Novel Concept of Energy in Bipolar Single-Valued Neutrosophic Graphs with Applications

: The energy of a graph is defined as the sum of the absolute values of its eigenvalues. Recently, there has been a lot of interest in graph energy research. Previous literature has suggested integrating energy, Laplacian energy, and signless Laplacian energy with single-valued neutrosophic graphs (SVNGs). This integration is used to solve problems that are characterized by indeterminate and inconsistent information. However, when the information is endowed with both positive and negative uncertainty, then bipolar single-valued neutrosophic sets (BSVNs) constitute an appropriate knowledge representation of this framework. A BSVNs is a generalized bipolar fuzzy structure that deals with positive and negative uncertainty in real-life problems with a larger do-main. In contrast to the previous study, which directly used truth and indeterminate and false membership, this paper proposes integrating energy, Laplacian energy, and signless Laplacian energy with BSVNs to graph structure considering the positive and negative membership degree to greatly improve decisions in certain problems. Moreover, this paper intends to elaborate on characteristics of eigenvalues, upper and lower bound of energy, Laplacian energy, and signless Laplacian energy. We introduced the concept of a bipolar single-valued neutrosophic graph (BSVNG) for an energy graph and discussed its relevant ideas with the help of examples. Furthermore, the significance of using bipolar concepts over non-bipolar concepts is compared numerically. Finally, the application of energy, Laplacian energy, and signless Laplacian energy in BSVNG are demonstrated in selecting renewable energy sources, while optimal selection is suggested to illustrate the proposed method. This indicates the usefulness and practicality of this proposed approach in real life.


Introduction
The graph spectrum is applicable in statistical physics and mathematical combinatorial optimization problems. Pattern recognition, modelling virus spread in computer networks, and safeguarding personal data in databases all benefit from the spectrum of a graph. The concept of graph energy is related to a graph's spectrum. This concept was originally introduced by Gutman [1] in 1978. It is defined as the sum of the absolute values of the eigen values of the graph's adjacency matrix. By linking the edge of a graph to the electron energy of a type of molecule, the energy of a graph is employed in quantum theory and many other applications in the context of energy. Later, Gutman and Zhou [2] defined the Laplacian energy of a graph as the sum of the absolute values of the differences of average vertex degree of G to the Laplacian eigenvalues of G. Details on the properties of graph energy and Laplacian energy can be found in [3][4][5][6][7][8][9][10][11]. ergy, Laplacian energy, and signless Laplacian energy in BSVNG, to investigate the properties on characteristics of eigenvalues, upper and lower bound of energy, Laplacian energy and signless Laplacian energy and to present the relationship among them.
The outline of this study is organized as follows: Section 2 gives the basic concepts related to neutrosophic and bipolar sets. Section 3 defines the concepts of energy in BSVNG while the concept of Laplacian energy in BSVNG is discussed in Section 4. On the other hand, Section 5 presents the concepts of signless Laplacian energy in BSVNG and the relation between energy, Laplacian energy, and signless Laplacian energy presented in Section 6. Moreover, to implement our proposed study, we discuss the application of the energy of BSVNG in the selection of renewable energy sources in Section 7, while Section 8 provides comparative results. Finally, this study is concluded by mentioning future potential research work in Section 9. Table 1. Significance influences towards energy, Laplacian energy and signless Laplacian energy graph.

Author and References Year Fuzzy/Neutrosophic Sets Significance Influences
Akram and Naz [21] 2018 Pythagorean fuzzy sets Introduce the energy and Laplacian energy in Pythagorean fuzzy graphs and Pythagorean fuzzy digraphs. Rajeshwari et al. [22] 2018 Bipolar fuzzy sets Introduce Laplacian energy for a bipolar fuzzy graph.
Naz et al. [23] 2018 Bipolar fuzzy sets Introduce the concept of energy in bipolar fuzzy graph (BFG) and bipolar fuzzy digraphs (BFDGs).
Mohsin et al. [27] 2019 Complex neutrosophic set Compute Laplacian energy of a complex neutrosophic graph in terms of its adjacency matrix.
Ramesh and Basha [24] 2020 Intuitionistic fuzzy sets Solve decision-making problem by signless Laplacian energy of intuitionistic fuzzy graph and cosine similarity measure. Ramesh and Basha [25] 2020 Intuitionistic fuzzy sets Solve group decision-making problem by signless Laplacian energy of intuitionistic fuzzy graph. Mullai and Broumi [29] 2020 Single-valued neutrosophic sets Introduce dominating sets and dominating numbers for energy graphs in single-valued neutrosophic graphs.

