computing network mentioned in Fig. 2. These are the proved results used for the modeling means scalability, and reducing latency. Because these modeling results have best characteristics for the development of enhanced cloud networks by the help of KBSO invariants.”

        (9)

                          (10)

Eq. (9) and Eq. (10) represent the proven results of the graph of the cloud computing network mentioned in Fig. 2. These are the results that have been validated and are utilised for modelling methods that reduce latency. Because these modelling outcomes exhibit the best qualities for the creation of improved cloud networks using CQI invariants.

D (G) =                                                                         (11)

RD (G) =                                                                      (12)

Eq. (11) and Eq. (12) represent the proven results of Dharwad invariants of the graph of the cloud computing network mentioned in Fig. 2.

Comment 4. In line number 44,  the term "pay-all-the-more just as expenses emerge" needs to be explained.

Response to Comment: Cloud computing relies upon sharing of resources to achieve clarity and regularly uses a "pay-all-the-more just as expenses emerge" model which can help in diminishing capital expenses yet may similarly provoke frightening working expenses for clueless clients [3]. It also means to reduce cost by using best characteristics cloud computing network as it expanded effectively. A model for conveying Web-based utility registering administrations, cloud computing has swiftly emerged. The rapid expansion of the cloud computing industry with its wide range of clients, from small businesses to major corporations, has made it difficult for cloud service providers to manage the vast amount of data and other resources in the cloud. Ineffective asset management can taint cloud computing's appearance. Therefore, resources should be distributed consistently to different partners without compromising the association's benefit or the satisfaction of clients.

Comment 5. In the background section 2.1, the cited paper [19] is not relevant to the explanation given.

Response to Comment: We have changed the reference as “S. El Kafhali, I. El Mir, and M. Hanini, “Security Threats, Defense Mechanisms, Challenges, and Future Directions in Cloud Computing,” Arch Computat Methods Eng, vol. 29, no. 1, pp. 223–246, 2022”

Comment 6. In general please avoid using "in this work" while referring to other literature. It is misleading to the readers that you are saying something related to the current paper. example, line numbers 145, 154-155, 175, 208, 244, 266, and 276. Please use a third-person point of view while writing the literature review.

Response to Comment: The researchers give a functional Dynamic Resource Allocation (DRA) concentrate on a cloud computing climate. It represents the unique part of the cloud computing climate and how tended in the writing. Additionally, it gives the scientific categorizations of approaches, planning types, and streamlining measurements. Their study assists researchers in understanding the powerful part of asset distribution in the cloud, consequently further developing its presentation [27].

The researchers focus on important cloud load-balancing tools and provides a comparative analysis of important recently suggested as well as existing load-balancing tools. In addition, they will look at load-balancing plans divided into three categories [29].

Comment 7. "The research work concentrates on the topological properties and solutions of cloud networks for interconnection networks, power generation networks, chemical compounds, robotics, etc through topological invariants". The above statement is repeated multiple times in the paper.

Response to Comment: We have removed the repetition now.

Comment 8. In the research methodology, Please explain C, r, s in 'C_r,s '. Similarly, explain what is KBSO(G) and similar terms. Also, explain how did you obtain equation (1) and other equations. Readers without background knowledge may not have any clue about how the equations are obtained.

Response to Comment: The Cr,s  is representing a graph of a cloud network where ‘C’ is the name of the cloud graph, ‘r’ and ‘s’ are their parameters which represent the rth number of Big clouds and sth number of small clouds. As figure 2 shows one cloud network with one big cloud consisting of s number of times small clouds but the study generates results and equations for any number of small or large clouds.  

Comment 9. Figure 2 cloud network graph is repeated in Figure 3 as well.

Response to Comment: Figure 2 is Cloud graph but Figure 3 showing extraction of cloud graph from one of the cloud computing network

Comment 10. Figures 4, 5, and 6 need to be explained.

Response to Comment:

Figure 4: Results of KBSO and KBSO_red invariants for cloud computing network

Fig. 4 shows the results (Equations 7 & 8) of KBSO and KBSOred invariants in red and blue colors respectively in the 3D version which shows the sharp upper and lower bounds of a cloud network. There are also some irregularities are present in the graph of cloud network because lower and upper bounds are not quite separate and straight line. These irregularities can be found by the help of Irregularity Indices as a future work.

Figure 5: CQI and QCI for cloud network

Fig. 5 shows the results with clear upper and lower bounds (Equations 9 & 10) of CQI and QCI in red and blue colors respectively in the 3D version. As seen from the Figure 5 separation and straight line started from 1 to 3 for r and s parameters. It would be used accordingly during the construction of cloud computing networks.

Figure 6: Dharwad and Dharwad_red invariants for cloud network

Fig. 6 shows the results (Equations 9 & 10) of Dharwad and Dharwadred in red and blue colors respectively in the 3D version which shows sharp upper and lower bounds of a cloud network. It is quite straight line for high values of parameters ‘r’ and ‘s’ means large number of clouds can be attached for best characteristics.

Comment 11. All the equations need some brief explanations.

