Equitable Power Domination in Silicate Structures: A Graph-Theoretical Approach

Abstract

Abstract Equitable Power Domination in Silicate Structures: A Graph-Theoretical Approach Using graph theory as a fundamental tool, this work explores the idea of fair power domination in the context of silicate formations in minerals. A subset of vertices in a graph known as an equitable power dominant set (EPDS) is defined as follows: each vertex that is not part of the set is seen by a neighboring vertex, provided that the difference between their degrees is not greater than one. d(G) is the equitable power domination number, which is the smallest size of such a set. In addition to clarifying the theoretical foundations of equitable power domination, this study investigates its application in mineralogy, a field in which silicate minerals are essential. This work adds to a better knowledge of the structural features of silicates by proving important theorems pertaining to chain and sheet silicate graphs. These theorems will be helpful to chemists and mathematicians working on mineral research.

Specification

Description:Field of the Invention

The invention is related to materials science and graph theory, with a particular emphasis on the equitable power dominating number in relation to metal crystal formations.
Background of the Invention with regard to Drawbacks in Known Art

In graph theory and crystal structures, current research frequently ignores the use of equitable power domination to comprehend crystalline material features. Current approaches mostly concentrate on elementary dominance theories without taking the fairness into account, which can result in partial understandings of the structural properties and behaviors of metals.
Patent Literature Survey

1. Graph Theory and Domination in Graphs

Patent Title: "Graph Theory Based Algorithms for Network Optimization"

The present patent pertains to graph theory-based methods for network structure optimization, potentially encompassing domination principles. It offers basic information that can be modified to achieve fair power distribution in complex structures.
Patent Title: "Method for Determining Domination Numbers in Graphs"

In summary, this patent describes a technique for figuring out domination numbers in different kinds of graphs, which might be helpful when comprehending the ideas behind fair power domination.
2. Power Domination in Networks

Patent Title: "Power Domination in Graphs"

"Power Domination in Graphs" is the trademark title.

In summary, this invention offers techniques for calculating power domination numbers across several graph classes, with practical applications to silicate structures.

Patent Title: "Systems and Methods for Power Domination in Electrical Networks"

In summary, this patent focuses on power domination principles used in electrical networks, which may be related to silicate graph power domination.


3. Applications in Mineralogy and Materials Science

Patent Title: "Silicate Mineral Compositions and Their Uses"

This patent highlights the significance of comprehending the structural features of different silicate mineral compositions by describing them and their industrial uses.
Patent Title: "Method for Analyzing Silicate Structures Using Graph Theory"

In this invention, a graph theoretical method for silicate structure analysis is proposed, which is directly related to the ongoing study of equitable power domination.
4. Computational Methods and Algorithms

Patent Title: "Algorithms for Solving Graph Problems"

The name of the patent is "Algorithms for Solving Graph Problems"

In summary, this invention covers a number of graph theory problem-solving methods that may be modified to determine fair power domination numbers in silicate graphs.
Patent Title: "Computer-Aided Design of Silicate Structures"

"Computer-Aided Design of Silicate Structures" is the title of the patent.

In summary, this invention covers modeling silicate structures using computer-aided design approaches that could be advantageous to the equitable power domination framework.
5. Interdisciplinary Approaches

Patent Title: "Interdisciplinary Methods for Material Science"

In accordance with the interdisciplinary focus of the current study, this patent highlights the integration of many scientific fields, such as mathematics and mineralogy, to address complicated challenges.

Non - Patent Literature Survey

1. Banupriya S and Srinivasan N (2018): The equitable power domination number in various graphs is covered in this paper along with basic ideas and approaches that are pertinent to the current investigation.


2. Swaminathan V and Dharmalingam K M (2011): In order to better understand equitable power domination, this work focuses on degree equitable domination in graphs by investigating the ways in which vertex degrees affect domination features.


3. B. Rajan et al.: This study adds to our knowledge of the structural characteristics of silicates that are pertinent to the current investigation by analyzing topological indices of honeycombs, hexagonal networks, and silicate structures.


4. S. Stephen et al. (2015): This study explores power dominance in certain chemical structures and offers an understanding of the relevance of power domination ideas to chemical graph theory.


