partitioned Neutrosophic fuzzy graph, Eigen Values, Energy of FQNFG, Laplacian energy of FQNFG, Multi-criteria decision making.

Fuzzy Set Theory concept was initiated by Zadeh (1965) [19]. Fuzzy graph theory merges the concepts of fuzzy sets with the structure of graphs and has widespread applications in various fields. Fuzzy graphs helps in modelling and analyzing the complex systems with uncertain or imprecise relationships. In 1975, Rosenfeld created the theory of Fuzzy Graphs. [14]. An idea about intutionistic fuzzy set relationships was initially introduced by Attanssov. They have putforth a number of applications, theorems and properties [16].

The classical, fuzzy and intuitionistic fuzzy sets were basic extensions of neutrosophic set. Smarandache generalized the fuzzy set, leads to development of the neutrosophic set [18]. It can handle any real world issue with ambiguous, inconsistent, uncertain and indeterminate data. Every item in a neutrosophic set has three membership grades: truth, indeterminacy and false. These three membership tiers fall between [0, 1] and are always independent.

Senapati et al. (2020) [15] presented the novel idea called Fermatean Fuzzy Set. Based on the Fermatean Neutrosophic fuzzy set, Said Broumi et al. (2022) [2] presented a novel framework for Fermatean neutrosophic graphs and their applications. Later V. Divya and J. Jesintha Rosline presented an innovative concept known as the Fermatean Quadripartitioned neutrosophic Fuzzy Graph. This Graph helps in evaluating complex decisions with multiple criteria and uncertain outcomes.

I.Gutman [4] define “Graph Energy” as sum of the magnitudes of eigen values of the adjacency matrix in graph. The energy bounds are covered in [3, 5, 8]. Energy of different graphs like regular, non-regular and circulant are studied in [6, 7, 17]. The graph’s energy is expanded to energy of fuzzy graph [9]. Later, energy of fuzzy graph broadened to intuitionistic and Neutrosophic graph [12, 10, 11]. The energy is applied to picture fuzzy graph, Pythagorean Fuzzy Graph, single valued neutrosophic fuzzy graph etc. Fuzzy graph Laplacian energy was introduced by Sadegh Rahimi Sharbaf et al. (2014) [13]. Later this was extended to Intuitionistic and Neutrosophic fuzzy graph [1, 10]. The energy of fuzzy graph has found applications in various fields like Chemical Graph Theory, Social Network analysis, Decision Making under uncertainty. The ability to model and analyse these uncertainties using fuzzy graph energy offers valuable insights and tools for decision making analysis and understand complex systems.

This study examines the energy and Laplacian energy of FQNFG. The paper is structured as follows. Section 2 presents prelimnaries . Section 3 defines energy of FQNFG. The lower and upper bounds of energy for FQNFG were also derived. In Section 4, we define and characterize Laplacian energy of FQNFG. Section 5 presents an application of determining the best Laptop involving four characteristics employing multi-criteria decision-making technique. Section 6 ends with a discussion of further research.

[2] A Fermatean Neutrosophic Graph (FNG) on a universal set X is a pair G = (P,Q) where P is Fermatean neutrosophic set on X and Q is a Fermatean Neutrosophic relation on X so that:

TQ(u, v) ≤ min(Tp(u), Tp(v)) IQ(u, v) ≥ max(Ip(u), Ip(v)) FQ(u, v) ≥ max(Fp(u), Fp(v))

3 3 3

with 0 ≤ TQ(u, v) + IQ(u, v) + FQ(u, v) ≤ 2 ∀u, v ∈ X, such that, TQ : X × X → [0, 1], IQ : X × X → [0, 1] and FQ : X × X → [0, 1] indicates degree of membership, indeterminacy-membership and non-membership of Q, Correspondingly, where P and Q are Fermatean Neutrosophic vertex and edge set of G. [9] Consider a fuzzy graph, G = (V, σ, µ) with adjacency matrix A. The energy G is termed as sum of the magnitudes of its eigenvalues. [1] Consider a fuzzy graph, G = (σ, µ) with |V | = n vertices and µ1 ≥ andµ2 ≥ · · · µn are Laplacian eigen values of G = (σ, µ). Laplacian energy of G is denoted by