Response to Comment:

5.1.2 Theorem 1

Let Cr, s = G be a graph of the cloud computing network, then, KBSO and KBSOred indices are

KBSO (G) =

            (7)    

KBSO red (G) =

                             (8)                                                

Eq. (7) and Eq. (8) represent the proven results of the graph of C r, s of the cloud computing network mentioned in Fig. 2. These are the proved results used for the modeling means scalability, and reducing latency. Because these modeling results have best characteristics for the development of enhanced cloud networks by the help of KBSO invariants.

5.1.4 Theorem 2

Let C r, s = G be a graph of the cloud computing network, then, CQI and QCI indices are

        (9)

                          (10)

Eq. (9) and Eq. (10) represent the proven results of the graph of the cloud computing network mentioned in Fig. 2. These are the results that have been validated and are utilized for modeling methods that reduce latency. Because these modeling outcomes exhibit the best qualities for the creation of improved cloud networks using CQI invariants.

5.1.5Theorem 3

 Let C r, s = G be a graph of the cloud computing network, then, Dharwad and Dharwadred indices are

D (G) =                                                                         (11)

RD (G) =                                                                      (12)

Eq. (11) and Eq. (12) represent the proven results of Dharwad invariants of the graph of the cloud computing network mentioned in Fig. 2

Comment 11. Please add a section to explain how the equations help to solve at least any one of the cloud computing challenges or in the discussion section. We have no clue how the equations will be used to solve the mentioned cloud networks.

Response to Comment: The efficiency, load balancing, and latency delay are also dependent on the topology of the network [37]. This is the reason why the study solved the topology of a cloud network. The deduced mathematical results are used for the construction of new best characteristics cloud networks including efficiency, load balancing, and less latency delay.

Comment 12. Are all the contributions listed in section 2.3 explained or proved?

Response to Comment: Yes Dear reviewer we have The expected contribution of this research is to analyze how existing cloud networks can be improved by optimizing their adaptability.

• During the said research certain cloud networks were modeled through deduced results by topological invariants. These results will be developed over the solution of networks graphically by freshly prepared topological indices. Yes we provide the mathematical results for the modeling of the cloud networks
• Existing networks will be studied for topological perspectives and QSPR and QSAR models will be developed and analyzed. Yes, we developed the model in the Methodology section.
• The relation between the lower bounds and upper bounds of the network or graph will be discovered. Further, these relationships will be defined through optimization. Yes, we provide the graphical predicted results and lower and upper bounds of the cloud graph.
• Cloud networks and other certain computer networks are solved and evaluated with the help of topological invariants. Equations 7, 8, 9, 10, 11, and 12 are the mathematical solutions of the cloud networks with the help of topological invariants.

The outcomes of the research will provide design guidelines for advanced cloud networks and their applications in interconnection networks, power generation interconnection networks, chemical compound interconnection networks, and robotics. Yes, graphical and mathematical results provide guidelines for the network engineers and architectural engineers during the modeling of the cloud network and all other networks in which the cloud network is used.

Comment 13. In line number 314, it is mentioned as validation and optimization have been done by Ml-based tools. I don't see any optimization or ML tools being used. Please clarify.

Response to Comment: Predicted graphical results and mathematical results are proved with the help of Maple tool

Comment 14. Please check the formatting on line number 311.

Response to Comment: I have checked all the manuscript’s formatting

Comment 15. While referring to the table, please use Table instead of Tab. Also, a brief explanation about edge partition will be helpful.

Response to Comment: We have updated as Table 1. By dividing a graph's collection of nodes into mutually exclusive groups, a cloud graph partition reduces the network to a smaller graph. The partitioned graph will have edges made up of original graph edges that cross over into the groups. The partitioned graph may be more useful for analysis and problem-solving than the original if the total number of edges is lower than in the case of the original graph.

Comment 16. The conclusion should be mapped to your research questions and show that your work has found answers to those questions.

Response to Comment:

TIs have lots of uses and implementations in many fields of computer science, chemistry, biology, informatics, arithmetic, material sciences, and many more, especially in cloud networks and other network architectures. But the utmost significant application is in the non-exact QSPR and QSAR. TIs are associated with the structure of cloud networks used in cloud computing. The study discusses the KBSO invariants, CQIs, and Dharwad invariants and their reduced forms which are freshly presented and have numerous prediction qualities for different variants of cloud computing networks for improvements in context with scalability, efficiency, higher throughput, best-fit topology, and latency in context to the topology. The study achieves improvements in all mentioned characteristics through the best-fit topology of the cloud network. For this purpose solve the existing network by converting it into a graph through topological invariants and get the solution in mathematical and graphical form. Graphical results show the irregularities in cloud networks as mentioned by the KBSO, CQI, and its reduced forms. Equations 7, 8, 9, 10, 11, and 12 are the mathematical solutions of cloud networks with the help of topological invariants and provide modeling tools and instructions for network engineers. The study established the model and provided the graphical anticipated results and lower and upper bounds.

Author Response File: Author Response.docx

Reviewer 3 Report

The authors present a research study on contemporary topic in the field of distributed computing systems.

Suggestions for manuscript improvement:

1. Title is not clear; it should be rewritten.

2. There is a need for serious language improvements (construction of sentences, tenses, articles). The suggestion is to engage an English language professional for language correction.

3. Paper objective should be clearly stated in the abstract.

4. In introduction section should be clearly stated motivation for research and based on that clear objective.

5. The section related work is too wide and there is no evident connection to the theme of the manuscript. It should be rewritten to point out the relation to the authors’ work.