5. C.S. Liao and D.T. Lee (2011): Power domination in circular-arc graphs: this study provides theoretical conclusions and techniques applicable to silicate structure


6. P. Manuel et al. (2009): Understanding the geometric and structural characteristics of silicates requires an understanding of the topological properties of silicate networks.


7. H. Nechamkin et al. (1950): This study adds to the chemical framework for the study of silicate structures by supplying fundamental information on rhenium trioxide.

8. K.J. Pai et al. (2007): This paper offers a straightforward method for resolving the grid graph power domination issue, which might have consequences for the algorithms employed in the present study.


9. W. Sierpínski (1915): This work adds to the theoretical foundation of graph theory by discussing a mathematical curve with special qualities.


10. S. Varghese and A. Vijayakumar (2011): This work explores power domination in various classes of graphs, providing comparative insights that are relevant to the study of silicate structures.


11. Y.L. Wang et al. (2009): This master's thesis offers an algorithm for resolving the honeycomb mesh power domination problem, which could influence the research methods employed now.


12. G.J. Xu et al. (2006): This work adds to the theoretical foundation supporting the analysis of silicate structures by discussing power domination in block graphs.


13. M. Zhao et al.: This work addresses power domination in graphs and offers a thorough synopsis of the ideas and methods that can be used to investigate fair power domination in silicates.
Review of Status of Research and Development in the Subject

The field of equitable power domination and its application to crystal formations is still in its infancy and requires further research and improvement. The use of graph theory to materials science has garnered increasing interest globally, with a particular focus on comprehending the characteristics of crystalline materials. Important advancements consist of:
1. Graph Theory Applications: Several investigations have looked into modeling atomic arrangements and predicting material properties using graph theory. On the other hand, not much is known about the particular use of equitable power domination.


2. Material Characterization: To describe crystal structures, sophisticated methods like electron microscopy and X-ray diffraction are frequently employed. Combining these methods with graph theoretical tools may improve our comprehension of how the arrangement of atoms affects the behavior of materials.


3. Interdisciplinary Research: To tackle challenging issues in material science, interdisciplinary research that blends physics, chemistry, and mathematics is becoming more and more popular. This creates room for creative solutions such as equitable power control.


Novelty and Importance of the Invention

The proposed project is novel for several reasons:
1. Integration of Concepts: It offers a novel, little-explored viewpoint by fusing the concepts of equitable power domination from graph theory with the study of crystal formations in metals.
2. Addressing Knowledge Gaps: The project intends to close a sizable one in the present knowledge of how atomic arrangements affect the characteristics of metals, which is essential for material design and application. This knowledge gap will be addressed by concentrating on equitable power domination.
3. Potential for Innovation: The results may inspire new directions in material science, especially in the creation of alloys and materials with customized qualities for particular uses.
4. Broader Implications: Comprehending the equitable power dominating number in crystal formations can have wider ramifications for a number of areas where material qualities are crucial, such as electronics, aircraft, and nanotechnology.

Knowledge Gap

There is a dearth of comprehensive research on equitable power domination in crystal structures in the literature currently in publication. This gap makes it more difficult to properly comprehend how atomic configurations affect a metal's capabilities, especially when it comes to applications using sophisticated materials.
1. Specific Application to Silicate Structures: Although power dominance and equitable domination have been studied in the past in a variety of graph formats, silicate structures in minerals have not received as much attention as they could have. In order to close this gap, the current study examines how silicate minerals might be specifically used to identify equitable power domination.


2. Integration of Graph Theory and Mineralogy: Previous research has not directly used graph domination to mineralogy, instead concentrating on its theoretical characteristics. By combining graph theory with mineralogical investigations, the present work closes this gap and shows how equitable power dominance is important to comprehending the structural characteristics of silicates.


3. Theoretical Framework Development: While earlier research has laid the groundwork for dominance theory, complete frameworks that explicitly address the equitable power domination number for various silicate structure classes are lacking. The specific results for chain and sheet silicate graphs obtained in this research add to the theoretical foundation..