2 µ(uiuj) LE(G) = µi −

A Fermatean Quadripartitioned Neutrosophic Fuzzy Graph (FQNFG) an Universal Set X is a pair GF Q = (σF Q, µF Q) , σF Q : X → [0, 1] represents Fermatean Quadripartitioned Neutrosophic Set on X and µF Q : X × X → [0, 1] denotes Fermatean Quadripartitioned Neutrosophic mapping on X × X so that

Tµ (uv) ≤ min(Tσ (u), Tσ (v)) Cµ (uv) ≤ min(Cσ (u), Cσ (v)) Uµ (uv) ≥ max(Uσ (u), Uσ (v)) Fµ (uv) ≥ max(Fσ (u), Fσ (v))

3 3 3 3

with 0 ≤ Tµ (uv) + Cµ (uv) + Uµ (uv) + Fµ (uv) ≤ 3 ∀u, v ∈ X, where Tµ : XxX → [0, 1], Cµ :

F Q F Q F Q F Q F Q F Q

X ×X → [0, 1], Uµ : X ×X → [0, 1], Fµ : X ×X → [0, 1], indicates degree of truth, contradiction, ignorance and false membership of µFQ. σFQ and µFQ indicates Fermatean Quadripartitioned Neutrosophic vertex and edge set of GFQ.

The Adjacency matrix A(GFQ) of FQNFG, GFQ = (σFQ, µFQ) is termed as square matrix A(GFQ) = [aij],

aij= Tµ (pipj), Cµ (pipj), Iµ (pipj), Fµ (pipj) where Tµ (pipj), Cµ (pipj), Iµ (pipj) and Fµ (pipj)

F Q F Q F Q F Q F Q F Q F Q F Q

indicates the strength of relationship, undecided relationship, unknown relationship and non-relationship of pi and pj correspondingly.

The adjacency matrix A(GF Q) of FQNFG involves 4 matrices they are truth, contradiction, ignorance and falsity membership values. ie)

A(GF Q) = A(Tµ (pipj)), A(Cµ (pipj)), A(Iµ (pipj)), A(Fµ (pipj))

F Q F Q F Q F Q

The "spectrum of adjacency matrix" of a FQNFG A(GF Q) is termed as (Q, R, S, T) where Q, R, S and T repre-sents eigen values of A(Tµ (pipj)), A(Cµ (pipj)), A(Iµ (pipj)) and A(Fµ (pipj)) respectively. Example

F Q F Q F Q F Q

1: Consider a Graph GF Q = (σF Q, µF Q), where σF Q = {p1, p2, p3, p4} and µF Q = {p1p2, p2p3, p3p4, p4p1, p1p3}.

σF Q p1 p2 p3 p4 Tσ 0.5 0.7 0.4 0.6 Cσ 0.6 0.5 0.6 0.4 Iσ 0.2 0.3 0.2 0.4 Fσ 0.4 0.1 0.3 0.2 µF Q p1p2 p2p3 p3p4 p4p1 p1p3 Tµ 0.4 0.3 0.5 0.4 0.3 Cµ 0.3 0.4 0.5 0.5 0.5 Iµ 0.4 0.4 0.6 0.5 0.3 Fµ 0.5 0.4 0.4 0.6 0.6

The adjacency matrix of a FQNFG is given by   (0.0, 0.0, 0.0, 0.0) (0.4, 0.3, 0.4, 0.5) (0.3, 0.5, 0.3, 0.6) (0.4, 0.5, 0.5, 0.6) (0.4, 0.3, 0.4, 0.5) (0.0, 0.0, 0.0, 0.0) (0.3, 0.4, 0.4, 0.4) (0.0, 0.0, 0.0, 0.0) (0.3, 0.5, 0.3, 0.6) (0.3, 0.4, 0.4, 0.4) (0.0, 0.0, 0.0, 0.0) (0.5, 0.5, 0.6, 0.4) (0.4, 0.5, 0.5, 0.6) (0.0, 0.0, 0.0, 0.0) (0.5, 0.5, 0.6, 0.4) (0.0, 0.0, 0.0, 0.0)

From Figure 2, Spectrum of FQNFG is spec(Tµ (pipj)) = {0.9701, −0.6703, −0.3296, 0.0299} spec(Cµ (pipj)) = {1.1490, −0.6556, −0.5, 0.0066} spec(Iµ (pipj)) = {1.1244, −0.8270, −0.3031, 0.0057} spec(Fµ (pipj)) = {1.3018, −0.7830, −0.5218, 0.003} spec(Gµ (pipj)) = {(0.9701, 1.1490, 1.1244, 1.3018), (−0.6703, −0.6556, −0.8270, −0.7830), (−0.3296, −0.5, −0.3031, −0.5218), (0.0299, 0.0066, 0.0057, 0.003)} The energy of a FQNFG, GF Q = (σF Q, µF Q) is defined as

