EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 4, 2024, 2586-2620 ISSN 1307-5543 – ejpam.com Published by New York Business Global Site Selection for Thermal Power Plant Based on Sombor Index in Neutrosophic Graphs Mohammed Alqahtani1, Murugan Kaviyarasu2,∗, Murugesan Rajeshwari3 1 Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, P.O. Box 93499, Riyadh 11673, Saudi Arabia 2 Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai, Tamil Nadu, India-600 062 3 Department of Mathematics, Presidency University, Bangalore, 560064, India Abstract. A graph or molecular graph’s overall resilience and connectedness are gauged by the Sombor Index (SOI). It is an improved version of the conventional SOI that is designed to capture ambiguity in the relationships between nodes or atoms. This indicator aids in assessing the network’s resilience and stability in erratic circumstances. We have investigated the charac- teristics of the SOI of neutrosophic graphs (NGs) in this study. The association between the neutrosophic first Zagreb Index (NFZI) of NGs and the neutrosophic Sombor Index (NSI) was revealed in the study. The application of SOI based on Site Selection for Thermal Power Plants in NGs is finally covered. 2020 Mathematics Subject Classifications: 05C72, 05C09, 03B52 Key Words and Phrases: Fuzzy logic, Neutrosophic logic, NSOI, NFZI, NSZI 1. Introduction 1.1. Fuzzy Graphs(FGs) Real-world problems are rarely strict and often involve fuzziness and roughness. With the use of FG theory, real-world problems may be efficiently and understandably mathematically explained. Zadeh used this collection’s membership function in the work that was given in [59]. This function assigns a membership value(MV), from zero to one, to each member. Expanding the traditional knowledge of set theory was his goal. Human views, judgment and assessment, according to Zadeh and Goguen, reduce fuzziness. Fuzzy sets (FSs) are useful for solving situations where the fault is caused by random variables rather than class membership. Scientists can study ambiguous conceptual issues with the use of this mathematical technique. The 1960s and 1970s saw ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i4.5461 Email addresses: m.alqahtani@seu.edu.sa (M. Alqahtani), kavitamilm@gmail.com (M. Kaviyarasu), rajeshwari@presidencyuniversity.in (M. Rajeshwari) https://www.ejpam.com 2586 Copyright: © 2024 The Author(s). (CC BY-NC 4.0) M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2587 a modest increase in the acceptance of this notion. Researchers were interested in this area when fuzzy systems built around rules were used to control technological processes in the late 1970s. Its effective application in washing machines, video cameras, subway trains and other places inspired many to carry out study in this field. Prior to 2000, there were more than 30,000 publications on FS theory. It has expanded over the last 10 years to encompass both an application-focused theory and an actual concept. The writers of [62] concluded after analyzing FS theory that it may be used to bridge the gap between formal and natural models and explain deterministic uncertainty. According to the author of this page, a FS is defined as follows: M = {(γ, λM (γ)), γ ∈ M} is the collection of ordered pairs and is denoted as such if M is the collection of objects represented by γ. In this case, λM (γ) denotes the MV, and 0 ≤ λM (γ) ≤ 1. A variable can be connected to both the possibility distribution and the probability distribution, according to research published in 1978 by authors of [60]. In [19], the relationship between the finite valued FS, the Zadeh FS and the n-dimensional FS is clarified. The approaches and methods of FG theory were presented by the authors of [53] for the analysis of multi-species fishing dynamics. The patterns of data can be obtained by using the above work. Fuzzy refers to not being able to hear or see clearly. FGs are useful in modeling most of the difficulties that arise in our daily lives discussed in [46] along with an introduction to some of their characteristics. Few rooted trees that encapsulate a particular FG, an algorithm and complexity analysis are presented [11]. In [12] and [14] discusses FGs that account for the fuzziness of vertex and edge existence, edge weight and connectedness. The hyper-wiener Index’s associated with various of graph has been given in [29]. The amazing notion of fuzzy cognitive map structure is developed by researcher in [44] through establishing a concept of output issues and lowering the amount of concepts and connections between them. 1.2. Intuitionistic Fuzzy Graphs (IFGs) The traditional FS theory and graphs are extended into IFSs and IFGs. In order to handle circumstances like ambiguity and hesitation as well as the need for a more accom- modating model of the degree of MV, non-membership value (NMV) and reluctance, Atanassov originally proposed them in the 1980s. Developing an uncertain model: IFSs allow for a more complex depiction of uncertainty by incorporating the concept of resis- tance. To better reflect confusing or poorly understood information, IFS adds NMV and objects membership to MVs [37]. Making decisions in ambiguous circumstances: IFSs provide a framework for decision-making in ambiguous circumstances [[54], [55]]. Making decisions based on both MV and NMV helps decision-makers assess the level of resistance associated with various choices and reach more precise conclusions. Handling ambiguous and precise data: Graphs and IFSs can be useful in modelling ambiguous or imprecise data. By addressing both the degree of MV and the degree of ambiguity or uncertainty associated with the details, they increase the flexibility of information characterisation [[32], [36]]. Formal methodologies and procedures for the research and implementation of IFSs and IFGs are made possible by the robust mathematical underpinnings of the intu- M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2588 itionistic fuzzy objects. For use with IFSs and graphs, a wide range of operations, aggregation techniques and algorithms have been created. Applications encompass net- working, control systems, image recognition, clustering, and decision-making. Fuzzy sets and graphs with intuitive properties are employed in these and other domains. It has been demonstrated that they increase the precision and efficacy of decision-making processes and offer a helpful toolkit for handling ambiguous or incomplete data in a variety of sectors. Thus, by adding the concept of hesitation or indeterminacy, IFSs and IFGs enhance conventional FS theory and graphs. For managing ambiguity and incomplete information, they offer a more sophisticated framework that facilitates more precise analysis and taking decisions in a variety of domains [8]. The phrase strong IFG graph is used by the authors in [3], who also discuss various postulations about line graphs and self-complimentary. The scientists in [42] looked at IFG elements and used these concepts to look at other kinds of IFG elements. In [41] discusses an enhanced technique for identifying dominant vertex set IFGs. IFG theory is used to explain and evaluate the connectivity of uncertain networks; in addition, [10] looks at the vertex connectivity inside an IFG. In [49], various product operations are defined on IFG and some key concepts are illustrated on these graphs. In [45], the fuzzy graph energy idea is expanded to include IFG. The clustering of fuzzy and IFG vertices is the topic of the essay [34]. The same page also introduces a few IFG-related parameters. One can analyze [48] to get a sense of how connected IFG is. One can review [16] in order to analyze Index concurrently with connection Index. An intuitionistic fuzzy model has been used to analyze and make decisions for several individuals based on multiple factors. The two main pieces of information employed in [7] are the expert dependability ratings and the assessments of their methods. In [30], scientists computed the third and fourth iterations of the SOI for various graph families inside an IFG setting. They then provided an application that makes use of these indices to enhance the efficiency of