6. Methodology is not clearly described / clear statement of research questions, details on methods.

7. Results and conclusions are not clearly presented, which is the consequence of the poor design of the research study. Results section should be rewritten in order to be more comprehensible.

Since these improvements relate to serious drawbacks, I suggest major revision of the manuscript.

Author Response

Response 1: It is revised as “Cloud Computing Network Empowered by Modern Topological Invariants”

Response 2: We have tried to improve and proofread manuscript as “Cloud computing is the on-demand availability of workstation structure resources, especially data capacity limit (cloud limit) and computing power, without a direct unique organization by the client [1]. Gigantic clouds regularly have limits coursed over various regions, each region being a server ranch [2]. Cloud computing relies upon sharing of resources to achieve clarity and regularly uses a "pay-all-the-more just as expenses emerge" model which can help in diminishing capital expenses yet may similarly provoke frightening working expenses for clueless clients [3]. A model for conveying Web-based utility registering administrations, cloud computing has swiftly emerged. The rapid expansion of the cloud computing industry with its wide range of clients, from small businesses to major corporations, has made it difficult for cloud service providers to manage the vast amount of data and other resources in the cloud. Ineffective asset management can taint cloud computing's appearance. Therefore, resources should be distributed consistently to different partners without compromising the association's benefit or the satisfaction of clients. One of the most significant and constantly evolving cloud computing paradigms is a framework as a service (IaaS). Scalability, administration style, greatest utility, lower costs, increased throughput, decreased idle time, the particular environment, cost viability, and a softer connection point are several examples of the core cloud computing components. Additionally, modern information-focused organizations have noticed a rapid increase in the asset requirements of contemporary apps. Due to this, more cloud servers have been provisioned, raising concerns about manageability, scalability, and flexibility-related issues in part. Energy efficiency, heterogeneity, load balancing, task scheduling, resource management, quality of service, workload management, the enormous volume of data, the provision of affordable, simple, and flexible services, scalability, dynamic resource allocation, quality of service, optimum utility, decreased overheads, and higher throughput is just a few of the issues that need to be addressed, and reduce costs are among the problems and challenges with cloud computing, flexibility, capacity, scalability, and dependability”  and from first paragraph of 2.2 Literature Review as” An analysis in light of big data and cloud computing innovation is suggested for meeting the correspondence requirements of photoelectric hybrid network design. The main focus of this study is the examination of big data and cloud computing innovation. To do this, it investigates specialized attributes, makes use of topological optical connections, sidesteps data structures, and other techniques, and, in the end, develops an exploration methodology for big data and cloud computing. The results of the exploratory work indicate that loads of the optical connections are 60, 50, and 20, respectively. Hub B reaches the paths of the six objective nodes at the point where loads of optical connections start to become more modest. The photoelectric hybrid network structure makes it possible to communicate in more ways by using optical connections and  controlling the scope of nearby connections. The challenges with photoelectric hybrid network structure in communication can be examined in the context of cloud computing and big data innovation [20][21] [22]. The overall reception of cloud server farms (CDCs) has brought about an omnipresent interest in facilitating application administrations on the cloud.” and from third paragraph of 2.2 Literature Review as “To modify load in the context of cloud computing, this study presents a crossover metaheuristic-based asset designation system called RAFL. The objective is to proactively reduce the heap lopsidedness among dynamic actual machines and in their asset limit-thinking (e.g., CPU and RAM). This avoids overloading or underloading dynamic physical machines and makes fair use of their asset limit consideration. In the proposed system, a phasor molecule swarm improvement and dragonfly calculation based half breed streamlining calculation named PPSO-DA is utilized to create an ideal asset portion plan for adjusting the heap. Recreation tests are performed utilizing the CloudSim test system to quantify the measurements of burden unevenness across dynamic actual machines and among their thought about asset limits. Results show that the proposed PPSO-DA calculation beats phasor molecule swarm streamlining, dragonfly calculation, exhaustive learning molecule swarm enhancement, memory-based half-breed dragonfly calculation, sine cosine calculation, and elephant grouping advancement, in tracking down an ideal asset assignment for adjusting the heap. The measurable examination and benchmark testing likewise approve the general predominance of PPSO-DA [24].” and from fifth paragraph of 2.2 Literature Review as “The review is additionally fundamental because of the dynamic assignment of assets lately, organizations have utilized the cloud computing worldview to run different computing and stockpiling responsibilities. The cloud offers quicker and more beneficial administration. In any case, the issue of asset designation is difficult for cloud suppliers. The extreme utilization of assets has raised the requirement for better administration of them. What's more, the assets required may surpass those accessible in the cloud as interest and limit differ after some time. Hence, dynamic asset designation procedures permit utilizing the accessible limit all the more productively. This paper gives a functional Dynamic Resource Allocation (DRA) concentrate on a cloud computing climate. It represents the unique part of the cloud computing climate and how tended in the writing. Additionally, it gives the scientific categorizations of approaches, planning types, and streamlining measurements. This study assists researchers in understanding the powerful part of asset distribution in the cloud, consequently further developing its presentation [27].” and from seventh paragraph of 2.2 Literature Review as “These days, cloud computing is pulled into wide consideration as it can convey IT administrations and assets on an interesting premise over the Internet. Load balancing is a vital test in cloud computing. Because of the perplexing structure of cloud computing, it is troublesome and expensive to assess the way of behaving of load-balancing procedures on various cloud assets in light of QoS boundaries in a genuine cloud climate. Subsequently, to beat what was going on, cloud computing devices are utilized for reproduction to test the way of behaving of load balancing strategies in the cloud framework under various circumstances in a rehashed way by changing different boundaries. Today, a variety of tools are available, including CloudSim, WorkflowSim, CloudSim4DWf, GreenCloud, and CloudAnalyst. Each tool has a different trademark, engineering, boundary, and outcome evaluation. Therefore, it is crucial to choose a capable load-balancing device that complies with the QoS requirements. This study focuses on important cloud load-balancing tools and provides a comparative analysis of important recently suggested as well as existing load-balancing tools. In addition, we will look at load-balancing plans divided into three categories [29].”