E(GF Q) = E(Tµ (pipj)), E(Cµ (pipj)), E(Iµ (pipj)), E(Fµ (pipj))

F Q F Q F Q F Q

n n n n

X X X X =  |αi|, |βi|, |λi|, |ηi|

i=1 i=1 i=1 i=1 αi∈Q βi∈R λi∈S ηi∈T

Equienergetic FQNFGs are defined as those having equal number of vertices and identical energy values.

Theorem 3.1. Consider a FQNFG, GF Q = (σF Q, µF Q) and adjaceny matrix A(GF Q). If α1 ≥ α2 ≥ · · · ≥ αn, β1 ≥ β2 ≥ · · · ≥ βn, λ1 ≥ λ2 ≥ · · · ≥ λn, and η1 ≥ η2 ≥ · · · ≥ ηn represent eigen values of A(Tµ (pipj)), A(Cµ (pipj)), A(Iµ (pipj)) and A(Fµ (pipj)) respectively. Then,

P n P n P n P n (1) αi = 0, βi = 0, λi = 0, ηi = 0

i=1 i=1 i=1 i=1 α∈Q β∈R λ∈S η∈T

P n P P n P

2 2 2 2

(2) α = 2 Tµ (pipj) , β = 2 Cµ (pipj) ,

i F Q i F Q i=1 1≤i≤j≤n i=1 1≤i≤j≤n αi∈Q βi∈R

P n P P n P

2 2 2 2

λ = 2 Iµ (pipj) , η = 2 Fµ (pipj) .

i F Q i F Q i=1 1≤i≤j≤n i=1 1≤i≤j≤n λi∈S ηi∈S

Proof : (1) W.k.t. A(GF Q) denotes a symmetric matrix with a trace of zero (its eigen values sum to zero).

i.e.) αi = 0

αi∈Q

Similarly we get for contradiction, ignorance and false membership.

P n P (2) To prove: α = 2 Tµ (pipj)

i F Q i=1 1≤i≤j≤n αi∈Q

Based on a Matrix Property,

tr((A(Tµ (pipj))) ) = αi

αi∈Q P 2 2 P 2 (3) 2 I (p p ) + n(n − 1)|I| n ≤ E(I (p p )) ≤ 2n I (p p )

µF Q i j µF Q i j µF Q i j 1≤i≤j≤n 1≤i≤j≤n

P 2 2 P 2 (4) 2 F (p p ) + n(n − 1)|F| n ≤ E(F (p p )) ≤ 2n F (p p )

µF Q i j µF Q i j µF Q i j 1≤i≤j≤n 1≤i≤j≤n

where |T|, |C|, |I| and |F| represent determinant of A(Tµ (pipj)), A(Cµ (pipj)), A(Iµ (pipj)), and A(Fµ (pipj)) respectively.

Proof : (1) Upper Bound (UB) Utilizing Cauchy-Schwarz inequality to the n numbers 1, 1, · · · 1 and |λ1|, |λ2|, · · · |λn|, we get

n u n X √ uX |λi| ≤ n |λi| (1)

i=1 i=1

X X X λi = |λi| + 2 λiλj (2)

i=1 i=1 1≤i≤j≤n

Equating coefficients of λ in the characteristic polynomial

(λ − λi) = |A(GF Q) − λI|

then

λiλj = − Tµ (pipj) (3)

1≤i≤j≤n 1≤i≤j≤n

Substituting (3) into (2) gives

|λi| = 2 Tµ (pipj) (4)

i=1 1≤i≤j≤n

Substituting (4) into (2) yields n s s X √ X X

2 2 2

|λi| ≤ n 2 Tµ (pipj) = 2n Tµ (pipj)

i=1 1≤i≤j≤n 1≤i≤j≤n

E(Tµ (pipj)) ≤ 2n Tµ (pipj)

(2) Lower bound (LB)

E(Tµ (pipj)) = |λi|

= |λi| + 2 |λiλj|

i=1 1≤i≤j≤n

X 2n(n − 1) = 2 Tµ (pipj) + AM{|λiλj|}

W.k.t. AM{|λiλj|} ≥ GM{|λiλj|}, 1 ≤ i ≤ j ≤ n E(Tµ (pipj)) ≥ 2 Tµ (pipj) + n(n − 1)GM{|λiλj|} and then