immunization facilities. The kinds of intu- itionistic fuzzy rough graphs are specified in [61]. Nodes and links are the representation of physical networks, such as those found in circuits in electronics, biological intricate systems, digital networks and social networks. In these networks, things are represented by nodes and the relationships between them are shown by links. Cities may be seen as nodes in the transportation system, for instance and the routes connecting them as connections. Harry Wiener first used topological indices in 1947 when he examined how pure structural change affected paraffin’s boiling temperature in [58]. The paraffin boil- ing temperatures are found using the linear formula sγ = pα+qβ+r, where α is the sum of the distances between any two carbon atoms in a molecule. This was how the Wiener Index was first presented. Numerous requirements are met in [17] in terms of various graph features as measured by the Wiener and Harary Indices. The SOI, a novel graph constant, is defined in [25]. The relations between the SOI and topological indices is discussed in the paper [18]. [13] Discusses the features of Sombor Indices, examines the extreme values of many graphical networks and suggests possible uses. The 3rd, 4th, 5th and 6th iterations of SOI were established by Ivan Gutman in [26]. In [27], the geometric arithmetic Index and the atomic bond connectivity Index for M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2589 various graph networks are calculated. Using indices of topology is one method of deter- mining the correlation between a chemical compound’s structure and its characteristics [57]. A number of features for various graph families and applications pertaining to these indices are described in [[24]-[9]], which also define Sombor Index and topological indices of degree-two. A FGwhich makes it simple to define the fuzzy relationship between any item, is one of the most versatile mathematical tools available. In [33], a number of fundamental topological metrics are established in the context of FGs, this study covers limits of indices and its applications. In reference [1], the fuzzy Randic Index and fuzzy harmonic Index are introduced, their maximum values are derived and an application pertaining to cybercrime showcased. The novel fuzzy Wiener Index and connection Index for bibolar fuzzy incidence graphs are defined and their interrelationship s are examined in research article [23]. In the fuzzy structure, a few topological indices are specified and in [38], the corresponding properties were discussed for pizza-graph. Together with a cybercrime application shown. The intersection, union and extremals of bibolar FG indices are computed, established and analyzed in [47]. The distance between the IFSs is a well- known regularly utilized information metric to enhance decision-making performance. Conversely, using different distance metrics produces different numerical results. As a result, it is worthwhile to thoroughly research the procedure for selecting an appropriate formula for calculating distance. Two types of connection indices are defined by the IFG design and [40] shows examples of their use on the transport network and internet routing system. In reference [43], the complements of an IFG and a self-complimentary IFG are delineated, their characteristics are examined and some functions are also implemented for these graphs. In [20], a number of degree, order and size attributes are presented; the identical article also defines full and regular IFGs. It is confirmed that operations on a strong IFG produce another strong IFGs in [51], where three products are specified. Illustrations are used to define and describe a few products and references [[52], [2]] provide calculations for the degree of vertices of IFGs that resulted from this process. [[15], [56]] provide the notions of constant IFG, the 2nd type of IFG and generalized IFG. In [21], a few different kinds of irregular graphs are defined and some findings on completely irregular IFGs are also covered. 1.3. Neutrosophic Graphs(NGs) Regarding the neutrosophic sets (NSs), Smarandache gave them the go-ahead. The truth membership value (T MV), falsity membership value (FMV) and indeterminacy membership value (IMV) are all included. By this work, some of the features of the strong NG proposed in [39] have been examined and an example is shown. Estab- lishments are made about definitions, propositions, homomorphism and isomorphism theorems in strong neutrosophic graphs. [22] Investigated the Wiener Index in neutro- sophic graphs by Masoud Ghods and Zahra Rostami. A significant topological Index is the Wiener Index. With reference to the geodesic distance between two vertices, this Index is a distance-based Index. Specifically, following the definition of the Wiener Index M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2590 in neutrosophic graphs. [35] In addition to providing some applications for computer networks, highway systems and transport network flow, this author employed the notion of connectedness indices in NGs. The first six Sombor numbers for single-valued neu- trosophic fuzzy networks are defined by Anwar et al. in [6]. The six Sombor numbers for these basic graph families are then calculated for single-valued neutrosophic fuzzy graphs and the degree of vertices of various graph families is established in a single- valued neutrosophic fuzzy framework. AL-Omeri and Kaviyarasu used the NG idea in [4] to locate the Japan Earthquake Response Center. Complex NGs are constructed by fusing concepts from graph theory with complex NSs, as stated Alqahtani et al. in [5]. This offers an adaptable framework for handling challenging situations involving the solution of problems. A number of procedures, including union, join and composition are studied in detail to enhance the management of complex NGs. 1.4. Research Gaps The SOI was developed to extend the analysis toward molecular descriptors’ calcu- lation while the prior strategies pay their most attention to crisp graphs; however, crisp graphs are not well suited in uncertain or fuzzy scenarios. This gap emerges since, in real-world graphically modeled systems e.g. for decision- making in molecular chemistry and urban planning, there are inevitable uncertainties and ambiguities that crisp graph theory does not consider or handle well enough. The research question to address this gap could be: As a contribution to answering these questions, this paper aims to demonstrate the potential of certain formalisms in dealing with uncertainty in networks-neutrosophic graph theory and NSOI in particu- lar. 1.5. Motivation of Study This research is motivated by the increasing demand for sophisticated methodologies for system analysis where formal uncertainty prevails. Sombor Indices and FGs have some limitations in describing the indeterminacy, therefore, the proposed neutrosophic Sombor Indices expand a range of Indices and contribute to the investigation of the graph properties. Extending the Sombor Indices of IFGs, it provides fresh approaches toward uncertainty management in networks and can serve as a base for potential future studies in a variety of diverse disciplines ranging from thermal power plant site selection to brand optimization. 