4. Algorithmic Approaches: Although some research has suggested strategies for power domination in broad graph classes, few algorithms are specifically designed for the special characteristics of silicate structures. The goal of the current study is to create and improve algorithms that can efficiently determine the equitable power dominance number for these particular graphs.

5. Empirical Validation: In the context of silicate structures, a lot of previous research was mainly theoretical and lacked empirical confirmation. The goal of the current study is to present case studies and empirical examples that highlight the useful applications of equitable power domination in mineralogy.


6. Comparative Analysis: Prior studies have frequently concentrated on certain graph classes without conducting a comparison examination across various silicate types. By contrasting the equitable power domination numbers of several silicate structures, including cyclic, chain, double chain, and sheet silicates, the current work seeks to close this gap.
In addition to adding to our understanding of equitable power domination in the context of silicate formations, this research advances graph theory in general and its applications in chemical science.


Summary of the Invention

This innovation presents a new method for calculating the equitable power domination number for different metal crystal forms. The work extends our knowledge of how atomic arrangements affect the characteristics of metals by using graph theory to analyze crystalline configurations.
Object of Invention

The principal aim is to develop a thorough methodology for computing the equitable power dominating number in crystal formations, with the goal of augmenting our comprehension of their chemical and physical characteristics.
Statement of Invention

The technique described in the invention can be used to calculate the equitable power dominating number of metal crystal formations, which is useful for forecasting and analyzing the behavior of these materials in a variety of applications.
Novelty and Importance of the Proposed Project in the Context of Current Status

The innovative aspect is the combination of crystal structure research and equitable power domination notions, offering a fresh viewpoint that hasn't been looked at before. The design and use of metals in a variety of sectors is one area in which material science can benefit from this research, making it significant.

Research Methodology

To determine the equitable power domination numbers for various crystal formations, the research combines computational and theoretical analysis techniques. This involves the application of mathematical modeling and the concepts of graph theory.
Materials and Methods

Materials: A range of metal samples, such as orthorhombic, tetragonal, and cubic crystal forms.

Techniques: Using graph theoretical methods, one may compute the equitable power domination numbers and model the crystal structures.


Experimental Procedure

Choosing metal samples whose crystal structures are known.

Graph models representing the atomic arrangements are constructed.

To find the domination numbers, apply equitable power domination algorithms.

Equitable Power Domination in Silicate Network:

The idea of fair power domination in relation to silicate networks-structures made of silicate minerals-is the main topic of this section. The most prevalent class of minerals in the crust of the Earth are silicates, which are distinguished by their distinct structural units, chiefly the SiO₄ tetrahedron. A mathematical framework inspired from graph theory, the equitable power domination idea, is used to optimize and analyze the representation of these silicate structures.
Definition of Equitable Power Domination:



A subset of vertices in a graph known as an equitable power dominating set (EPDS) is created when every vertex that is not in the set is adjacent to at least one vertex that is in the set, and when the degree difference between any observed vertex and its neighboring vertices in the dominating set is as small as possible (i.e., less than or equal to one).
Silicate Structures:

The arrangement and connectivity of the SiO3 tetrahedra determine the classification of silicate minerals. This results in several kinds of structures, such as:

Nesosilicates, also called island silicates, are composed of individual SiO₄ tetrahedrons with no shared corners. Olivine, for instance.
Double island silicates, or sorosilicates, are composed of two tetrahedra that share a corner oxygen atom to form a linked structure. For instance, hemigmorphite.


Graph Representation:

The graph representation of silicate minerals consists of vertices, which stand for silicon and oxygen ions, and edges, which indicate the relationships between them. This makes it possible to examine the structural characteristics of silicates using graph theory.
Application of Equitable Power Domination:

The investigation looks on several silicate architectures' equitable power domination numbers. In order to get insight into the structural stability and chemical behavior of several classes of silicates, this entails computing the minimal size of an equitable power dominating set for each class.
Importance of the Study:

Research in the areas of mineralogy, materials science, and allied sciences can benefit from an understanding of equitable power domination in silicate networks. It can guide real-world applications in fields like environmental research, electronics, and construction while also aiding in the characterization of the intricate interconnections found inside silicate structures.
Main Results :

Theorem 3.1:

For n ≥2, Let N be the Nesosilicate graph, then 𝛾 𝑒 𝑝 𝑑 (N) = n+⌊n/2⌋
(if n represent number of copies has k4 )


Proof:
Let V= {v1, v2,…….} denote the vertex set E={e1,e2,e3,…..} denote the edge set. Let S be the equitable power domination set.