Response 3: The objectives in the Abstract are improved and added in the following manner highlighted also as “Cloud computing networks used in IoT and other themes of network architectures can be investigated and improved by cheminformatics which is a combination of chemistry, computer, and mathematics. Cheminformatics involves graph theory and its tools. Any number that can be uniquely calculated by a graph is known as a graph invariant. In graph theory, networks are converted into graphs with workstations or routers or nodes as vertex and paths or connections as edges. Many topological indices have been developed for the determination of the physical properties of networks involved in cloud computing. The study computed newly prepared topological invariants, K-banhatti sombor (KBSO) invariants, Dharwad Invariants, Quadratic-contraharmonic invariants (QCI), and their reduced forms with other forms of cloud computing networks. These are used to explore and enhanced their characteristics such as scalability, efficiency, higher throughput, reduced latency, and best-fit topology. These attributes depend on the topology of the cloud, where different nodes, paths, and clouds are to be attached to achieve the best of the attributes mentioned before. The study only deals with a single parameter which is a topology of the cloud network. The improvement of  topology  improve the other characteristics as well which is the main objective of this study. Its prime objective is to develop formulas so that it can check the topology and performance of certain cloud networks without doing/performing experiments and also before developing them. The calculated results are valuable and helpful in understanding the deep physical behavior of the cloud’s networks. These results will also be useful for researchers to understand how these networks can be constructed and improved with different physical characteristics for enhanced versions.”

Response 4: We have improved the Introduction according to our objectives, added 1.1 Research Motivation, 1.2 Research questions about deals withas “Cloud computing is the on-demand availability of workstation structure resources, especially data capacity limit (cloud limit) and computing power, without a direct unique organization by the client [1]. Gigantic clouds regularly have limits coursed over various regions, each region being a server ranch [2]. Cloud computing relies upon sharing of resources to achieve clarity and regularly uses a "pay-all-the-more just as expenses emerge" model which can help in diminishing capital expenses yet may similarly provoke frightening working expenses for clueless clients [3]. A model for conveying Web-based utility registering administrations, cloud computing has swiftly emerged. The rapid expansion of the cloud computing industry with its wide range of clients, from small businesses to major corporations, has made it difficult for cloud service providers to manage the vast amount of data and other resources in the cloud. Ineffective asset management can taint cloud computing's appearance. Therefore, resources should be distributed consistently to different partners without compromising the association's benefit or the satisfaction of clients. One of the most significant and constantly evolving cloud computing paradigms is a framework as a service (IaaS). Scalability, administration style, greatest utility, lower costs, increased throughput, decreased idle time, the particular environment, cost viability, and a softer connection point are several examples of the core cloud computing components. Additionally, modern information-focused organizations have noticed a rapid increase in the asset requirements of contemporary apps. Due to this, more cloud servers have been provisioned, raising concerns about manageability, scalability, and flexibility-related issues in part. Energy efficiency, heterogeneity, load balancing, task scheduling, resource management, quality of service, workload management, the enormous volume of data, the provision of affordable, simple, and flexible services, scalability, dynamic resource allocation, quality of service, optimum utility, decreased overheads, and higher throughput is just a few of the issues that need to be addressed, and reduce costs are among the problems and challenges with cloud computing, flexibility, capacity, scalability, and dependability [4] [5] [6] [7] [8] [9]. All of these attributes depend on the cloud's topology, which determines how different nodes, pathways, and clouds should be connected to maximize the previously described attributes or characteristics. The study only addresses the topology of the cloud network to achieve the best of all attributes and resolve all the issues and challenges to the possible extent with the help of Topological invariants.”,

“1.1 Research Motivation:

The investigation of cloud computing networks' topological invariants is the primary goal of this study. The study determines the seriousness and intensity of topological indices in particular cloud networks. The paper demonstrates the advantages of some topological invariants, such as KBSO, CQI, and Dharwad, as well as their reduced forms. Its main goal is to provide formulas that can be used to evaluate the topology and performance of certain cloud networks both before they are manufactured and without doing experiments. The research yielded mathematical conclusions that are used to the modelling of specific cloud networks.

Due to its incremental and quick character, it is also uncovering new and significant formulas or solutions for modelling and creating specific cloud networks, for which no acceptable solution has yet been identified.

The idea is to create new, highly effective cloud networks with the best features while also enhancing the ones that already exist. This is because vendors and manufacturers require products that are reliable and effective. The study gives the ability to create the strongest, most reliable, and error-free specific networks.”