1.6. Objectives of the Study For crisp graphs, a wide range of topological indices have been investigated and shown to have several uses. However, it is seen in many real-world applications that many scenarios cannot be represented by crisp graphs and hence FGs can only handle MV. Determining a IFGs is necessary. To respond to this query, a IFGs must be defined. The definition of the Sombre index for IFGs and certain findings pertaining M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2591 to vertex and edge critical IFGs are presented in this work. The association between SOI of IFGs and the first Zagreb Index was also found. We used the first full SOI in site selection for thermal power plants at the conclusion of this work. 1.7. Structure of Article The following describes the structure of this research article: Section 2: Given the fundamental ideas behind neutrosophic graphs, Section 3 proposes a framework for SOI of NG and discusses their theorems and example. Applications for site selection for thermal power plant utilizing SOI of NG are described in Section 4. Section 5 provides findings and suggests ideas for further research at the end. Several symbols and their associated meanings are often used in this text. Table 1 below provides a summary of these symbols together with their explanations: Symbols Abbreviations Symbols Abbreviations SOI Sombor Index NGs Neutrosophic Graphs NFZI Neutrosophic First Zagreb Index NSZI Neutrosophic Second Zagreb Index FSs Fuzzy sets FGs Fuzzy Graph MV Membership Value IFGs Intuitionistic Fuzzy Graphs IFSs Intuitionistic Fuzzy Sets NMV Non Membership Value NSs Neutrosophic sets NGs Neutrosophic Graphs T MN Truth Membership Value IMN Indeterminacy Membership Value FMN Falsity Membership Value NSOI Neutrosophic Sombor Index NSOI Neutrosophic Sombor Index TSOI Truth Sombor Index ISOI Indeterminacy Sombor Index FSOI Falsity Sombor Index TFZI Truth First Zagreb Index IFZI Indeterminacy First Zagreb Index FFZI Falsity First Zagreb Index Table 1: List of symbols and abbreviations. 2. Preliminaries Definition 1. [26] Consider a simple graph G = (V, E). Let dα represent the degree of the vertex α ∈ V(G). I.Gutman developed a new vertex-degree-based topological Index called SOI, which is defined as follows: SOI(G) = ∑ α,β∈E(G) = √ (dα)2 + (dβ)2. Definition 2. [59] Assume D is a universal set. A fuzzy set S on D is a mapping σ : D → [0, 1]. σ represents the fuzzy set S membership function. S = (u, σ) represents a FS. Definition 3. [50] Given a FG such that (G,⊒, σ), the SOI of FG is stated that by M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2592 SOI(G) = ∑ α,β∈E(G) √ (ϖ(α)dG(α))2 + (ϖ(β)dG(β))2. Definition 4. [28] If G = (V,W,P) is a FG, the first Zagreb Index for FGs is stated below: NFZI(G) = ∑m i=1[ϖ(βi)dG(βi)] 2. Definition 5. [31] Let’s call G a FG. The fuzzy entire Zagreb Index of G is denoted by M z 1 , which is defined as M z 1 : ∑ β∈V (G)(ϖ(β)d(β))2 + ∑ ϱ∈E(G)(e(ϱ)d(ϱ)) 2. Definition 6. [31] Let (G,⊒, σ) be an FG. The second entire Zagreb Index of G is indicated by M z 2 and its defined by M z 2 : ∑ α,β is adjacent to eachother(w(α)d(α)w(β)d(β)) +∑ β is incidente to each orther(w(β)d(β)e(ϱ)d(ϱ)), where w is the MV of a vertex and e is the MV of an edge. Definition 7. [4] 1. A NG indicated as G = ((τα1, ıα2,𭟋α3), (τβ1, ıβ2,𭟋β3)) is sym- bolized as G∗ = (V, E), where V is the set of vertices and Eis the collection of edges. The functions τα1, ıα2 and 𭟋α3 are mappings from V to the closed interval [0, 1], various degrees of MV , IV and NMV,accordingly, for each element yi ∈ V. It holds that 0 ≤ τα1(yi) + ıα2(yi) +𭟋α3(yi) ≤ 3 for all yi ∈ V. 2. Furthermore in the framework of G∗, the functions τβ1 , ıβ2 and 𭟋β3 are mappings from V × V to the closed interval [0, 1], representing the degrees of MV , IV and NMV, accordingly for each edge. (yi, yj) ∈ E. τβ1(yi, yj) ≤ τα1(yi) ∧ τα1(yj), ıβ1(yi, yj) ≤ ıα1(yi) ∧ ıα1(yj), 𭟋β1(yi, yj) ≥ 𭟋α1(yi) ∧ Fα1(yj), 0 ≤ τβ1(yi, yj) + ıβ2(yi, yj) +𭟋β3(yi, yj) ≤ 3. 3. SOI of Neutrosophic Graphs Definition 8. Let G = (V, ϖ, ρ) be a NG then the SOI of neutrosophic graphs is define as, follows NSOI(G) = ∑ α,β∈E(G) √ (τϖ(α)τdG(α))2 + (τϖ(β)τdG(β))2 (3.1) + √ (ıϖ(α)ıdG(α))2 + (ıϖ(β)ıdG(β))2 + √ (𭟋ϖ(α)𭟋dG(α))2 + (𭟋ϖ(β)𭟋dG(β))2 Example 1. Let G be a NG with V(G) = {a, b, c, d} such that ϖ(a) = (0.7, 0.5, 0.3), ϖ(b) = (0.6, 0.4, 0.5), ϖ(c) = (0.5, 0.4, 0.2), ϖ(d) = (0.3, 0.2, 0.1), ρ(ab) = (0.6, 0.4, 0.5), ρ(bc) = (0.5, 0.4, 0.5), ρ(cd) = (0.3, 0.2, 0.2), ρ(ac) = (0.4, 0.5, 0.4), ρ(ad) = (0.3, 0.1, 0.4) dG(a) = M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2593 Figure 1: Neutrosophic graph (1.3, 1, 1.3), dG(b) = (1.1, 0.8, 1), dG(c) = (1.2, 1.1, 1.1), dG(d) = (0.6, 0.3, 0.6). NSOI(G) = ∑ α,β∈E(G) √ (τϖ(α)τdG(α))2 + (τϖ(β)τdG(β))2 + √ (ıϖ(α)ıdG(α))2 + (ıϖ(β)ıdG(β))2 + √ (𭟋ϖ(α)𭟋dG(α))2 + (𭟋ϖ(β)𭟋dG(β))2 NSOI(G) = √ ((0.7)(1.3))2 + ((0.6)(1.1))2 + ((0.5)(1))2 + ((0.4)(0.8))2 + ((0.3)(1.3))2 + ((0.5)(1.3))2 + √ ((0.7)(1.3))2 + ((0.5)(1.2))2 + ((0.5)(1))2 + ((0.4)(1.1))2 + ((0.3)(1.1))2 + ((0.2)(1.1))2 + √ ((0.7)(1.3))2 + ((0.3)(0.6))2 + ((0.5)(1))2 + ((0.2)(0.3))2 + ((0.3)(1.3))2 + ((0.1)(0.6))2 + √ ((0.6)(1.1))2 + ((0.5)(1.2))2 + ((0.4)(0.8))2 + ((0.4)(1.1))2 + ((0.5)(1))2 + ((0.2)(1.1))2 + √ ((0.5)(1.2))2 + ((0.3)(0.6))2 + ((0.4)(1.1))2 + ((0.2)(0.3))2 + ((0.2)(1.1))2 + ((0.1)(0.6))2 NSOI(G) = 6.5653. Theorem 1. Let the neutrosophic path graph be denoted by Pm̂. Consequently, NSOI(ρm̂) ≤ 6( √ 2(m̂− 3) + √ 5). Proof. A neutrosophic path G = Pm̂ has V(Pm̂) = {β1, β2, β3, ..βm̂} and E(Pm̂) = {ϱ1, ϱ2, ϱ3, ..ϱm̂−1}. Assume that ϖ1, ϖ2, ϖ3, ...., ϖm̂ and ρ1, ρ2, ρ3, ρm̂−1, respectively, denote the type ofMVs of Pm̂’s vertices and edges. Then clearly dG(β1) = ρ(ϱ1), dG(βm̂) = ρ(ϱm̂−1) and dG(βi) = ρ(ϱi) + ρ(ϱi+1),for 2 ≤ i ≤ m̂− 2. Therefore, NSOI(Pm̂) = TSOI(Pm̂) + ISOI(Pm̂) + FSOI(Pm̂) (3.2) TSOI(Pm̂) = ∑ α,β∈E(G) √ (τϖ(α)τdG(α))2 + (τϖ(β)τdG(β))2 M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2594 = √ (τϖ(β1)τdG(β1))2 + (τϖ(β2)τdG(β2))2 + √ (τϖ(βm̂−1)τdG(βm̂−1))2 + (τϖ(βm̂)τdG(βm̂))2 + ∑ βi,βj∈E(G)−{ϱ1,ϱm̂−1} √ (τϖ(βi)τdG(βi))2 + (τϖ(βj)τdG(βj))2 = √ (τϖ(β1)τρ(ϱ1))2 + (τϖ(β2)(τρ(ϱ1) + τρ(ϱ2)))2 + √ (τϖ(βm̂)τρ(ϱm̂−1))2 + (τϖ(βm̂−1)(τρ(ϱm̂−1) + τρ(ϱm̂−2)))2 + ∑ βi,βj∈E(G)−{ϱ1,ϱm̂−1} √ (τϖ(βi)(τρ(ϱi) + τρ(ϱj+1)))2 + (τϖ(βj)(τρ(ϱj) + ρ(ϱj+1)))2 = √ (τϖ(β1)(τρ(ϱ1))2 + (τϖ(β2)2(τρ(ϱ1)2 + τρ(ϱ2)2) + 2τρ(ϱ1)τρ(ϱ2)) + √ τϖ(βm̂)2τρ(ϱm̂−1)2 + (τϖ(βm̂−1)2(τρ(ϱm̂−1)2 + τρ(ϱm̂−2)2 + 2τρ(ϱm̂−1)τρ(ϱm̂−2))) + ∑ βi,βj∈E(G)−{ϱ1,ϱm̂−1} √√√√(τϖ(βi) 2(τρ(ϱi) 2 + τρ(ϱi+1) 2 + 2τρ(ϱi)τρ(ϱi+1))) + (τϖ(βj) 2(τρ(ϱj) 2 + τρ(ϱj+1) 2 + 2τρ(ϱj)τρ(ϱj+1))), since 0 ≤ τϖ(β) ≤ 1 and 0 ≤ τρ(ϱ) ≤ 1. Therefore, TSOI(Pm̂) = ∑ α,β∈E(G) √ 12.12 + 12(12 + 12) + 2(1))(1).1 + √ 12.12 + 12.12 + 12.12 + 2(12).1.1 = (m̂− 3) √ 12.12 + 12.12 + 2(12).1.1 + 12.12 + 12.12 + 2(12).1.1 TSOI(Pm̂) = 2 √ 2(m̂− 3) + 2 √ 5 TSOI(Pm̂) = 2( √ 2(m̂− 3) + √ 5) (3.3) ISOI(Pm̂) = ∑ α,β∈E(G) √ (ıϖ(α)ıdG(α))2 + (ıϖ(β)ıdG(β))2 = √ (ıϖ(β1)ıdG(β1))2 + (ıϖ(β2)ıdG(β2))2 + √ (ıϖ(βm̂−1)ıdG(βm̂−1))2 + (ıϖ(βm̂)ıdG(βm̂))2 + ∑ βi,βj∈E(G)−{ϱ1,ϱm̂−1} √ (ıϖ(βi)ıdG(βi))2 + (ıϖ(βj)ıdG(βj))2 = √ (ıϖ(β1)ıρ(ϱ1))2 + (ıϖ(β2)(ıρ(ϱ1) + ıρ(ϱ2)))2 + √ (ıϖ(βm̂)ıρ(ϱm̂−1))2 + (ıϖ(βm̂−1)(ıρ(ϱm̂−1) + ıρ(ϱm̂−2)))2 M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2595 + ∑ βi,βj∈E(G)−{ϱ1,ϱm̂−1} √ (ıϖ(βi)(ıρ(ϱi) + ıρ(ϱj+1)))2 + (ıϖ(βj)(ıρ(ϱj) + ρ(ϱj+1)))2 = √ (ıϖ(β1)(ıρ(ϱ1))2 + (ıϖ(β2)2(ıρ(ϱi)2 + ıρ(ϱ2)2) + 2ıρ(ϱ1)ıρ(ϱ2)) + √ ıϖ(βm̂)2ıρ(ϱm̂−1)2 + (ıϖ(βm̂−1)2(ıρ(ϱm̂−1)2 + ıρ(ϱm̂−2)2 + 2ıρ(ϱm̂−1)ıρ(ϱm̂−2))) + ∑ βi,βj∈E(G)−{ϱ1,ϱm̂−1} √√√√(ıϖ(βi) 2(ıρ(ϱi) 2 + ıρ(ϱi+1) 2 + 2ıρ(ϱi)ıρ(ϱi+1))) + (ıϖ(βj) 2(ıρ(ϱj) 2 + ıρ(ϱj+1) 2 + 2ıρ(ϱj)ıρ(ϱj+1))), since 0 ≤ ıϖ(α) ≤ 1 and 0 ≤ ıρ(ϱ) ≤ 1. Therefore, ISOI(Pm̂) = ∑ α,β∈E(G) √ 12.12 + 12(12 + 12) + 2(1))(1).1 + √ 12.12 + 12.12 + 12.12 + 2(12).1.1 = (m̂− 3) √ 12.12 + 12.12 + 2(12).1.1 + 12.12 + 12.12 + 2(12).1.1 ISOI(Pm̂) = 2 √ 2(m̂− 3) + 2 √ 5 ISOI(Pm̂) = 2( √ 2(m̂− 3) + √ 5) (3.4) and FSOI(Pm̂) = ∑ α,β∈E(G) √ (𭟋ϖ(α)𭟋dG(α))2 + (𭟋ϖ(β)𭟋dG(β))2 = √ (𭟋ϖ(β1)𭟋dG(β1))2 + (𭟋ϖ(β2)𭟋dG(β2))2 + √ (𭟋ϖ(βm̂−1)𭟋dG(βm̂−1))2 + (𭟋ϖ(βm̂)𭟋dG(βm̂))2 + ∑ βi,βj∈E(G)−{ϱ1,ϱm̂−1} √ (𭟋ϖ(βi)𭟋dG(βi))2 + (𭟋ϖ(βj)𭟋dG(βj))2 = √ (𭟋ϖ(β1)𭟋ρ(ϱ1))2 + (𭟋ϖ(β2)(𭟋ρ(ϱ1) +𭟋ρ(ϱ2)))2 + √ (𭟋ϖ(βm̂)𭟋ρ(ϱm̂−1))2 + (𭟋ϖ(βm̂−1)(𭟋ρ(ϱm̂−1) +𭟋ρ(ϱm̂−2)))2 + ∑ βi,βj∈E(G)−{ϱ1,ϱm̂−1} √ (𭟋ϖ(βi)(𭟋ρ(ϱi) +𭟋ρ(ϱj+1)))2 + (𭟋ϖ(βj)(𭟋ρ(ϱj) + ρ(ϱj+1)))2 = √ (𭟋ϖ(β1)(𭟋ρ(ϱ1))2 + (𭟋ϖ(β2)2(𭟋ρ(ϱi)2 +𭟋ρ(ϱ2)2) + 2𭟋ρ(ϱ1)𭟋ρ(ϱ2) + √ 𭟋ϖ(βm̂)2𭟋ρ(ϱm̂−1)2 + (𭟋ϖ(βm̂−1)2(𭟋ρ(ϱm̂−1)2 +𭟋ρ(ϱm̂−2)2 + 2𭟋ρ(ϱm̂−1)𭟋ρ(ϱm̂−2))) + ∑ βi,βj∈E(G)−{ϱ1,ϱm̂−1} √√√√(𭟋ϖ(βi) 2(𭟋ρ(ϱi) 2 +𭟋ρ(ϱi+1) 2 + 2𭟋ρ(ϱi)𭟋ρ(ϱi+1))) + (𭟋ϖ(βj) 2(𭟋ρ(ϱj) 2 +𭟋ρ(ϱj+1) 2 + 2𭟋ρ(ϱj)𭟋ρ(ϱj+1))), M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2596 since 0 ≤ 𭟋ϖ(β) ≤ 1 and 0 ≤ 𭟋ρ(ϱ) ≤ 1. Therefore, FSOI(Pm̂) = ∑ α,β∈E(G) √ 12.12 + 12(12 + 12) + 2(1))(1).1 + √ 12.12 + 12.12 + 12.12 + 2(12).1.1 = (m̂− 3) √ 12.12 + 12.12 + 2(12).1.1 + 12.12 + 12.12 + 2(12).1.1 FSOI(Pm̂) = 2 √ 2(m̂− 3) + 2 √ 5 FSOI(Pm̂) = 2( √ 2(m̂− 3) + √ 5). (3.5) Substitute (3.3), (3.4) and (3.5) in (3.2), NSOI(Pm̂) = 2( √ 2(m̂− 3) + √ 5) + 2( √ 2(m̂− 3) + √ 5) + 2( √ 2(m̂− 3) + √ 5) = 6( √ 2(m̂− 3) + √ 5). Hance proof is compete. Theorem 