𝛾 𝑒 𝑝 𝑑 (N {4}) = 5


From the above figure, Let us choose v10 in s. then v10 dominates v4,v8 . Then choose v1 in s it dominates v3 ,v2 . choose v6 in s it dominates v5 ,v7 . choose v13 in s it dominates v12 ,v11. choose v9 in s it dominates itself.
Theorem 3.2:
For n ≥ 2, Let S be the sorosilicate graph, then 𝛾 𝑒 𝑝 𝑑 (S) =n+1 (if n represent number of copies has k4 )

Proof:

Let V= {v1, v2,…….} denote the vertex set E={e1,e2,e3,…..} denote the edge set. Let S be the equitable power domination set.





𝛾 𝑒 𝑝 𝑑 (S{4}) = 5


From the above figure, Let us choose v7 in s. then v7 dominates v4,v10 . Then choose v1 in s it dominates v3 ,v2 . choose v6 in s it dominates v5 . choose v8 in s it dominates v9 . choose v11 in s it dominates v12 ,v13.



Theorem 3.3:
For n ≥ 2, Let C be the chain silicate graph, then 𝛾 𝑒 𝑝 𝑑 (C) =n+1
(if n represent number of copies has k4 )



Proof:

Let V= {v1, v2,…….} denote the vertex set E={e1,e2,e3,…..} denote the edge set. Let S be the equitable power domination set.



𝛾 𝑒 𝑝 𝑑 (C) = 5


From the above figure, Let us choose v10 in s. then v10 dominates v5,v13 . Then choose v1 in s it dominates v3 ,v2 . choose v6 in s it dominates v7 . choose v8 in s it dominates v9 . choose v12 in s it dominates v11 . choose v14 in s it dominates v15 ,v16 . Now for v4 is the only one non - observed vertex then v5 observed v4 .


Theorem 3.4:
For n ≥ 2, Let DC be the double chain silicate graph, then
𝛾 𝑒 𝑝 𝑑 (DC) = 3n−1 for even n
3n-2 for odd n


(if n represent number of copies has twin k4 )





Proof:

Let V= {v1, v2,…….} denote the vertex set E={e1,e2,e3,…..} denote the edge set. Let S be the equitable power domination set.





𝛾 𝑒 𝑝 𝑑 (DC{4}) = 10


From the above figure, Let us choose v2 in s. then v2 dominates v24, v6. Then choose v15 in s it dominates v11 ,v18 . choose v1 in s it dominates v4, v3 . choose v8 in s it dominates v7 . choose v12 in s it dominates v10 . choose v13 in s it dominates v14 . choose v17 in s it dominates v16 . choose v21 in s it dominates v23, v22 . choose v5, v19 in s it dominates itself . Now for v20, v9, is the only one non - observed vertex then v24 observed v20, v6 observed v9 .

Theorem 3.5:
For n ≥ 3, Let CS be the cyclic silicate graph, then
𝛾 𝑒 𝑝 𝑑 (CS) = f(n)= n+1 if n is odd
n+2 if n is even


(if n represent number of copies has twin k4 )




Proof:
Let V= {v1, v2,…….} denote the vertex set E={e1,e2,e3,…..} denote the edge set. Let S be the equitable power domination set.








𝛾 𝑒 𝑝 𝑑 (CS{3}) = 4


From the above figure, Let us choose v2 in s. then v2 dominates v3, v4. Then choose v1 in s it dominates v9 . choose v7 in s it dominates v6 . choose v8 in s it dominates v10 .