“1.2 Research Questions:

Our research questions deal with better cloud networks used in interconnection networks, parallel processing, power generation networks, integrated circuits, bioinformatics, chemical compound development, and robotics. The study focuses to provide mathematical results for modeling purposes before the manufacturing of the above-mentioned products by avoiding compromised cloud networks.

The following questions are arising from the said topic:

• How does the study solve the topology of the cloud networks involved in interconnection networks mathematically by graph theory?
• How modeled the interconnection networks with the help of deduced mathematical results.

How did the study enhance the existing interconnection networks, reduced their irregularities, and found error-free, failure-free, and efficient advanced cloud networks as compared to existing networks?”

Response 5: We have improved the section, divided into Background and Literature Review also. We have added the Authors Contributions and Scope of the Study in the manuscript as

“Expected Contributions:

The following are the expected contributions of said research:

• The expected contribution of this research is to analyze how existing cloud networks can be improved by optimizing their adaptability.
• During the said research certain cloud networks were modeled through deduced results by topological invariants. These results will be developed over the solution of networks graphically by freshly prepared topological indices.
• Existing networks will be studied for topological perspectives and QSPR and QSAR models will be developed and analyzed.
• The relation between the lower bounds and upper bounds of the network or graph will be discovered. Further, these relationships will be defined through optimization.
• Cloud networks and other certain computer networks are solved and evaluated with the help of topological invariants.

The outcomes of the research will provide designing guidelines for advanced cloud networks and their applications in interconnection networks, power generation interconnection networks, chemical compound interconnection networks, and robotics.” And

 “Scope:

The research work concentrates on the topological properties and solutions of cloud networks for interconnection networks, power generation networks, chemical compounds, robotics, etc through topological invariants. The topological properties include lower bounds, upper bounds, and prediction qualities of deduced mathematical results. Though cloud networks are modeled through these solved results. So, engineers and manufacturers foresee concerning products before manufacturing or developing them.”   

Response 6: We have added Methodology Diagram in the manuscript as

Response 7: We have improved the methodology section according to which results are developed as “This systematic study will take an existing cloud computing network associate it with a graph and solve the topology of the graph with the help of KBSO indices, QCI, Dharwad index, and their reduced forms. The concerning results in the form of formulas will compare with existing results. These deduced results will be used for the modeling and development of a best-fit network having the best feasible characteristics. This model is very concerning as it solved the topology of cloud computing networks in numeric and graphical form and gives accurate results. After analysis, a simulation tool maple is used for the verification and validation of results [35]. A ‘C_{r, s}’ is an existing cloud network which is under investigation, the study find vertices and edges of the given network,  define certain modern topological invariants KBSO, CQI, Dharwad and their redusced forms, then convert the cloud network into graph after mapping, afterward solve the mapped network graph through given topological invariants. At the end validation and optimization has been done by ML-based mathematical tool. And results are also improved as “The ‘C_{r, s}’ cloud network mapped, converted into graph, solved, validated and optimized the results according to the steps and methodology discussed in methodology section with the help of topological invariants mentioned in Eq. (1) to (6)

                                                                          (1)

                                                          (2)

Eq. (1) and Eq. (2) show the KBSO index and its reduced form which will be used for the solution of the cloud computing network. In the above equations, d_u and d_e are showing edge partitions where ‘u’ and ‘e’ are the vertices of a graph C _{r, s} under discussion.

                                                       (3)

                                                                    (4)

Eq. (3) and Eq. (4) show the CQI and QCI which will be used for the solution of the cloud computing network. In the above equations, d_u and d_v are showing edge partitions where the vertices of a graph C _{r, s} are "u" and "v" under discussion.

                                                    (5)

The Cloud computing network is converted into graphical form first, then associated with the graph. The graph is solved through the KBSO index, CQI, Dharwad index, and their other forms.

                                     (6)

Eq. (5) and Eq. (6) show the Dharwad index and its reduced form will also be used for the solution of the cloud computing network. In the above equations, d_u and d_v are showing edge partitions where the vertices of a graph C _{r, s} are "u" and "v" under discussion.“

 ”We improved the conclusion as “TIs have lots of uses and implementations in many fields of computer science, chemistry, biology, informatics, arithmetic, material sciences, and many more, especially in cloud networks and other network architectures. But the utmost significant application is in the non-exact QSPR and QSAR. TIs are associated with the structure of cloud networks used in cloud computing. The study discusses the KBSO invariants, CQIs, and Dharwad invariants and their reduced forms which are freshly presented and have numerous prediction qualities for different variants of cloud computing networks for improvements in context with scalability, efficiency, higher throughput, best-fit topology, and latency in context to the topology. The study achieves improvements in all mentioned characteristics through the best-fit topology of the cloud network. For this purpose solve the existing network by converting it into a graph through topological invariants and get the solution in mathematical and graphical form. Graphical results show the irregularities in cloud networks as mentioned by the KBSO, CQI, and its reduced forms. Future work is to deal with these irregularities. Mathematically deduced results from Eq. 3 to Eq. 5 will be used for the modeling and improvements of cloud networks used in cloud computing as well as in different chemical structure development with the best characteristics indeed.  ”

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

I would like to thank the authors for providing explanations for my previous comments. I know it is an arduous process to publish a quality paper. We are all learning to improve our writing quality while submitting and receiving reviews for each paper. Please find more comments to improve the readability of your paper.