2. Let Cm̂ denote a neutrosophic cycle graph respectively, then NSOI(Cm̂) ≤ 6 √ 2m̂. Proof. Let G = Cm̂ be a neutrosophic cycle with V(Cm̂) = {v1, v2, v3, ..., vm̂} and E(Cm̂) = {ϱ1, ϱ2, ϱ3, ..., ϱm̂−1}. Let ϖ1, ϖ2, ϖ3, ..., ϖm̂ and ρ1, ρ2, ρ3, ..., ρm̂−1 be the there of type of MVs of vertices and edges of Cm̂ respectively. Then, clearly dG(βi) = ρ(ϱi) + ρ(ϱi+1). Therefore, NSOI(Cm̂) = TSOI(Cm̂) + ISOI(Cm̂) + FSOI(Cm̂) (3.6) TSOI(Cm̂) = ∑ α,β∈E(G) √ (τϖ(α)τdG(α))2 + (τϖ(β)τdG(β))2 = ∑ βi,βj∈E(G) √ (τϖ(βi)(τρ(ϱi) + τρ(ϱi+1)))2 + √ (τϖ(βj)(τρ(ϱj) + τρ(ϱj+1)))2 + ∑ βi,βj∈E(G) √√√√(τϖ(βi) 2(τρ(ϱi) 2 + τρ(ϱi+1) 2 + 2τρ(ϱi)τρ(ϱi+1))) + (τϖ(βj) 2(τρ(ϱj) 2 + τρ(ϱj+1) 2 + 2τρ(ϱj)τρ(ϱj+1))), since 0 ≤ τϖ(β) ≤ 1 and 0 ≤ τρ(ϱ) ≤ 1. Therefore, TSOI(Cm̂) ≤ n √ 12.12 + 12.12 + 2(12).1.1 + 12.12 + 12.12 + 2(12).1.1 TSOI(Cm̂) ≤ 2 √ 2m̂ (3.7) ISOI(Cm̂) = ∑ α,β∈E(G) √ (ıϖ(α)ıdG(α))2 + (ıϖ(β)ıdG(β))2 M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2597 = ∑ βi,βj∈E(G) √ (ıϖ(βi)(ıρ(ϱi) + ıρ(ϱi+1)))2 + √ (ıϖ(βj)(ıρ(ϱj) + ıρ(ϱj+1)))2 + ∑ βi,βj∈E(G) √√√√(ıϖ(βi) 2(ıρ(ϱi) 2 + ıρ(ϱi+1) 2 + 2ıρ(ϱi)ıρ(ϱi+1))) + (ıϖ(βj) 2(ıρ(ϱj) 2 + ıρ(ϱj+1) 2 + 2ıρ(ϱj)ıρ(ϱj+1))), since 0 ≤ ıϖ(β) ≤ 1 and 0 ≤ ıρ(ϱ) ≤ 1. Therefore, ISOI(Cm̂) ≤ n √ 12.12 + 12.12 + 2(12).1.1 + 12.12 + 12.12 + 2(12).1.1 ISOI(Cm̂) ≤ 2 √ 2m̂. (3.8) FSOI(Cm̂) = ∑ α,β∈E(G) √ (𭟋ϖ(α)𭟋dG(α))2 + (𭟋ϖ(β)𭟋dG(β))2 = ∑ βi,βj∈E(G) √ (𭟋ϖ(βi)(𭟋ρ(ϱi) +𭟋ρ(ϱi+1)))2 + √ (𭟋ϖ(βj)(𭟋ρ(ϱj) +𭟋ρ(ϱj+1)))2 + ∑ βi,βj∈E(G) √√√√(𭟋ϖ(βi) 2(𭟋ρ(ϱi) 2 +𭟋ρ(ϱi+1) 2 + 2𭟋ρ(ϱi)𭟋ρ(ϱi+1))) + (𭟋ϖ(βj) 2(𭟋ρ(ϱj) 2 +𭟋ρ(ϱj+1) 2 + 2𭟋ρ(ϱj)𭟋ρ(ϱj+1))), since , 0 ≤ 𭟋ϖ(β) ≤ 1 and 0 ≤ 𭟋ρ(ϱ) ≤ 1. Therefore, FSOI(Cm̂) ≤ n √ 12.12 + 12.12 + 2(12).1.1 + 12.12 + 12.12 + 2(12).1.1 FSOI(Cm̂) ≤ 2 √ 2m. (3.9) Substitute (3.7), (3.8) and (3.9) in (3.6). NSOI(ρm̂) ≤ 2 √ 2m+ 2 √ 2m+ 2 √ 2m NSOI(ρm̂) ≤ 6 √ 2m. Hance proof is compete. Theorem 3. Let Km̂ denotes neutrosophic complete graph respectively, then NSOI(Km̂) ≤ 3( m̂(m̂−1) 2 √ 2m̂− 2). Proof. Let G = Km̂ be a neutrosophic complete graph with V(Km̂) = {β1, β2, β3, ..., βm̂} and E(Km̂) = {ϱ1, ϱ2, ϱ3, .., ϱm̂−1}. Let ϖ1, ϖ2, ϖ3, ..., ϖm̂ and ρ1, ρ2, ρ3, ..., ρm̂ be the there of type of MVs of vertices and edges of Km̂ respectively. Then clearly, dG(βi) = ∑ βi∼ϱj ρ(ϱj). Therefore, NSOI(Km̂) = TSOI(Km̂) + ISOI(Km̂) + FSOI(Km̂) (3.10) M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2598 TSOI(Km̂) = ∑ α,β∈E(G) √ (τϖ(α)τdG(α))2 + (τϖ(β)τdG(β))2 = ∑ βi,βj∈E(G) √ (τϖ(βi)2( ∑ βi∼ϱj τρ(ϱj)2)) + (τϖ(βj)2( ∑ βj∼ϱi τρ(ϱi)2)) ≤ ∑ βi,βj∈E(G) √ τϖ(βi)2(m̂− 1)τρ(ϱj)2 + τϖ(βj)2(m̂− 1)τρ(ϱi)2, since, 0 ≤ τϖ(β) ≤ 1 and 0 ≤ τρ(ϱ) ≤ 1. Therefore, TSOI(Km̂) ≤ ∑ βi,βj∈E(G) √ (m̂− 1) + (m̂− 1) TSOI(Km̂) = m̂(m̂− 1) 2 √ 2m̂− 2. (3.11) ISOI(km̂) = ∑ α,β∈E(G) √ (ıϖ(α)ıdG(α))2 + (ıϖ(β)ıdG(β))2 = ∑ βi,βj∈E(G) √ (ıϖ(βi)2( ∑ βi∼ϱj ıρ(ϱj)2)) + (ıϖ(βj)2( ∑ βj∼ϱi ıρ(ϱi)2)) ≤ ∑ βi,βj∈E(G) √ ıϖ(βi)2(m̂− 1)ıρ(ϱj)2 + ıϖ(βj)2(m̂− 1)ıρ(ϱi)2, since, 0 ≤ ıϖ(β) ≤ 1 and 0 ≤ ıρ(ϱ) ≤ 1. Therefore, ISOI(Km̂) ≤ ∑ βi,βj∈E(G) √ (m̂− 1) + (m̂− 1) ISOI(Km̂) = m̂(m̂− 1) 2 √ 2m̂− 2 (3.12) and FSOI(Km̂) = ∑ α,β∈E(G) √ (𭟋ϖ(α)𭟋dG(α))2 + (𭟋ϖ(β)𭟋dG(β))2 = ∑ βi,βj∈E(G) √ (𭟋ϖ(βi)2( ∑ βi∼ϱj 𭟋ρ(ϱj)2)) + (𭟋ϖ(βj)2( ∑ βj∼ϱi 𭟋ρ(ϱi)2)) ≤ ∑ βi,βj∈E(G) √ 𭟋ϖ(βi)2(m̂− 1)𭟋ρ(ϱj)2 +𭟋ϖ(βj)2(m̂− 1)𭟋ρ(ϱi)2, M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2599 since, 0 ≤ 𭟋ϖ(β) ≤ 1 and 0 ≤ 𭟋ρ(ϱ) ≤ 1. Therefore, FSOI(Km̂) ≤ ∑ βi,βj∈E(G) √ (m̂− 1) + (m̂− 1) FSOI(Km̂) = m̂(m̂− 1) 2 √ 2m̂− 2. (3.13) Substitute (3.11), (3.12) and (3.13) in (3.10), we get NSOI(Km̂) ≤ m̂(m̂− 1) 2 √ 2m̂− 2 + m̂(m̂− 1) 2 √ 2m̂− 2 + m̂(m̂− 1) 2 √ 2m̂− 2 NSOI(Km̂) ≤ 3( m̂(m̂− 1) 2 ) √ 2m̂− 2. Lemma 1. Let G = K1,m̂−1 be a neutrosophic star and statisties the condition τϖ(0) ≤ τϖ(β), ıϖ(0) ≤ ıϖ(β) and 𭟋ϖ(0) ≤ 𭟋ϖ(β) where 0 is the center of the neurotrophic star, then, NSOI(K1, m̂− 1) ≤ 3(m̂− 1) √ m̂2 − 2m̂+ 2. Proof. Let G = K1,m̂−1 be a neutrosophic star graph with V(K1,m̂−1) = {β1, β2, β3, .....βm̂} and E(K1,m̂−1) = {ϱ1, ϱ2, ϱ3, .....ϱm̂−1}. Let ϖ1, ϖ2, ϖ3, .....ϖm̂ and ρ1, ρ2, ρ3, .....ρm̂ be the MVs of vertices and edges of Km̂. Let β1 = 0 be the center of the star. It is given that ϖ(0) ≤ ϖ(β). Then, clearly dG(βi) = ϖ(0) and dG(0) = (m̂− 1)ϖ(0). Therefor, NSOI(K1, m̂− 1) = TSOI(K1, m̂− 1) + ISOI(K1, m̂− 1) + FSOI(K1, m̂− 1) (3.14) TSOI(K1, m̂− 1) = ∑ α,β∈E(G) √ (τϖ(α)τdG(α))2 + (τϖ(β)τdG(β))2 = ∑ βi,βj∈E(G) √ (τϖ(βi)2(τdG(0))2) + (τϖ(βj)2(τdG(βj))2) = ∑ βi,βj∈E(G) √ (τϖ(βi)2((m̂− 1)τϖ(0))2) + (τϖ(βj)2(τϖ(0)))2, since , 0 ≤ τϖ(β) ≤ 1 and 0 ≤ τρ(ϱ) ≤ 1. Therefore, TSOI(K1, m̂− 1) ≤ (m̂− 1)[ √ (12.((m̂− 1)2.12)) + (12.12)] = (m̂− 1) √ m̂2 − 2m̂+ 2 (3.15) M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2600 ISOI(K1, m̂− 1) = ∑ α,β∈E(G) √ (ıϖ(α)ıdG(α))2 + (ıϖ(β)ıdG(β))2 = ∑ βi,βj∈E(G) √ (ıϖ(βi)2(ıdG(0))2) + (ıϖ(βj)2(ıdG(βj))2) = ∑ βi,βj∈E(G) √ (ıϖ(βi)2((m̂− 1)ıϖ(0))2) + (ıϖ(βj)2(ıϖ(0)))2, since 0 ≤ ıϖ(β) ≤ 1 and 0 ≤ ıρ(ϱ) ≤ 1. Therefore, ISOI(K1, m̂− 1) ≤ (m̂− 1)[ √ (12.((m̂− 1)2.12)) + (12.12)] ISOI(K1, m̂− 1) = (m̂− 1) √ m̂2 − 2m̂+ 2 (3.16) FSOI(K1, m̂− 1) = ∑ α,β∈E(G) √ (𭟋ϖ(α)𭟋dG(α))2 + (𭟋ϖ(β)𭟋dG(β))2 = ∑ βi,βj∈E(G) √ (𭟋ϖ(βi)2(𭟋dG(0))2) + (𭟋ϖ(βj)2(𭟋dG(βj))2) = ∑ βi,βj∈E(G) √ (𭟋ϖ(βi)2((m̂− 1)𭟋ϖ(0))2) + (𭟋ϖ(βj)2(𭟋ϖ(0)))2, since 0 ≤ 𭟋ϖ(β) ≤ 1 and 0 ≤ 𭟋ρ(ϱ) ≤ 1. Therefore, FSOI(K1, m̂− 1) ≤ (m̂− 1)[ √ (12.((m̂− 1)2.12)) + (12.12)] FSOI(K1, m̂− 1) = (m̂− 1) √ m̂2 − 2m̂+ 2 (3.17) Substitute (3.15), (3.16) and (3.17) in (3.14), we get NSOI(K1, m̂− 1) = (m̂− 1) √ m̂2 − 2m̂+ 2 + (m̂− 1) √ m̂2 − 2m̂+ 2 + (m̂− 1) √ m̂2 − 2m̂+ 2 NSOI(K1, m̂− 1) = 3((m̂− 1) √ m̂2 − 2m̂+ 2). Hence the proof. Definition 9. Let G = (V, ϖ, ρ) be a NG then the first Zagreb Index of neutrosophic graph is defined as follows: NFZI(G) = ∑ α,β∈E(G) [(τϖ(α)τdG(α)) + (τϖ(β)τdG(β))] + [(ıϖ(α)ıdG(α)) + (ıϖ(β)ıdG(β))] (3.18) + [(𭟋ϖ(α)𭟋dG(α)) + (𭟋ϖ(β)𭟋dG(β))] M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2601 Theorem 4. Let G = (V, ϖ, ρ) be a NFZI of graph denoted the first Zagreb Index for neurotrophic graphs. Then NSOI(G) ≤ NFZI(G). Proof. Let G = (V,W,P) be a NG. By the definition of SOI for NGs we have, NSOI(G) = TSOI(G) + ISOI(G) + FSOI(G) (3.19) TSOI(G) = ∑ α,β∈E(G) √ (τϖ(α)τdG(α))2 + (τϖ(β)τdG(β))2 = ∑ α,β∈E(G) √ (τϖ(α)τdG(α)) + (τϖ(β)τdG(β))2 − 2τϖ(α)τϖ(β)τdG(α)τdG(β) ≤ ∑ α,β∈E(G) √ (τϖ(α)τdG(α) + τϖ(β)τdG(β))2 TSOI(G) = ∑ α,β∈E(G) (τϖ(α)τdG(α) + τϖ(β)τdG(β)) (3.20) ISOI(G) = ∑ α,β∈E(G) √ (ıϖ(α)ıdG(α))2 + (ıϖ(β)ıdG(β))2 = ∑ α,β∈E(G) √ (ıϖ(α)ıdG(α)) + (ıϖ(β)ıdG(β))2 − 2ıϖ(α)ıϖ(β)ıdG(α)ıdG(β) ≤ ∑ α,β∈E(G) √ (ıϖ(α)ıdG(α) + ıϖ(β)ıdG(β))2 ISOI(G) = ∑ α,β∈E(G) (ıϖ(α)ıdG(α) + ıϖ(β)ıdG(β)) (3.21) FSOI(G) = ∑ α,β∈E(G) √ (𭟋ϖ(α)𭟋dG(α))2 + (𭟋ϖ(β)𭟋dG(β))2 = ∑ α,β∈E(G) √ (𭟋ϖ(α)𭟋dG(α)) + (𭟋ϖ(β)𭟋dG(β))2 − 2𭟋ϖ(α)𭟋ϖ(β)𭟋dG(α)𭟋dG(β) ≤ ∑ α,β∈E(G) √ (𭟋ϖ(α)𭟋dG(α) +𭟋ϖ(β)𭟋dG(β))2 FSOI(G) = ∑ α,β∈E(G) (𭟋ϖ(α)𭟋dG(α) +𭟋ϖ(β)𭟋dG(β)) (3.22) M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2602 Substitute (3.20), (3.21) and (3.22) in (3.19), we get NSOI(G) = ∑ α,β∈E(G) (τϖ(α)τdG(α) + τϖ(β)τdG(β)) + ∑ α,β∈E(G) (ıϖ(α)ıdG(α) + ıϖ(β)ıdG(β)) + ∑ α,β∈E(G) (𭟋ϖ(α)𭟋dG(α) +𭟋ϖ(β)𭟋dG(β)) NSOI(G) = NFZI(G). Example 2. Let G be a NG with V(G) = {p, q, r, s, t} such that ϖ(p) = (0.7, 0.6, 0.4), ϖ(q) = (0.6, 0.5, 0.4), ϖ(s) = (0.5, 0.4, 0.3), ϖ(r) = (0.5, 0.4, 0.3), ϖ(t) = (0.8, 0.7, 0.4) and ϖ(pq) = (0.6, 0.4, 0.5), ϖ(qr) = (0.4, 0.4, 0.5), ϖ(rs) = (0.4, 0.3, 0.5), ϖ(st) = (0.5, 0.4, 0.5), ϖ(tq) = (0.6, 0.5, 0.5). dG(p) = (0.6, 0.4, 0.5), dG(q) = (1, 0.9, 1), dG(r) = (0.8, 0.7, 1), dG(s) = (0.9, 0.7, 0.5), Figure 2: Neutrosophic Graph dG(t) = (1.1, 0.9, 1). NSOI(G) = TSOI(G) + ISOI(G) + FSOI(G) (3.23) TSOI(G) = ∑ α,β∈E(G) √ (τϖ(α)τdG(α))2 + (τϖ(β)τdG(β))2 = √ (τϖ(p)τdG(p))2 + (τϖ(q)τdG(q))2 + √ (τϖ(q)τdG(q))2 + (τϖ(r)τdG(r))2 + √ (τϖ(r)τdG(r))2 + (τϖ(s)τdG(s))2 + √ (τϖ(s)τdG(s))2 + (τϖ(t)τdG(t))2 + √ (τϖ(t)τdG(t))2 + (τϖ(q)τdG(q))2 M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2603 TSOI(G) = √ ((0.7)(0.6))2 + ((0.6)(1))2 + √ ((0.6)(1))2 + ((0.5)(0.8))2 + √ ((0.5)(0.8))2 + ((0.5)(0.9))2 + √ ((0.5)(0.9))2 + ((0.8)(1.1))2 + √ ((0.8)(1.1))2 + ((0.6)(1))2 TSOI(G) = 4.6069 (3.24) ISOI(G) = ∑ α,β∈E(G) √ (ıϖ(α)ıdG(α))2 + (ıϖ(β)ıdG(β))2 = √ (ıϖ(p)ıdG(p))2 + (ıϖ(q)ıdG(q))2 + √ (ıϖ(q)ıdG(q))2 + (ıϖ(r)ıdG(r))2 + √ (ıϖ(r)ıdG(r))2 + (ıϖ(s)ıdG(s))2 + √ (ıϖ(s)ıdG(s))2 + (ıϖ(t)ıdG(t))2 + √ (ıϖ(t)ıdG(t))2 + (ıϖ(q)ıdG(q))2 ISOI(G) = √ ((0.6)(0.4))2 + ((0.5)(0.6))2 + √ ((0.5)(0.7))2 + ((0.4)(0.8))0.7 + √ ((0.4)(0.7))2 + ((0.4)(0.7))2 + √ ((0.4)(0.7))2 + ((0.7)(0.9))2 + √ ((0.7)(0.9))2 + ((0.5)(0.9))2 ISOI(G) = 2.7387 (3.25) and FSOI(G) = ∑ α,β∈E(G) √ (𭟋ϖ(α)𭟋dG(α))2 + (𭟋ϖ(β)𭟋dG(β))2 = √ (𭟋ϖ(p)𭟋dG(p))2 + (𭟋ϖ(q)𭟋dG(q))2 + √ (𭟋ϖ(q)𭟋dG(q))2 + (𭟋ϖ(r)𭟋dG(r))2 + √ (𭟋ϖ(r)𭟋dG(r))2 + (𭟋ϖ(s)𭟋dG(s))2 + √ (𭟋ϖ(s)𭟋dG(s))2 + (𭟋ϖ(t)𭟋dG(t))2 + √ (𭟋ϖ(t)𭟋dG(t))2 + (𭟋ϖ(q)𭟋dG(q))2 FSOI(G) = √ ((0.4)(0.5))2 + ((0.4)(1))2 + √ ((0.4)(1))2 + ((0.3)(1))0.7 + √ ((0.3)(1))2 + ((0.3)(0.5))2 + √ ((0.3)(0.5))2 + ((0.4)(0.1))2 + √ ((0.4)(1))2 + ((0.4)(1))2 FSOI(G) = 2.7387 (3.26) Substitute (3.24), (3.25) and (3.26) in (3.23), we get NSOI(G) = 4.6069 + 2.7387 + 2.7387 NSOI(G) = 9.2297. Now consider the first Zagreb Index of neutrosophic graphs. NFZI(G) = TFZI(G) + IFZI(G) + FFZI(G) (3.27) TFZI(G) = ∑ α,β∈E(G) [τϖ(α)τdG(α) + τϖ(β)τdG(β)] = (τϖ(p)τdG(p) + τϖ(q)τdG(q)) + (τϖ(q)τdG(q) + τϖ(r)τdG(r)) + (τϖ(r)τdG(r) M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2604 + τϖ(s)τdG(s)) + (τϖ(s)τdG(s) + τϖ(t)τdG(t)) + (τϖ(t)τdG(t) + τϖ(q)τdG(q)) = ((0.7)(0.6) + (0.6)(1)) + ((0.6)(1) + (0.5)(0.8)) + ((0.5)(0.8) + (0.5)(0.9)) + ((0.5)(0.9) + (0.8)(1.1)) + ((0.8)(1.1) + (0.6)(1)) TFZI(G) = 6.8 (3.28) IFZI(G) = ∑ α,β∈E(G) [ıϖ(α)ıdG(α) + ıϖ(β)ıdG(β)] = (ıϖ(p)ıdG(p) + ıϖ(q)ıdG(q)) + (ıϖ(q)ıdG(q) + ıϖ(r)ıdG(r)) + (ıϖ(r)ıdG(r) + ıϖ(s)ıdG(s)) + (ıϖ(s)ıdG(s) + ıϖ(t)ıdG(t)) + (ıϖ(t)ıdG(t) + ıϖ(q)ıdG(q)) = ((0.6)(0.4) + (0.5)(0.6)) + ((0.5)(0.7) + (0.4)(0.8)) + ((0.4)(0.7) + (0.4)(0.7)) + ((0.4)(0.7) + (0.7)(0.9)) + ((0.7)(0.9) + (0.5)(0.9)) IFZI(G) = 3.87 (3.29) FFZI(G) = ∑ α,β∈E(G) [𭟋ϖ(α)𭟋dG(α) +𭟋ϖ(β)𭟋dG(β)] = (𭟋ϖ(p)𭟋dG(p) +𭟋ϖ(q)𭟋dG(q)) + (𭟋ϖ(q)𭟋dG(q) +𭟋ϖ(r)𭟋dG(r)) + (𭟋ϖ(r)𭟋dG(r) +𭟋ϖ(s)𭟋dG(s)) + (𭟋ϖ(s)𭟋dG(s) +𭟋ϖ(t)𭟋dG(t)) + (𭟋ϖ(t)𭟋dG(t) +𭟋ϖ(q)𭟋dG(q)) FFZI(G) = ((0.4)(0.5) + (0.4)(1)) + ((0.4)(1) + (0.3)(1)) + ((0.3)(1) + (0.3)(0.5)) + ((0.3)(0.5) + (0.4)(0.1)) + ((0.4)(1) + (0.4)(1)) FFZI(G) = 3.04 (3.30) Substitute (3.28), (3.29) and (3.30) in (3.27), we get NFZI(G) = 6.8 + 3.87 + 3.04 NFZI(G) = 13.71. Thus, cleary NSOI(G) < NFZI(G). Theorem 5. let G = (V, ϖ, ρ) be a n-vertex NG with m-edges. Then NSOI(G) ≥ NSOI(G − e), where e ∈ E(G). Proof. Let G = (V,W,P) be a NG and H = G − e is a graph obtained by removing an edge e ∈ E(G). The MVs in G and H are given by the relationship. τϖG(β) ≥ τϖH(β), ıϖG(β) ≥ ıϖH(β) and𭟋ϖG(β) ≥ 𭟋ϖH(β) and τρG(ϱ) ≥ τρH(ϱ), ıρG(ϱ) ≥ ıρH(ϱ) and 𭟋ρG(ϱ) ≥ 𭟋ρH(ϱ). This show that τdG(β) ≥ τdH(β), ıdG(β) ≥ ıdH(β) and 𭟋dG(β) ≥ 𭟋dH(β). Now, NSOI(G) = TSOI(G) + ISOI(G) + FSOI(G) (3.31) TSOI(G) = ∑ α,β∈E(G) √ (τϖ(α)τdG(α))2 + (τϖ(β)τdG(β))2 M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2605 ≥ ∑ α,β∈E(G) √ (τϖ(α)τdH(α))2 + (τϖ(β)τdH(β))2 = TSOI(H) = TSOI(G − e) (3.32) ISOI(G) = ∑ α,β∈E(G) √ (ıϖ(α)ıdG(α))2 + (ıϖ(β)ıdG(β))2 ≥ ∑ α,β∈E(G) √ (ıϖ(α)ıdH(α))2 + (ıϖ(β)ıdH(β))2 = ISOI(H) ISOI(G) = ISOI(G − e) (3.33) FSOI(G) = ∑ α,β∈E(G) √ (𭟋ϖ(α)𭟋dG(α))2 + (𭟋ϖ(β)𭟋dG(β))2 ≥ ∑ α,β∈E(G) √ (𭟋ϖ(α)𭟋dH(α))2 + (𭟋ϖ(β)𭟋dH(β))2 = FSOI(H) FSOI(G) = FSOI(G − e) (3.34) Substitute (3.32), (3.33) and (3.34) in (3.31), we get NSOI(G) ≥ NSOI(G − ϱ). Example 3. Form the Example 2, we can obtained, NSPI(G)= 9.2297. Now, Let H = G − {bc} be a graph obtained by removing an edge bc ∈ E(G). The MVs of the vertices of H will remain same as in G but there is a change in degree of the b and c in H. Then dG(b) = (1.2, 0.9, 0.9) and then dG(c) = (0.4, 0.3, 0.5). Now, NSOI(G) = TSOI(H) + ISOI(H) + FSOI(H) (3.35) TSOI(G) = ∑ α,β∈E(H) √ (τϖ(α)τdH(α))2 + (τϖ(β)τdH(β))2 = √ (τϖ(a)τdH(a))2 + (τϖ(b)τdH(b))2 + √ (τϖ(b)τdH(b))2 + (τϖ(e)τdH(e))2 + √ (τϖ(ϱ)τdH(ϱ))2 + (τϖ(d)τdH(d))2 + √ (τϖ(d)τdH(d))2 + (τϖ(c)τdH(c))2 TSOI(G) = √ ((0.7)(0.6))2 + ((0.6)(1.2))2 + √ ((0.6)(1.2))2 + ((0.8)(1.1))2 M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2606 Figure 3: NG on 5-vertices + √ ((0.8)(1.1))2 + ((0.5)(0.9))2 + √ ((0.5)(0.9))2 + ((0.5)(0.4))2 = 0.833 + 1.137 + 0.988 + 0.4924 TSOI(G) = 3.4504 (3.36) ISOI(G) = ∑ α,β∈E(H) √ (ıϖ(α)τdH(α))2 + (ıϖ(β)ıdH(β))2 = √ (ıϖ(a)ıdH(a))2 + (ıϖ(b)ıdH(b))2 + √ (ıϖ(b)ıdH(b))2 + (ıϖ(e)ıdH(e))2 + √ (ıϖ(e)ıdH(e))2 + (ıϖ(d)ıdH(d))2 + √ (ıϖ(d)ıdH(d))2 + (ıϖ(c)ıdH(c))2 ISOI(G) = √ ((0.6)(0.4))2 + ((0.5)(0.9))2 + √ ((0.5)(0.9))2 + ((0.7)(0.9))2 + √ ((0.7)(0.9))2 + ((0.4)(0.7))2 + √ ((0.4)(0.7))2 + ((0.4)(0.3))2 = 0.51 + 0.774 + 0.689 + 0.3046 ISOI(G) = 2.2776 (3.37) FSOI(G) = ∑ α,β∈E(H) √ (𭟋ϖ(α)τdH(α))2 + (𭟋ϖ(β)𭟋dH(β))2 = √ (𭟋ϖ(a)𭟋dH(a))2 + (𭟋ϖ(b)𭟋dH(b))2 + √ (𭟋ϖ(b)𭟋dH(b))2 + (𭟋ϖ(e)𭟋dH(e))2 + √ (𭟋ϖ(e)𭟋dH(e))2 + (𭟋ϖ(d)𭟋dH(d))2 + √ (𭟋ϖ(d)𭟋dH(d))2 + (𭟋ϖ(c)𭟋dH(c))2 M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2607 FSOI(G) = √ ((0.4)(0.5))2 + ((0.4)(1.0))2 + √ ((0.4)(1.0))2 + ((0.4)(1))2 + √ ((0.4)(1))2 + ((0.3)(1))2 + √ ((0.3)(1))2 + ((0.3)(0.5))2 = 0.447 + 0.565 + 0.5 + 0.334 FSOI(G) = 1.846 (3.38) Substitute (3.36), (3.37) and (3.38) in (3.35), we get NZI(G) = 3.4504 + 2.2776 + 1.846 NZI(G) = 7.574. Thus clearly NSOI(G) > NSOI(G)− e. Figure 4: Neutrosophic Graph G − {bc} Theorem 6. Let G = (V, ϖ, ρ) be a n-vertex neutrosophic graph. Then NSOI(G) ≥ NSOI(G − β), where β ∈ V(G). Proof. Let G = (V, ϖ, ρ) be a NG and H = G − β is a graph obtained by removing an edge β ∈ V(G). The MVs in G and H are given by the relationship. τϖG(β) ≥ τϖH(β), ıϖG(β) ≥ ıϖH(β) and𭟋ϖG(β) ≥ 𭟋ϖH(β) and τρG(ϱ) ≥ τρH(ϱ), ıρG(ϱ) ≥ ıρH(ϱ) and 𭟋ρG(ϱ) ≥ 𭟋ρH(ϱ). Now, This show that dG(β) ≥ dH(β) and dG(ρ) ≥ dH(ρ) NSOI(G) = TSOI(H) + ISOI(H) + FSOI(H) (3.39) TSOI(G) = ∑ α,β∈E(G) √ (τϖ(α)τdH(α))2 + (τϖ(β)τdH(β))2 ≥ ∑ α,β∈E(H) √ (τϖ(α)τdH(α))2 + (τϖ(β)τdH(β))2 = TSOI(H) M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2608 TSOI(G) = TSOI(G − β) (3.40) ISOI(G) = ∑ α,β∈E(G) √ (ıϖ(α)ıdH(α))2 + (ıϖ(β)ıdH(β))2 ≥ ∑ α,β∈E(H) √ (ıϖ(α)ıdH(α))2 + (ıϖ(β)ıdH(β))2 = ISOI(H) ISOI(G) = ISOI(G − β) (3.41) FSOI(G) = ∑ α,β∈E(G) √ (𭟋ϖ(α)𭟋dH(α))2 + (𭟋ϖ(β)𭟋dH(β))2 ≥ ∑ α,β∈E(H) √ (𭟋ϖ(α)𭟋dH(α))2 + (𭟋ϖ(β)𭟋dH(β))2 = FSOI(H) FSOI(G) = FSOI(G − β) (3.42) Substitute (3.40), (3.41) and (3.42) in (3.39), we get NSOI(G) = TSOI(G − β) + ISOI(G − β) + FSOI(G − β) NSOI(G) > NSOI(G − β) Thus clearly NSOI(G) > NSOI(G − β). Example 4. Let H = G − {d} be a NG obtained by removing a vertex d from figure 4. Clearly, the memberships values of H remains same for the vertices and there is a change in degree of vertices e and c. Then dH(c) = (0.4, 0.4, 0.5) and dH(ϱ) = (0.6, 0.7, 0.4). NSOI(G) = TSOI(H) + ISOI(H) + FSOI(H) (3.43) TSOI(H) = √ (τϖ(a)τdH(a))2 + (τϖ(b)τdH(b))2 + √ (τϖ(b)τdH(b))2 + (τϖ(c)τdH(c))2 + √ (τϖ(b)τdH(b))2 + (τϖ(e)TdH(e))2 = √ ((0.7)(0.6))2 + ((0.6)(1.6))2 + √ ((0.6)(1.6))2 + ((0.6)(1.6))2 + √ ((0.5)(0.4))2 + ((0.8)(0.6))2 = 1.047 + 0.9806 + 0.52 TSOI(H) = 2.5476 (3.44) M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2609 ISOI(H) = √ (ıϖ(a)ıdH(a))2 + (ıϖ(b)ıdH(b))2 + √ (ıϖ(b)ıdH(b))2 + (ıϖ(c)ıdH(c))2 + √ (ıϖ(b)ıdH(b))2 + (ıϖ(e)ıdH(e))2 = √ ((0.6)(0.4))2 + ((0.5)(1.3))2 + √ ((0.5)(1.3))2 + ((0.4)(0.4))2 + √ ((0.5)(1.3))2 + ((0.7)(0.5))2 = 0.6928 + 0.6694 + 0.7382 ISOI(H) = 2.1004 (3.45) FSOI(H) = √ (𭟋ϖ(a)𭟋dH(a))2 + (𭟋ϖ(b)𭟋dH(b))2 + √ (𭟋ϖ(b)𭟋dH(b))2 + (𭟋ϖ(c)𭟋dH(c))2 + √ (𭟋ϖ(b)𭟋dH(b))2 + (𭟋ϖ(e)𭟋dH(e))2 = √ ((0.4)(0.5))2 + ((0.4)(1.4))2 + √ ((0.4)(1.4))2 + ((0.3)(0.5))2 + √ ((0.4)(1.4))2 + ((0.4)(0.5))2 = 0.5946 + 0.5797 + 0.5946 FSOI(H) = 1.7689.............(44). (3.46) Substitute (3.44), (3.45) and (3.46) in (3.43), we get NSOI(G) = 2.5476 + 2.1004 + 1.7689 = 6.4169 NSOI(G) > NSOI(G − α) Thus clearly NSOI(G) > NSOI(G − α). Figure 5: Neutrosophic Graph G − {d} 4. Site Selection for Thermal Power Plant by using SOI in NGs Now a day without power plants, people wouldn’t have reliable access to electricity, which would make it difficult to live and work in the ways we’re accustomed to today. Power plants M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2610 help distribute energy to large populations, making it possible for society to function smoothly and efficiently. Selecting an appropriate location for a thermal power plant is a crucial choice that affects the facility’s overall performance, operational expenses, and efficiency over time. To make sure that the chosen site satisfies all technical and financial needs while also taking social and environmental aspects into account, a thorough examination of many different elements must be conducted. Finding a site that fully satisfies every ideal criteria is difficult, but the objective is to choose a location that gives the best possible balance of these characteristics, assuring the plant’s long- term survival and financial rationale by using neutrosophic graph. 