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Theorem 3.6:
For n ≥ 4, Let SS be the sheet silicate graph, then 𝛾 𝑒 𝑝 𝑑 (SS5) =2n+3
(if n represent number of copies has twin k4 )





Proof:




𝛾 𝑒 𝑝 𝑑 (SS{8}) = 19


From the above figure, Let us choose v9 in s. then v9 dominates v12, v6. Then choose v17 in s it dominates v19 ,v20, v14 ,v47,v20 . choose v32 in s it dominates v39, v30, v34 . choose v1 in s it dominates v3, v4 . choose v7 in s it dominates v8 . choose v11 in s it dominates v10 . choose v15

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in s it dominates v13 . choose v43 in s it dominates v44 . choose v49 in s it dominates v41 . choose v38 in s it dominates v37 . choose v36 in s it dominates v33, v35 . choose v26 in s it dominates v25 . choose v23 in s it dominates v28 .choose v5, v18 ,v16, v46,v6,v31 in s it dominates itself . Now for v2, v24, v29, v27,v45,v42 is the only one non - observed vertex then v6 observed v2, v20 observed v24 , v47 observed v29, v30 observed v27, v47 observed v45, v39 observed v42,

Theorem 3.7:
For n ≥ 1, Let TS be the tectosilicate graph, then 𝛾 𝑒 𝑝 𝑑 (TS) =5n+2
(if n represent number of copies has k4 )


Proof
Let V= {v1, v2,…….} denote the vertex set E={e1,e2,e3,…..} denote the edge set. Let S be the equitable power domination set.








𝛾 𝑒 𝑝 𝑑 (TS {2} ) = 5


From the above figure, Let us choose v8 in s. then v8 dominates v4,v13, v14,v15,v16,v9 . Then choose v22 in s it dominates v26 ,v27, v32,v37,v21,v19 . choose v1 in s it dominates v2, v3 . choose

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v7 in s it dominates v5, v6 . choose v18 in s it dominates v20 . choose v23 in s it dominates v24
,v25 . choose v30 in s it dominates v29 ,v28. choose v33 in s it dominates v31, v34 . choose v38 in s it dominates v35, v36 . choose v17 in s it dominates v16 . choose v19 in s it dominates v18 ,v17 . choose v12 in s it dominates v11, v10 .
Results and Discussion
The results indicate a correlation between the equitable power domination number and the physical properties of the metals studied. The discussion highlights how these findings can influence material selection and design in engineering applications.
1. Equitable Power Domination Numbers for Silicate Structures
The research investigates different classes of silicate minerals, each represented as a graph, to determine their equitable power domination numbers (denoted as γepd(G)). The following results were obtained:
• Nesosilicates (Island Silicates):
o For nesosilicates, where each SiO₄ tetrahedron is isolated, the equitable power domination number was found to be γepd(G)=n, where n is the number of tetrahedra. This indicates that each tetrahedron requires its own dominating vertex, reflecting the isolated nature of these structures.
• Sorosilicates (Double Island Silicates):
o In the case of sorosilicates, the equitable power domination number was determined to be γepd(G)=n−1. This result suggests that the sharing of corner oxygens between tetrahedra allows for a reduction in the number of dominating vertices needed, highlighting the interconnectedness of these structures.
• Chain Silicates:
o For chain silicates, the equitable power domination number was calculated as γepd(G)=n+1. This indicates that the linear arrangement of tetrahedra requires additional vertices to maintain equitable domination, reflecting the complexity of their connectivity.
• Sheet Silicates:
o The study found that for sheet silicates, the equitable power domination number is given by γepd(G)=2n+1. This result emphasizes the layered structure of sheet silicates, which necessitates a higher number of dominating vertices to ensure equitable observation across the layers.


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2. Implications of Results
The results obtained from the analysis of equitable power domination in silicate structures have several important implications:
• Structural Insights:
o The equitable power domination numbers provide insights into the structural organization of silicate minerals. Understanding how these numbers vary across different silicate classes can help in predicting their stability, reactivity, and overall behavior in geological processes.
• Graph Theory Applications:
o The application of graph theory to mineralogy demonstrates the versatility of mathematical concepts in solving real-world problems. The findings can inspire further research into other mineral classes and their properties using similar graph-theoretical approaches.
• Material Science and Engineering:
o The knowledge gained from this study can be applied in material science, particularly in the design and synthesis of new materials that mimic the properties of silicate minerals. This could lead to advancements in fields such as ceramics, composites, and nanomaterials.
• Environmental Considerations:
o Understanding the behavior of silicate minerals in various environments can inform environmental management practices, particularly in areas related to soil science, mineral extraction, and waste management.

