Seidel Laplacian Energy of Fuzzy graphs

The energy of a graph is related to its spectrum, which is equal to the total of the latent values of the pertinent adjacency matrix. In this research work, we proposed some of the features and the energy of the Seidel Laplacian of a fuzzy graph. Also, the lower and upper bounds for the energy of the Seidel Laplacian of a fuzzy graph were studied with suitable illustrative examples.


Introduction
Many difficulties in daily life are solved using graph theory.For instance, Euler resolved the infamous "Konigsberg bridges" problem in 1735.Although it is the first platform to employ concepts from graph theory, it is difficult to glean precise information about it due to the system's size and complexity [1].Fuzzy graphs are used in such systems to solve them.
Zadeh proposed fuzzy sets in 1965, whose concept is only a membership function.Based on his fuzzy relationship, Kaufman suggested the fuzzy graph in 1973.Fuzzy sets were used in real-life problems with uncertainty.Fuzzy logic can be used in many applications, like tracking the maximum power from solar power voltaic and controller circuit design applications [2][3][4][5].
Later, Rosenfeld elaborated on the definition of graph theory, which includes fuzzy vertices and fuzzy edges.One of the most current research areas in what has become a promising multidisciplinary field is fuzzy graph theory.Graph theory depicts a group of items and their relationships through pictures or diagrams.There are numerous applications for graph theory in different categories, such as Mathematics, Information technology, Computer science, Chemistry, Modelling, Networking, Physics and Biology to mention a few.Now a days, a greater number of works are involved in some particular topics like labelling, indices, and hubset [6][7][8].Akalyadevi et al. defined the Spherical neutrosophic cubic fuzzy models in a bipolar environment and studied their operations [9][10][11][12].The pair of sets (V, E) that represent the formal definition of a graph are V, which is the set of nodes, and E, which is the set of edges linking the vertices.Ivan Gutman initiated the energy concept in 1978.The energy of the graph is calculated using the eigen (latent) values of the relevant adjacency matrix.A graph's overall absolute value of its latent values as determined by its adjacency matrix.Mcclellaud also investigated the graph's energy boundaries, for which he developed the Mcclellaud inequality.For fuzzy graphs, Sunil Mathew et al. defined the adjacency matrix and energy.Also, some conclusions concerning the energy bounds for weighted and simple graphs are enhanced.
Fuzziness is added to energy calculations to increase their resilience, adaptability, and comprehensiveness, which eventually results in more efficient and sustainable energy management.The relevance of applying energy ideas to fuzzy graphs is that it enables a more precise, thorough, and realistic description of complex systems with uncertain or imprecise interactions.This allows to learn more about the dynamics, behaviour and optimisation of these systems across a variety of academic disciplines.The concept of a graph's energy is expanded to include a fuzzy graph's energy.For further study, refer to [13][14][15][16][17][18][19][20][21].Here the concept of a Seidel Laplacian (Sl) graph's energy is expanded to a fuzzy graph's energy for Seidel Laplacian.

Preliminaries
Throughout this article, consider Ҁ is a fuzzy graph with 'q' edges and 'p' nodes with an order 'p' matrix.

Definition :2.1[2]
A fuzzy set 'Ґ' of a set is defined as a function, and the value of Ґ describes a degree of membership for all Ґ in X.

Definition :2.2[18]
Ҁ = ( , Ґ, σ) be a fuzzy graph that consists of a non-empty set and two functions , such that for all x, y in , . is the fuzzy node set and is the fuzzy edge set on the graph Ҁ.
is the fuzzy relation on .
The latent value of the adjacency matrix, also known as the latent (eigen) value of a graph, is represented by the letters e1, e2, e3, and so on.A graph's energy is the total of its latent values' absolute values.

Result:2.3
The following relations were met by the eigenvalues of the ordinary and Laplacian matrices:

Definition:2.5[17]
The Seidel matrix of a graph is the real symmetric matrix S(Ҁ) = (sij), where sij = 1 for adjacent vertices and sij = 1 for non-adjacent vertices and sij = 0 for nodes with i = j.

Proof:
Using the result [17], Suppose that is a graph that has 'q' edges and 'p' nodes.Then where Let and Then This implies that Checking the theorem using example 3.1, Theorem 3.8 Assume that has 'q' edges and 'p' nodes and is a fuzzy graph and be same as in Theorem 3.6.Then .
(i) In the adjacency matrix, the latent values added together equal 0. (ii) The adjacency matrix's latent value square sum equals 2 * the number of edges.(iii) The entire sum of the latent values of the Laplacian matrix is equal to 2 times the number of edges.(iv) The laplacian matrix's total latent values are calculated as (2* number of edges) + (sum of squares of all degrees).