4.1. Essential Conditions for Building and Functioning: Several essential requirements must be met for a thermal power plant to be built and oper- ated. The soil stability at the location and the availability of water supplies are two of the most crucial elements. 1. Fuel Supply(F): One of the most important considerations when choosing a location is its closeness to a consistent fuel source. Fossil fuels including oil, gas, and coal are usually used in thermal power plants. Long-distance fuel transportation can result in considerable cost increases, which lower the plant’s overall productivity and profitability. Thus, it is best to locate the facility near important fuel supplies or transit corridors like pipelines or railroads. This close proximity guarantees a consistent supply of fuel to the plant while reducing the risk of supply disruptions and transportation expenditures. 2. Soil type and geology(SG): The construction and stability of the power plant depend heavily on the site’s geology and soil composition. For the plant’s safety and to prevent structural problems, the foundation needs to be placed on solid ground. It is best to stay away from areas with unstable soils, such as those that are vulnerable to landslides, erosion, or subsidence. To reduce the danger of earthquakes, the location should also have little seismic risk or be devoid of seismic activity. To determine if the soil and underlying rock formations are suitable for sustaining large buildings like cooling towers, boilers and turbines, a thorough geotechnical assessment is necessary. 3. Water Availability(WA): For thermal power plants, where it is mostly utilized for steam generation and cooling, water is an essential resource. A steady and sufficient supply of water is essential to the plant’s productivity and stability of operations. Thus, locations close to big bodies of water, such lakes, rivers, or reservoirs, are frequently optimal. Water availability must be weighed against environmental factors, such as the effect on aquatic ecosystems and water rights, though. Alternative cooling techniques or water sources, including seawater or treated wastewater, may be required in desert places where water is scarce. 4. Land Availability(LA): A thermal power plant needs land for auxiliary infrastructure, such as fuel storage, water treatment facilities and waste disposal sites, in addition to the primary facilities. The location should have enough room for upcoming improvements or expansions. For equipment placement and to save building expenses, the terrain should also be level or moderately sloping. The ownership and present usage of the property should also be taken into account, as clearing and acquiring land may be expensive and time-consuming. 5. Facilities for Transportation(FT): Building and running a thermal power plant requires an effective transportation infrastructure. Heavy equipment, building supplies, and plant parts need to be carried to the construction site. Fuel and other required supplies must be supplied on a regular basis once the plant is operating. Being close to ports, railroads and highways may greatly lower logistical difficulties and transportation costs. Additionally, efficient M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2611 transportation networks are essential for plant staff mobility and for enabling emergency services. In the process we can identify the order of key components by using neutrosophic SOI of thermal thermal power plant. The relationship among the key components like Fuel Supply, Soil type and geology, Water Availability, Land Availability and Facilities for Transportation of the thermal power plant is constructed as Neutrosophic Graphs and it is given below. Figure 6: NG based on Fuel Supply(F) d(𭟋1) = (1.0, 1.0, 0.9), d(𭟋2) = (0.9, 0.9, 1.2), d(𭟋3) = (1.2, 1, 1.5), d(F4) = (1.4, 1.2, 1.3), d(F5) = (1.3, 1.1, 1.1). NSOI(G) = √ [((0.6)(1.0))2 + ((0.5)(0.9))2] + [((0.5)(1.0))2 + ((0.4)(0.9))2] + [((0.3)(0.9))2 + ((0.5)(1.2))2] = √ [((0.5)(0.9))2 + ((0.7)(1.2))2] + [((0.4)(0.9))2 + ((0.6)(1))2] + [((0.5)(1.2))2 + ((0.8)(1.5))2] = √ [((0.7)(1.2))2 + ((0.8)(1.4))2] + [((0.6)(1))2 + ((0.7)(1.2))2] + [((0.8)(1.5))2 + ((0.6)(1.3))2] = √ [((0.8)(1.4))2 + ((0.7)(1.3))2] + [((0.7)(1.2))2 + ((0.6)(1.1))2] + [((0.6)(1.3))2 + ((0.5)(1.1))2] = √ [((0.7)(1.3))2 + ((0.6)(1.0))2] + [((0.6)(1.1))2 + ((0.5)(1.0))2] + [((0.5)(1.1))2 + ((0.3)(0.9))2] NSOI(G) = 8.747. d(W1) = (1.5, 1.3, 1.2), d(W2) = (1.9, 1.6, 1.6), d(W3) = (1.2, 1, 0.9), d(W4) = (2.0, 1.8, 1.5), d(W5) = (1.2, 1.1, 1.0). NSOI(G) = √ [((0.8)(1.5))2 + ((0.7)(1.9))2] + [((0.7)(1.3))2 + ((0.8)(1.6))2] + [((0.6)(1.2))2 + ((0.5)(1.6))2] = √ [((0.7)(1.9))2 + ((0.6)(1.9))2] + [((0.8)(1.6))2 + ((0.5)(1.6))2] + [((0.5)(1.6))2 + ((0.4)(1.6))2] = √ [((0.6)(1.9))2 + ((0.9)(2))2] + [((0.5)(1.6))2 + ((0.7)(1.8))2] + [((0.4)(1.6))2 + ((0.5)(1.5))2] = √ [((0.9)(2))2 + ((0.6)(1.2))2] + [((0.7)(1.8))2 + ((0.6)(1.1))2] + [((0.5)(1.5))2 + ((0.4)(1.0))2] = √ [((0.9)(2))2 + ((0.8)(1.5))2] + [((0.7)(1.8))2 + ((0.7)(1.3))2] + [((0.5)(1.5))2 + ((0.6)(1.2))2] M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2612 Figure 7: NG basesd on Water Availability(WA) = √ [((0.6)(1.2))2 + ((0.7)(1.9))2] + [((0.7)(1.8))2 + ((0.8)(1.6))2] + [((0.4)(1))2 + ((0.5)(1.6))2] NSOI(G) = 15.847 Figure 8: NG basesd on Soil type and geology(SG) d(S1) = (1.2, 1.0, 0.9), d(S2) = (1.2, 1, 1.7), d(S3) = (1.1, 0.9, 0.7), d(S4) = (1.0, 0.8, 0.9), d(S5) = (1.1, 0.9, 1.0). NSOI(G) = √ [((0.7)(1.2))2 + ((0.6)(1.2))2] + [((0.8)(1))2 + ((0.5)(1))2] + [((0.4)(0.9))2 + ((0.4)(0.7))2] = √ [((0.6)(1.2))2 + ((0.7)(1.1))2] + [((0.5)(1))2 + ((0.6)(0.9))2] + [((0.4)(0.7))2 + ((0.4)(0.7))2] = √ [((0.7)(1.1))2 + ((0.5)(1))2] + [((0.6)(0.9))2 + ((0.4)(0.8))2] + [((0.4)(0.7))2 + ((0.3)(0.9))2] = √ [((0.5)(1))2 + ((0.8)(1.1))2] + [((0.4)(0.8))2 + ((0.6)(0.9))2] + [((0.3)(0.9))2 + ((0.5)(1.0))2] = √ [((0.8)(1.1))2 + ((0.7)(1.2))2] + [((0.6)(0.9))2 + ((0.8)(1))2] + [((0.5)(1))2 + ((0.4)(0.9))2] NSOI(G) = 7.0375 M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2613 Figure 9: NG basesd on Land Availability(LA) d(L1) = (1.4, 1.2, 1.6), d(L2) = (1.4, 1.3, 1.4), d(L3) = (1.3, 1.2, 1.2), d(L4) = (1.2, 1, 1.2), d(L5) = (1.3, 1.1, 1.4). NSOI(G) = √ [((0.7)(1.4))2 + ((0.8)(1.4))2] + [((0.6)(1.2))2 + ((0.7)(1.3))2] + [((0.8)(1.6))2 + ((0.5)(1.4))2] = √ [((0.8)(1.4))2 + ((0.7)(1.3))2] + [((0.7)(1.3))2 + ((0.8)(1.2))2] + [((0.5)(1.4))2 + ((0.6)(1.2))2] = √ [((0.7)(1.3))2 + ((0.6)(1.2))2] + [((0.8)(1.2))2 + ((0.5)(1.0))2] + [((0.6)(1.2))2 + ((0.4)(1.2))2] = √ [((0.6)(1.2))2 + ((0.8)(1.3))2] + [((0.5)(1.0))2 + ((0.7)(1.1))2] + [((0.4)(1.2))2 + ((0.6)(1.4))2] = √ [((0.8)(1.3))2 + ((0.7)(1.4))2] + [((0.7)(1.1))2 + ((0.7)(1.3))2] + [((0.6)(1.4))2 + ((0.8)(1.6))2] NSOI(G) = 10.6409 Figure 10: NG basesd on Facilities for Transportation(FT) d(FT1) = (1.3, 1.1, 1.1), d(FT2) = (1.1, 0.9, 0.9), d(FT3) = (1.0, 0.9, 0.7), d(FT4) = (1.3, 1.2, 1.2), d(FT5) = (1.3, 1.2, 1.2). NSOI(G) = √ [((0.7)(1.3))2 + ((0.6)(1.1))2] + [((0.6)(1.1))2 + ((0.5)(0.9))2] + [((0.5)(1.1))2 + ((0.4)(0.9))2] = √ [((0.6)(1.1))2 + ((0.5)(1.0))2] + [((0.5)(0.9))2 + ((0.4)(0.9))2] + [((0.4)(0.9))2 + ((0.3)(0.7))2] = √ [((0.5)(1.0))2 + ((0.6)(1.3))2] + [((0.4)(0.9))2 + ((0.6)(1.2))2] + [((0.3)(0.7))2 + ((0.5)(1.2))2] M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2614 = √ [((0.6)(1.3))2 + ((0.8)(1.3))2] + [((0.6)(1.2))2 + ((0.7)(1.2))2] + [((0.5)(1.2))2 + ((0.6)(1.2))2] = √ [((0.8)(1.3))2 + ((0.7)(1.3))2] + [((0.7)(1.2))2 + ((0.6)(1.1))2] + [((0.6)(1.2))2 + ((0.5)(1.1))2] NSOI(G) = 7.9165 4.2. Decision Making Key Components NSOI Valus Fuel Supply(F) 8.747 Water Availability(WA) 15.847 Soil type and geology(SG) 7.0375 Land Availability(LA) 10.6409 Facilities for Transportation(FT) 7.9165 Table 2: Neutrosophic SOI Values Figure 11: Graphical Representation of SOI Values The Neutrosophic SOI values presented in table 1 Water Availability(WA) plays a vital role in selecting suitable place to establish thermal power plant. The various attributes taken into considerations to calculate SOI are graphically represented in Fig 10. Since the graphical networks are selected through defining the connection between elements of the system for fuel supply, water and the kind of the soil. These factors are crucial because the firm’s power plant operations depend on them as well as the structure’s stability. Experiences indicate that geotechnical factors such as the type of Water Availability(WA) features are critical in the design of foundations required to support the construction of the plant in the long run. In practical use, unsound ground presents a variety of structural hazards which is why they pose critical importance in large-scale power plant construction such as thermal power plants. M. Alqahtani, M. Kaviyarasu, M. Rajeshwari / Eur. J. Pure Appl. Math, 17 (4) (2024), 2586-2620 2615 The dependencies can be represented using neutrosophic graphs and particularly when dealing with uncertainties. 