1.     On Line 55, I believe IaaS should be Infrastructure as a Service instead of the framework as a service.

2.     There are repeated words in Lines 62-27 and grammar must be checked.

3.     The Grammar must be corrected in line 102, “Gutman in 2021, define”.

4.     The background section (2.1) looks not relevant to this work.

5.     On lines 191-192, it is written as “we propose man-made brainpower (AI) based all- 191 encompassing asset-the-board strategy for maintainable cloud computing called HUNTER”. It can be written as, “man-made brain-power-based (AI) all-encompassing asset-based strategy for maintainable cloud computing called Hunter has been proposed [25]. Writing, “We propose” in the literature survey may confuse the readers.

6.     Similarly on line 224, it is enough to write, “To address these concerns, the RATS-HM technique, which combines asset distribution security with expert task planning for cloud computing is proposed [27]”. Stating “In this study” in the literature work cause confusion.

7.     Again, on line 257, it is written as “We proposed”. Is the current manuscript proposed AMOCDARA? It may confuse the readers.

8.     On line 299, punctuations must be verified.

9.     On line 326, the font must be corrected in “defines”.

10.  It is mentioned on line 419 that Figure 4 shows the upper and lower bounds of a cloud network. It would be better to write specifically. For example, upper and lower bounds of vertices or edges or subgraphs or whatever.

11.  It has been mentioned on line 429 that these are the results that have been validated and are utilized for modeling methods that reduce latency. Is there any evidence or proof for that?

12.  It is mentioned in the research methodology (line no329) that “In the end validation and optimization have been done by ML-based mathematical tools” and in Figure 1 as well. But I don’t see any section or explanation on what ML tool has been used to validate and optimize the taken topological invariants.  If you have provided it, please make sure to mention that. On line 329, there is an unnecessary comma after “In”.

13.  In general, stating the problem precisely, existing solutions, gaps, how the proposed work fills the gap in solving the problem, explaining the method used, and proving the results will help the readers to understand the manuscript well. Removing irrelevant statements/references and repeated statements will make clear the intention of the writing.

Author Response

Response to Comments

Manuscript ID: applsci- 2061428

Title: Cloud Computing Network Empowered IoT by Modern Topological Invariants

Reviewer 2

Dear Reviewer,

Thank you very much for giving us the opportunity to revise the manuscript. We would like to thank the editor and all the reviewers for their valuable comments and suggestions. Based on the feedback, we have revised our manuscript. We marked the revisions in the manuscript as red color. For clarity, we have marked our responses in blue. We also highlight the revised content in the draft to facilitate the reviewer and editor.

Comment 1. On Line 55, I believe IaaS should be Infrastructure as a Service instead of the framework as a service

Response to Comment: It has been updated as “One of the most significant and constantly evolving cloud computing paradigms is a Infrastructure as a service (IaaS). Scalability, administration style, greatest utility, lower costs, increased throughput, decreased idle time, the particular environment, cost viability, and a softer connection point are several examples of the core cloud computing components.”

Comment 2.  There are repeated words in Lines 62-27 and grammar must be checked..

Response to Comment: We have removed the repeated words and sentences

Comment 3. The Grammar must be corrected in line 102, “Gutman in 2021, define”

Response to Comment: We have updated as  

“Gutman defines the sombor indices and their different forms in 2021.”

Comment 4. The background section (2.1) looks not relevant to this work.

Response to Comment: We Explained and updated as “Cloud computing enjoys many benefits; however, it likewise has a large number of gambles in context security, scalability and latency delay, etc. These cannot easily overlook. For a fruitful Cloud Computing reception in an enterprise, legitimate preparation and familiarity with arising chances, dangers, weaknesses, and potential arrangements are essential.”

and as “ In previous studies, cloud networks have some issues regarding security and latency delay, etc. These issues with some other issues linger on in this research theme.”

Comment 5.  On lines 191-192, it is written as “we propose man-made brainpower (AI) based all- 191 encompassing asset-the-board strategy for maintainable cloud computing called HUNTER”. It can be written as, “man-made brain-power-based (AI) all-encompassing asset-based strategy for maintainable cloud computing called Hunter has been proposed [25]. Writing, “We propose” in the literature survey may confuse the readers.

Response to Comment: We have updated as “To address these constraints, man-made brainpower (AI) based all-encompassing asset-the-board strategy for maintainable cloud computing called HUNTER has been proposed by S. Tuli and his team.”

Comment 6 Similarly on line 224, it is enough to write, “To address these concerns, the RATS-HM technique, which combines asset distribution security with expert task planning for cloud computing is proposed [27]”. Stating “In this study” in the literature work cause confusion.

Response to Comment: We have updated according to your instructions as “To address these concerns, the RATS-HM technique, which combines asset distribution security with expert task planning for cloud computing, is proposed [27]”

Comment 7. Again, on line 257, it is written as “We proposed”. Is the current manuscript proposed AMOCDARA? It may confuse the readers.

Response to Comment: We have updated as “To settle these issues, it proposed a versatile market-situated combinatorial twofold sale asset portion (AMO-CDARA) model that designates administrations to clients given different boundaries like less value, QoS, and supplier positioning.”