Energy of Bipolar Single-Valued Neutrosophic Graphs
In this section, we define and investigate the energy of a graph within the frameworks of BSVNG theory and discuss its properties. The adjacency matrix of a BSVNG can be expressed as six matrices. The first matrix contains the elements as positive truth-membership values; the second matrix contains the elements as positive indeterminacy-membership values; the third matrix contains the elements as positive falsity-membership values; the fourth matrix contains the elements as negative truth-membership values; the fifth matrix contains the elements as negative indeterminacy-membership values; and the sixth matrix contains the elements as negative falsity-membership values, i.e.,

Proof.
i. Upper bound: Apply Cauchy-Schwarz inequality to the vectors     Figure 1,
Proof. The proof is similar to that of Theorem 1.    .

Example 5. Consider a BSVNG,
Similarly, it is easy to show that  iii. Proof. By using Cauchy-Schwarz inequality, we obtain Substituting (6) into (5), we obtain

is a degree matrix of a BSVNG and  
A G is an adjacency matrix.   v v ,v v 5 7 6 7 as shown in Figure 3, defined by Tables 6 and 7 as follows: Table 6. Signless Laplacian energy of BSVNG set on V.   v v 2 4 v v 2 5 v v 3 5 v v 4 5 v v 4 6 v v 4 7 v v 5 7 v v 6 7 v v Then, Combining (9) and (10), yields Similarly, we can show that

Application of Energy of BSVNG
A group decision-making problem concerning the selecting of the most compatible renewable energy alternatives is solved to illustrate the applicability of the proposed concepts of energy of BSVNGs in practical scenarios. However, in order to reflect the relationship among the alternatives, we need to make pairwise comparisons for all the alternatives in the process of decision-making. If every element in the preference relations is a bipolar single-valued neutrosophic number (BSVNN), then the concept of the bipolar single-valued neutrosophic preference relation (BSVNPR) can be put forth as follows: Here for all p,q , , ,n.  1 2 

Selection of the Most Compatible Renewable Energy Source
Renewable energy sources involve biomass energy, geothermal energy, ocean energy, solar energy, wind energy, and hydropower energy. They have an enormous potential to meet the energy needs of the world. By doing that, the world's energy security can be powered by modern conversion technologies by reducing the long-term price of fuels from conventional sources and decreasing the use of fossil fuels. Using renewable energies does not only impact reducing air pollution, safety risks, and greenhouse gas emissions in the atmosphere but also are recycled in nature. Furthermore, it reduces dependence on imported fuels, creates new jobs, and provides regional employment.
We considered an issue, taken from [41], as an application for the proposed method in the present paper. The issue given is that the managers of a municipal close to sea cost want to invest in renewable energy technologies to self-meet their energy needs. After numerous consultations, six renewable energy sources were considered as an alternative.  Tables 8-10.    3 7606 3 0000 3 7606 3 0000 3 1361 3 0000   4 0388 3 0000 4 0388 3 4825 3 0000 3 4825   3 9621 3 0000 3 9621 3 3062 3 0000 3 3062 Then, the weight of each expert can be determined as:    Similarly, we can calculate other aggregated values using BSVNWA. Table 11 presents overall aggregated values.  Tables 12-14 while Table 15 presents overall aggregated values.

Comparative Study
In this section, the proposed energy BSVNG method is compared with BNSs developed by Deli et al. [30]. First, we construct the pair-wise comparison matrix provided by the decision-maker in Tables 8-10. Then, we compute weighted average operators and calculate the score function for each alternative. Lastly, we rank all the alternatives according to the score function. Table 16 shows the score function and rank for each alternative. Table 16. The score function and rank for each alternative adopted from Deli et al. [30].      . As we can see, after the ranking of alternatives according to score function in descending order, A6 is still the best alternative as in our proposed energy BSVNG method. Hence, this comparative method shows the availability and effectiveness of our proposed method.

Conclusions
The energy of graphs has many mathematical properties that have been investigated. Certain bounds (upper and lower) on energy had been studied by many researchers. This paper proposed integrating bipolar single-valued neutrosophic set and the energy graph, Laplacian energy graph and signless Laplacian energy graph. Specifically, this paper developed the new concept of energy in BSVNG. It investigated its properties such as the characteristics of eigenvalues, lower and upper bound of energy graph, Laplacian energy graph and signless Laplacian energy graph. Moreover, the relation between them is also discussed, and the proposed method was applied to renewable energy sources selection in which the optimal solution is suggested. In this application, we suggest that a wind power plant (A6) is the optimal alternative. This paper has proceeded with a comparative analysis whereing the rank of alternatives, using the proposed method, is similar to the method of comparison. Hence, it implies that the proposed method is valid and effective. In short, this study implies several significant contributions and modifications to the energy graph, Laplacian energy graph, and signless Laplacian energy graph. In future work, it is suggested to extend the graph energy to: (1) interval-bipolar neutrosophic graphs, (2) neutrosophic vague, (3) dominating energy in neutrosophic graph, etc. Furthermore, this paper does not evaluate the sensitivity of the experts' weights to the evaluation outcomes. As a result, any new sensitivity analysis, particularly on the experts' weights, could be pursued as a research topic in the future.