4.3. Comparative Analysis The SOIs for the fuzzy and IFGs have been defined, the usage is more limited, as compared to the critical path, strong, complete, complementary, wheel and star graphs. Weight for MVs, IMVs and NMVs assigned for vertices, can be from 0 to 1 as the domain of X is defined more thoroughly over closed interval [0, 1]. This is similar to the Neutrosophic framework which has not been designed in this manner but is based on capturing the value of the unknown component of an expression as well as the truth value. Sombor topological Indices in NG theory are somewhat less formalized than crisp graph theory since each vertex can be only one value, and yet they are more versatile and can be applied wherever some decision has to be made, from fighting cyber-crime to diagnosing diseases and planning for roads. We cannot do that in crisp graph theory because vertices can have only one value assigned to them in this mathematical concept. It is also more all-encompassing from this perspective, for our work is wider and more inclusive. 4.4. Sensitivity Analysis The evaluation of NSOI values, through sensitivity analysis, determines the significance of multiple factors involved in decision making for the selection of an ideal site for the establishment of thermal power plant. From Table 2, Fuel Supply (F) has NSOI of 8.747, Water Availability (WA)15.847, Soil Type and Geology (SG) 7.0375, Land Availability (LA) 10.6409 and Facilities for Transportation (FT) 7.9165. Due to the rather high NSOI value with respect to the factors of Soil Type and Geology (SG), it can be stated that these factors have a strong influence on selecting a site for genset construction, pointing to the fact that these conditions are critical for obtaining the required thermal power plant efficiency. The various attributes that have been used to develop the NSOI model have been illustrated in Fig. 11 in order to create a more informative visual context for decision makers. This analysis would put forward that SG has to be considered side by side some other significant aspects like Land Availability or Fuel Supply while screening potential locations for thermal power plant. Due to this, the decision-makers can develop solutions that can improve the energy infrastructure’s robustness and endurance under conditions of uncertainty and variability of environmental factors reducing the vulnerability as noted in the research. Consequently, it enables comprehensive decision-making leading to the development of thermal power plants. 4.5. Advantages and limitations As a result of our examination, the following are the main benefits and Limitations: • The NSOI is very effective in a real-life application since it allows the use of imprecise and uncertain data. • However, some practitioners might notice that NSOI is less friendly, especially in the sense that the computations may require some understanding. • The support of multiple parameters (MV, IMV and NMV) enables NSOI to provide a more refined understanding of relations between characteristics. • In contrast, numbers that may be objectively obtained may significantly influence the results which in turn will significantly vary from one analysis to another. REFERENCES 2616 • However, its cross-disciplinary usefulness means that the NSOI is important; however, the lack of substantial empirical research to support the validity of the technique means that there may be worries over its reliability in terms of real-life implementation. 5. Conclusion In this research, we proposed the SOI for neutrosophic graphs to measure the overall struc- tural and communication resilience of networks under uncertainty. Apart from setting up con- straints for specific neutrosophic graphs, the research posited the neutrosophic first and second Zagreb Indices. Based on our main findings, SOI can serve as a helpful instrument to reveal structural characteristics in conditions when their nature remains unclear. The numerical anal- ysis revealed that the maximum value of the SOI corresponds to the critical components that should be given more focus when choosing the right location of thermal power plants taking into account the former’s operation efficiency with the later’s operational constraints. NGs could be employed to achieve rational criteria in the decision-making processes, particularly in complex ones. This research is important because it bridges the gap between works proposing such Indices and their feasible realizations. 5.1. Future Work Wiener Index, Zagreb Indices, Randic type Indices, Schult type Indices, Dominating Indice of neutrosophic graph and its applications. References [1] U. Ahmad, N.K. Khan, and A.B. Saeid. Fuzzy topological indices with application to cybercrime problem. Granular Computing, 8:967–980, 2023. [2] M. Akram and R. Akmal. Operations on intuitionistic fuzzy graph structures. Fuzzy Infor- mation and Engineering, 8(4):389–410, 2016. [3] M. Akram and B. Davvaz. Strong intuitionistic fuzzy graphs. Filomat, 26(1):177–196, 2012. [4] W.F. AL-Omeri and M. Kaviyarasu. Study on neutrosophic graph with application on earthquake response center in japan. Symmetry, 16:743, 2024. [5] M. Alqahtani, M. Kaviyarasu, A. Al-Masarwah, and M. Rajeshwari. Application of complex neutrosophic graphs in hospital infrastructure design. Mathematics, 12:719, 2024. [6] S. Anwar, M. Azeem, M.K. Jamil, and et al. Single-valued neutrosophic fuzzy sombor numbers and their applications in trade flows between different countries via sea route. Journal of Supercomputing, 80:19976–20019, 2024. [7] K. Atanassov. Generalized nets and intuitionistic fuzziness as tools for modeling of data mining processes and tools. Notes on Intuitionistic Fuzzy Sets, 26(4):9–52, 2020. [8] M. Azam, M.S.A. Khan, S. Yang, S.U. Jan, T. Senapati, S. Moslem, and W.K. Mashwani. Novel dual partitioned maclaurin symmetric mean operators for the selection of computer network security system with complex intuitionistic fuzzy setting. IEEE Access, 11:85050– 85066, 2023. REFERENCES 2617 [9] M. Azeem, M.F. Nadeem, A. Khalil, and A. Ahmad. On the bounded partition dimension of some classes of convex polytopes. Discrete Mathematics, Science and Cryptography, 2020. [10] J. Bera, K.C. Das, S. Samanta, and J.