Comment 8. On line 299, punctuations must be verified.

Response to Comment: The Cr,s  is representing a graph of a cloud network where ‘C’ is the name of the cloud graph, ‘r’ and ‘s’ are their parameters which represent the rth number of Big clouds and sth number of small clouds. As figure 2 shows one cloud network with one big cloud consisting of s number of times small clouds but the study generates results and equations for any number of small or large clouds. 

Comment On line 326, the font must be corrected in “defines”.

Response to Comment:We have updated as superscript was pressed by mistake now “After analysis, a simulation tool maple is used for the verification and validation of results [34]. A ‘C_{r, s}’ is an existing cloud network that is under investigation, the study finds vertices and edges of the given network, defines certain modern topological invariants KBSO, CQI, Dharwad, and their reduced forms, then converts the cloud network into graph after mapping, afterward solve the mapped network graph through given topological invariants as mentioned in Figure 1.”

Comment 10. It is mentioned on line 419 that Figure 4 shows the upper and lower bounds of a cloud network. It would be better to write specifically. For example, upper and lower bounds of vertices or edges or subgraphs or whatever.

Response to Comment: We have updated as “Figure 4 shows the results (Equations 7 & 8) of KBSO and KBSO_red invariants in red and blue colors respectively in the 3D version which shows the upper and lower bounds of a graph of the cloud network. These are not fully separated and not showing sharp bounds.”

Comment 11.   It has been mentioned on line 429 that these are the results that have been validated and are utilized for modeling methods that reduce latency. Is there any evidence or proof for that?

Response to Comment: The reason for the validation is as “The results are validated because Theorem 2 is proven and graphical results are also showing quite sharp upper and lower bounds of the graph of the cloud network by CQI and QCI. So, these modeling outcomes exhibit the best qualities for the creation of improved cloud networks using CQI invariants.”

Comment 12.  It is mentioned in the research methodology (line no329) that “In the end validation and optimization have been done by ML-based mathematical tools” and in Figure 1 as well. But I don’t see any section or explanation on what ML tool has been used to validate and optimize the taken topological invariants.  If you have provided it, please make sure to mention that. On line 329, there is an unnecessary comma after “In”.

Response to Comment: We have updated as “In the end validation and optimization have been done by ML-based mathematical tools. Mapping and predicted graphical results based on proven mathematical results are portrayed with the help of the ML-Based mathematical tool Maple.”

We also updated the regarding punctuations.

Comment 13.  In general, stating the problem precisely, existing solutions, gaps, how the proposed work fills the gap in solving the problem, explaining the method used, and proving the results will help the readers to understand the manuscript well. Removing irrelevant statements/references and repeated statements will make clear the intention of the writing.

Response to Comment: We have explained as

This systematic study will take an existing cloud computing network associate it with a graph and solve the topology of the graph with the help of KBSO indices, QCI, Dharwad index, and their reduced forms. The concerning results in the form of formulas will compare with existing results. These deduced results will be used for the modeling and development of a best-fit network having the best feasible characteristics. This model is very concerning as it solved the topology of cloud computing networks in numeric and graphical form and gives accurate results. After analysis, a simulation tool maple is used for the verification and validation of results [34]. A ‘C_{r, s}’ is an existing cloud network that is under investigation, the study finds vertices and edges of the given network, defines certain modern topological invariants KBSO, CQI, Dharwad, and their reduced forms, then converts the cloud network into graph after mapping, afterward solve the mapped network graph through given topological invariants as mentioned in Figure 1. In the end validation and optimization have been done by ML-based mathematical tools. Mapping and predicted graphical results based on proven mathematical results are portrayed with the help of the ML-Based mathematical tool Maple. The C_r,s  is representing a graph of a cloud network where ‘C’ is the name of the cloud graph, ‘r’ and ‘s’ are their parameters which represent the r^th number of Big clouds and s^th number of small clouds. As figure 2 shows one cloud network with one big cloud consisting of s number of times small clouds but the study generates results and equations for any number of small or large clouds. The highlighted one is the proposed method which generate mathematical results for the modeling of the enhanced networks with best characteristics and also investigated irregularities if found. These are our main concern of the study. These results are also providing the guidelines for the engineers.  

Author Response File: Author Response.docx

Reviewer 3 Report

The authors answered all stated suggestions in the revised manuscript. However, there is a need for more improvements of the manuscript.

New suggestions for minor manuscript improvement:

1. In introduction section, just before subsection 1.1 Research Motivation, there is alone reference [18]. It is not clear the purpose of that reference. Some adjustments of the text to include that reference is needed.

2. Research questions at page 4 should be clearly marked as:

RQ1. How does the study solve the topology of the cloud networks involved in interconnection networks mathematically by graph theory?

RQ2. …

The second research question lacks “?”

Based on these research question, short answers should be included in a new and the last subsection in the section “5 Experimental Results”., for example “5.2. Discussion of research questions”.

In this subsection explicit answers on all RQs should be provided.

3. Methodology diagram at page 9 is not marked as Figure with the proper number. After labeling this diagram with the number, it should be referenced from the manuscript text and described with few sentences.

4. In Diagram at page 9, there is mistake in the 4th block, the text is “Eperimentation”. This should be fixed, and the whole diagram should be checked.