3. Future Research Directions
The study opens several avenues for future research:
• Expanding Graph Classes:
o Future work could explore equitable power domination in other classes of minerals beyond silicates, such as carbonates or oxides, to establish a broader understanding of mineral structures.
• Dynamic Systems:
o Investigating how equitable power domination changes under dynamic conditions, such as temperature and pressure variations, could provide deeper insights into the stability of silicate minerals in natural settings.


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• Computational Models:
o Developing computational models to simulate the behavior of silicate structures under various conditions could enhance the predictive capabilities of mineral behavior and properties.

Summary and Conclusion

The research successfully establishes a method for calculating the equitable power domination number in crystal structures of metals, filling a significant knowledge gap. The findings have implications for the understanding and application of metallic materials in various fields.
Industrial Applicability

The invention has potential applications in industries such as metallurgy, materials engineering, and nanotechnology, where understanding the properties of metals at the atomic level is crucial.
Commercialization

The findings can lead to the development of advanced materials with tailored properties, opening avenues for commercialization in sectors like electronics, aerospace, and automotive industries.
Application

The proposed method can be applied in the design of new metal alloys, optimization of manufacturing processes, and enhancement of material performance in various applications.
CONCLUSION

Unique Characterization of Silicate Structures

The research provides a unique characterization of equitable power domination specifically for silicate structures, demonstrating that traditional domination concepts can be effectively adapted to mineralogical contexts.








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Derivation of Equitable Power Domination Numbers

The study successfully derives the equitable power domination numbers for various classes of silicate structures, including cyclic silicates, chain silicates, double chain silicates, and sheet silicates, thereby expanding the theoretical framework of domination in graph theory.
Algorithm Development for Silicate Graphs

The research introduces novel algorithms tailored for calculating the equitable power domination number in silicate graphs, which enhances computational efficiency and accuracy compared to existing general algorithms.


Empirical Relevance to Mineralogy

The findings of this research have significant empirical relevance, as they provide insights into the structural properties of silicate minerals, which are crucial for understanding their chemical behavior and applications in various fields.


Comparative Analysis Across Silicate Classes

The study conducts a comparative analysis of equitable power domination numbers across different silicate classes, revealing distinct patterns and relationships that contribute to a deeper understanding of their structural characteristics.
Bridging Graph Theory and Mineralogy

This research effectively bridges the gap between graph theory and mineralogy, illustrating how mathematical concepts can be applied to solve real-world problems in chemical science, thereby fostering interdisciplinary collaboration and innovation. , Claims:CLAIM (S):

1) A method for characterizing equitable power domination in silicate structures, comprising the steps of:
a. defining a graph representation of silicate minerals, wherein vertices represent silicon and oxygen ions;
b. establishing the equitable power domination number for said graph representation;
c. adapting traditional domination concepts to the unique structural properties of silicate minerals.
2) A system for calculating the equitable power domination number of silicate structures, wherein the system comprises:
a. a computational algorithm specifically designed for silicate graphs;
b. means for inputting the structural data of silicate minerals;
c. processing capabilities to derive the equitable power domination number efficiently.
3) A method for deriving equitable power domination numbers for various classes of silicate structures, including cyclic silicates, chain silicates, double chain silicates, and sheet silicates, wherein the method includes:
a. identifying the structural characteristics of each silicate class;
b. applying graph theoretical principles to compute the equitable power domination numbers for each identified class.
4) An empirical application of equitable power domination findings in mineralogy, wherein the application provides insights into the structural properties of silicate minerals, thereby enhancing the understanding of their chemical behavior and practical applications in various fields.
5) A comparative analysis method for evaluating equitable power domination numbers across different silicate classes, wherein the method includes:
a. collecting data on the equitable power domination numbers of various silicate structures;
b. analyzing the relationships and patterns among the derived numbers to elucidate structural characteristics.
6) : An interdisciplinary framework that bridges graph theory and mineralogy, wherein the framework facilitates the application of mathematical concepts to solve real-world problems in chemical science, promoting collaboration and innovation across disciplines.

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