-G. Lee. Connectivity status of intuitionistic fuzzy graph and its application to merging of banks. Mathematics, 11(8):1949, 2023. [11] A. Bhattacharya and M. Pal. A fuzzy graph theory approach to the facility location problem: A case study in the indian banking system. Mathematics, 11(13):2992, 2023. [12] M. Blue, B. Bush, and J. Puckett. Unified approach to fuzzy graph problems. Fuzzy Sets and Systems, 125(3):355–368, 2002. [13] K.C. Das, A.S. Cevik, I.N. Cangul, and Y. Shang. On sombor index. Symmetry, 13(1):140, 2021. [14] R. Das, L. Sahoo, S. Samanta, V. Simic, and T. Senapati. Identifying the shortest path of a semidirected graph and its application. Mathematics, 10(24):4807, 2022. [15] B. Davvaz, N. Jan, T. Mahmood, and K. Ullah. Intuitionistic fuzzy graphs of nth type with applications. Journal of Intelligent Fuzzy Systems, 36(4):3923–3932, 2019. [16] J. Dinar, Z. Hussain, S. Zaman, and S.U. Rehman. Wiener index for an intuitionistic fuzzy graph and its application in water pipeline network. Ain Shams Engineering Journal, 14(1):101826, 2023. [17] L. Feng, X. Zhu, and W. Liu. Wiener index, harary index and graph properties. Discrete Applied Mathematics, 223:72–83, 2017. [18] S. Filipovski. Relations between sombor index and some degree-based topological indices. Iranian Journal of Mathematical Chemistry, 12(1):19–26, 2021. [19] Y. g. Shang, X. h. Yuan, and E.S. Lee. The n-dimensional fuzzy sets and zadeh fuzzy sets based on the finite valued fuzzy sets. Computers Mathematics with Applications, 60(3):442– 463, 2010. [20] A.N. Gani and S.S. Begum. Degree, order and size in intuitionistic fuzzy graphs. Interna- tional Journal of Algorithms, Computing and Mathematics, 3(3):11–16, 2010. [21] A.N. Gani, R.J. Hussain, and S.Y. Mohamed. Irregular intuitionistic fuzzy graph. IOSR Journal of Mathematics (IOSR-JM), 9:47–51, 2014. [22] Masoud Ghods and Zahra Rostami. Wiener index and applications in the neutrosophic graphs. Neutrosophic Sets and Systems, 46:229–245, 2021. [23] S. Gong and G. Hua. Topological indices of bipolar fuzzy incidence graph. Open Chemistry, 19(1):894–903, 2021. [24] I. Gutman. On the origin of two degree-based topological indices. Bulletin of the Serbian Academy of Sciences and Arts, 39:39–52, 2014. [25] I. Gutman. Geometric approach to degree-based topological indices: Sombor indices. MATCH Communications in Mathematical and Computer Chemistry, 86(1):11–16, 2021. REFERENCES 2618 [26] I. Gutman. Sombor indices-back to geometry. Open Journal of Discrete Applied Mathemat- ics, 5(2):1–5, 2022. [27] S. Hayat and M. Imran. Computation of topological indices of certain networks. Applied Mathematics and Computation, 240:213–228, 2014. [28] S.R. Islam and M. Pal. First zagreb index on a fuzzy graph and its application. Journal of Intelligent Fuzzy Systems, 40:10575–10587, 2020. [29] S.R. Islam and M. Pal. Hyper-wiener index for fuzzy graph and its application in share market. Journal of Intelligent Fuzzy Systems, 41(1):2073–2083, 2021. [30] M.K. Jamil, S. Anwar, M. Azeem, and I. Gutman. Intuitionistic fuzzy sombor indices: A novel approach for improving the performance of vaccination centers. Communications in Combinatorics and Optimization, pages 1–31, 2024. [31] U. Jana and G. Ghorai. First entire zagreb index of fuzzy graph and its application. Axioms, 12:415, 2023. [32] L.S. Jin, R.R. Yager, C. Ma, L.M. Lopez, R.M. Rodriguez, T. Senapati, and R. Mesiar. Intuitionistic fuzzy type basic uncertain information. Iranian Journal of Fuzzy Systems, 20(5):189–197, 2023. [33] S. Kalathian, S. Ramalingam, S. Raman, and N. Srinivasan. Some topological indices in fuzzy graphs. Journal of Intelligent Fuzzy Systems, 39(5):6033–6046, 2020. [34] M.G. Karunambigai, M. Akram, S. Sivasankar, and K. Palanivel. Clustering algorithm for intuitionistic fuzzy graphs. International Journal of Uncertainty, Fuzziness and Knowledge- Based Systems, 25(03):367–383, 2017. [35] M. Kaviyarasu, M. Aslam, F. Afzal, and et al. The connectivity indices concept of neu- trosophic graph and their application of computer network, highway system and transport network flow. Scientific Reports, 14:4891, 2024. [36] M.S.A. Khan, S.U. Jan, R. Jan, T. Senapati, and S. Moslem. Complex interval-valued intuitionistic fuzzy decision support system with application to covid-19 healthcare facilities. Complex Intelligent Systems, 9:7103–7132, 2023. [37] P. Majumder, P. Bhowmik, A. Das, T. Senapati, V. Simic, and D. Pamucar. An intuitionistic fuzzy based hybrid decision-making approach to determine the priority value of indicators and its application to solar energy feasibility analysis. Optik - International Journal for Light and Electron Optics, 295:171492, 2023. [38] Z.S. Mufti, A. Tabraiz, Q. Xin, B. Almutairi, and R. Anjum. Fuzzy topological analysis of pizza graph. AIMS Mathematics, 8(6):12841–12856, 2023. [39] M. Mulla, S. Broumi, and R. Jeyabalan. Homomorphism and isomorphism in strong neu- trosophic graphs. International Journal of Neutrosophic Science, pages 8–20, 2019. [40] T. Naeem, A. Gumaei, M.K. Jamil, A. Alsanad, and K. Ullah. Connectivity indices of intuitionistic fuzzy graphs and their applications in internet routing and transport network flow. Mathematical Problems in Engineering, 2021:1–16, 2021. REFERENCES 2619 [41] N. Nazir, T. Shaheen, L.S. Jin, and T. Senapati. An improved algorithm for identification of dominating vertex set intuitionistic fuzzy graphs. Axioms, 12(3):289, 2023. [42] R. Parvathi and M.G. Karunambigai. Intuitionistic fuzzy graphs. In Computational Intel- ligence, Theory and Applications: International Conference 9th Fuzzy Days in Dortmund, Germany, volume 18, pages 139–150. 2006. [43] R. Parvathi, M.G. Karunambigai, and K.T. Atanassov. Operations on intuitionistic fuzzy graphs. In 2009 IEEE International Conference on Fuzzy Systems. IEEE, 2009. [44] K. Poczeta, L. Kubu’s, and A. Yastrebov. Analysis of an evolutionary algorithm for complex fuzzy cognitive map learning based on graph theory metrics and output concepts. Biosys- tems, 179:39–47, 2019. [45] B. Praba, V.M. Chandrasekaran, and G. Deepa. Energy of an intuitionistic fuzzy graph. Italian Journal of Pure and Applied Mathematics, 32:431–444, 2014. [46] K. Prakash, M. Parimala, H. Garg, and M. Riaz. Lifetime prolongation of a wireless charg- ing sensor network using a mobile robot via linear diophantine fuzzy graph environment. Complex Intelligent Systems, 8(3):2419–2434, 2022. [47] M. Rajeshwari, R. Murugesan, M. Kaviyarasu, and C. Subrahmanyam. Bipolar fuzzy graph on certain topological indices. Journal of Algebra and Statistics, 13(3):2476–2481, 2022. [48] H. Rashmanlou, M. Pal, S. Raut, F. Mofidnakhaei, and B. Sarkar. Novel concepts in intu- itionistic fuzzy graphs with application. Journal of Intelligent Fuzzy Systems, 37(3):3743– 3749, 2019. [49] H. Rashmanlou, S. Samanta, M. Pal, and R.A. Borzooei. Intuitionistic fuzzy graphs with categorical properties. Fuzzy Information and Engineering, 7(3):317–334, 2015. [50] D.T. Hanumantha Reddy, M.V. Chakradhara Rao, and S.M. Hosamani. Sombor index of fuzzy graphs and its applications. Journal of Propulsion Technology, 44(6), 2023. [51] S. Sahoo and M. Pal. Different types of products on intuitionistic fuzzy graphs. Pacific Science Review A: Natural Science and Engineering, 17(3):87–96, 2015. [52] S. Sahoo and M. Pal. Product of intuitionistic fuzzy graphs and degree. Journal of Intelligent Fuzzy Systems, 32(1):1059–1067, 2017. [53] S.B. Saila. Application of fuzzy graph theory to successional analysis of a multispecies trawl fishery. Transactions of the American Fisheries Society, 121(2):211–233, 1992. [54] T. Senapati, G. Chen, R. Mesiar, and R.R. Yager. Intuitionistic fuzzy geometric aggregation operators in the framework of aczel-alsina triangular norms and their application to multiple attribute decision making. Expert Systems with Applications, 212:118832, 2023. [55] T. Senapati, V. Simic, A. Saha, M. Dobrodolac, Y. Rong, and E.B. Tirkolaee. Intuition- istic fuzzy power aczel-alsina model for prioritization of sustainable transportation sharing practices. Engineering Applications of Artificial Intelligence, 119:105716, 2023. [56] A. Shannon and K.T. Atanassov. On a generalization of intuitionistic fuzzy graphs. Notes on Intuitionistic Fuzzy Sets, 12(1):24–29, 2006. REFERENCES 2620 [57] M.I. Stankevich, I.V. Stankevich, and N.S. Zefirov. Topological indices in organic chemistry. Russian Chemical Reviews, 57(3):191, 1988. [58] H. Wiener. Structural determination of paraffin boiling points. Journal of the American Chemical Society, 69(1):17–20, 1947. [59] L.A. Zadeh. Fuzzy set, information and control. Information and Control, 8:338–353, 1965. 66. MA Mascrenghe. [60] L.A. Zadeh. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1(1):3–28, 1978. [61] J. Zhan, H.M. Malik, and M. Akram. Novel decision-making algorithms based on intuition- istic fuzzy rough environment. International Journal of Machine Learning and Cybernetics, 10:1459–1485, 2019. [62] H.-J. Zimmermann. Fuzzy set theory. Wiley Interdisciplinary Reviews: Computational Statistics, 2(3):317–332, 2010.