Author Response

Comment 1: In introduction section, just before subsection 1.1 Research Motivation, there is alone reference [18]. It is not clear the purpose of that reference. Some adjustments of the text to include that reference is needed.

Response 1: It is revised as “In 2021 V. R Kulli presented some topological degree-based indices following Gutman's sombor indices. These indices are called Dharwad indices. It has a couple of different structures like diminished Dharwad, decreased Dharwad remarkable, and δ-Dharwad index which is utilized to tackle the geography of sweet-smelling compounds called aromatic compounds[16-17]. It can also be used for the verification, improvements and exploring irregularities of ferroelectric materials [18].”

Comment 2: Research questions at page 4 should be clearly marked as:

RQ1. How does the study solve the topology of the cloud networks involved in interconnection networks mathematically by graph theory?

RQ2. …

The second research question lacks “?”

Based on these research question, short answers should be included in a new and the last subsection in the section “5 Experimental Results”., for example “5.2. Discussion of research questions”.

In this subsection explicit answers on all RQs should be provided.

Response 2: We have updated manuscript as

“1.2 Research Questions:

Our research questions deal with better cloud networks used in interconnection networks, parallel processing, power generation networks, bioinformatics, chemical compound development, and robotics. The study focuses to provide mathematical results for modeling purposes before the manufacturing of the above-mentioned products by avoiding compromised cloud networks.

The following questions are arising from the said topic:

RQ1 How does the study solve the topology of the cloud networks involved in interconnection networks mathematically by graph theory?

RQ2 How modeled the interconnection networks with the help of deduced mathematical results.

RQ3 How did the study enhance the existing interconnection networks, reduced their irregularities, and found error-free, failure-free, and efficient advanced cloud networks as compared to existing networks?” and also added 5.2 Discussion section as “

5.2 Discussion

According to the aforementioned findings, topological invariants, such as K-banhatti sombor invariants, Contraharmonic-quadratic invariants, Dharwad invariants, and their reduced forms, allow us to gather information about cloud networks in the form of algebraic structures and provide us with a mathematical technique to infer the hidden properties of various structures, such as particular networks. The degree-based topological indices and the distance-based topological indices are the two primary classes of topological indices that clash, but in the present study, the networks were solved using the degree-based topological indices, and the best results are displayed in the graphs. Through the application of topological invariants, the research focuses on the topological characteristics and solutions of cloud networks for interconnection networks used in the internet, power generating networks, chemical compounds, robotics, etc. Lower bounds, higher bounds, and the ability to predict the outcomes of derived mathematical operations are among the topological traits. Although these solved solutions serve to model interconnection networks. Engineers and producers, therefore, anticipate issues with items before creating or developing them.”

Comment 3: Methodology diagram at page 9 is not marked as Figure with the proper number. After labeling this diagram with the number, it should be referenced from the manuscript text and described with few sentences.

Response 3: We have updated Methodology diagram and referred in the text “4 Research Methodology

This systematic study will take an existing cloud computing network associate it with a graph and solve the topology of the graph with the help of KBSO indices, QCI, Dharwad index, and their reduced forms. The concerning results in the form of formulas will compare with existing results. These deduced results will be used for the modeling and development of a best-fit network having the best feasible characteristics. This model is very concerning as it solved the topology of cloud computing networks in numeric and graphical form and gives accurate results. After analysis, a simulation tool maple is used for the verification and validation of results [32]. A ‘C_{r, s}’ is an existing cloud network that is under investigation, the study finds vertices and edges of the given network, _defines certain modern topological invariants KBSO, CQI, Dharwad, and their reduced forms, then converts the cloud network into graph after mapping, afterward solve the mapped network graph through given topological invariants as mentioned in Figure 1. In, the end validation and optimization have been done by ML-based mathematical tools.

Figure 1: Methodology Flow Diagram

”

Comment 4: In Diagram at page 9, there is mistake in the 4th block, the text is “Eperimentation”. This should be fixed, and the whole diagram should be checked.

Response 4: We have updated as “5 Experimentation and Results

The ‘C_{r, s}’ cloud network was mapped, converted into the graph, solved, validated, and optimized the results according to the steps and methodology discussed in the methodology section with the help of topological invariants mentioned in Eq. (1) to (6)

                                                                                        (1)

                                                       (2)

Eq. (1) and Eq. (2) show the KBSO index and its reduced form which will be used for the solution of the cloud computing network. In the above equations, d_u and d_e are showing edge partitions where ‘u’ and ‘e’ are the vertices of a graph C _{r, s} under discussion.

                                                                    (3)

                                                                                     (4)

Eq. (3) and Eq. (4) show the CQI and QCI which will be used for the solution of the cloud computing network. In the above equations, d_u and d_v are showing edge partitions where the vertices of a graph C _{r, s} are "u" and "v" under discussion.

                                                                            (5)

The Cloud computing network is converted into graphical form first, then associated with the graph. The graph is solved through the KBSO index, CQI, Dharwad index, and their other forms.

                                                           (6)

Eq. (5) and Eq. (6) show the Dharwad index and its reduced form will also be used for the solution of the cloud computing network. In the above equations, d_u and d_v are showing edge partitions where the vertices of a graph C _{r, s} are "u" and "v" under discussion.”

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