Y n(n − 1) GM{|λ λ |} =  |λ λ |

i j i j

Y n(n − 1) = |λi|

= |λi|

= |T|n

E(Tµ (pipj)) ≥ 2 Tµ (pipj) + n(n − 1)|T|n

∴ t2 Tµ (pipj) + n(n − 1)|T|n ≤ E(Tµ (pipj))

≤ 2n Tµ (pipj)

Correspondingly, it follows that

t2 Cµ (pipj) + n(n − 1)|C|n ≤ E(Cµ (pipj))

≤ 2n Cµ (pipj)

t2 Iµ (pipj) + n(n − 1)|I|n ≤ E(Iµ (pipj))

≤ 2n Iµ (pipj)

t2 Fµ (pipj) + n(n − 1)|F|n ≤ E(Fµ (pipj))

≤ 2n Fµ (pipj)

Example 3: From Figure 2, Let’s calculate LB and UB for FQNFG.

E(Tµ (pipj)) = 1.9999, LB = 1.5684 and UB = 2.4495.

∴ 1.5684 ≤ 1.9999 ≤ 2.4495 E(Cµ (pipj)) = 2.3112, LB = 1.6125 and UB = 2.8284.

∴ 1.6125 ≤ 2.3112 ≤ 2.8284 E(Iµ (pipj)) = 2.2602, LB = 1.5875 and UB = 2.8566.

∴ 1.5875 ≤ 2.2602 ≤ 2.8566 E(Fµ (pipj)) = 2.6096, LB = 1.7493 and UB = 3.2125.

∴ 1.7493 ≤ 2.6096 ≤ 3.2125

Consider GFQ = (σFQ, µFQ) is a FQNFG with n vertices. Then degree matrix D(GFQ) = (D(Tµ (pipj)),

FQ

D(Cµ (pipj)), D(Iµ (pipj)), D(Fµ (pipj))) = [dij], of GFQ is a n × n diagonal matrix termed as

FQ FQ FQ

dG (pi) if i = j

FQ

dij = 0 otherwise

The Laplacian matrix (LM) of a FQNFG GF Q = (σF Q, µF Q) is termed as L(GF Q) = (L(Tµ (pipj)), L(Cµ (pipj)), L(Iµ (pipj)), L(Fµ (pipj))) = D(GF Q) − A(GF Q), where A(GF Q) and D(GF Q) represent an adjacency matrix and degree matrix of FQNFG. The spectrum of LM of FQNFG, L(GF Q) is denoted by (QL, RL, SL, TL), where QL, RL, SL and TL are sets of Laplacian eigen values of L(Tµ (pipj)), L(Cµ (pipj)), L(Iµ (pipj)) and L(Fµ (pipj)) respectively. Example 4:

Let GF Q = (σF Q, µF Q) be a FQNFG, where σF Q = (p1, p2, p3, p4) and µF Q = (p1p2, p2p3, p3p1, p3p4, p2p4)

σF Q p1 p2 p3 p4 Tσ 0.4 0.5 0.8 0.5 Cσ 0.6 0.7 0.6 0.7 Iσ 0.5 0.1 0.4 0.4 Fσ 0.3 0.3 0.1 0.2

The adjacency matrix of a FQNFG is given by   (0.0, 0.0, 0.0, 0.0) (0.3, 0.5, 0.5, 0.6) (0.3, 0.5, 0.7, 0.3) (0.0, 0.0, 0.0, 0.0) (0.3, 0.5, 0.5, 0.6) (0.0, 0.0, 0.0, 0.0) (0.2, 0.5, 0.5, 0.3) (0.4, 0.6, 0.5, 0.4) (0.3, 0.5, 0.7, 0.3) (0.2, 0.5, 0.5, 0.3) (0.0, 0.0, 0.0, 0.0) (0.4, 0.5, 0.5, 0.3) (0.0, 0.0, 0.0, 0.0) (0.4, 0.6, 0.5, 0.4) (0.4, 0.5, 0.5, 0.3) (0.0, 0.0, 0.0, 0.0) µF Q p1p2 p2p3 p3p1 p3p4 p2p4 Tµ 0.3 0.2 0.3 0.4 0.4 Cµ 0.5 0.5 0.5 0.5 0.6 Iµ 0.5 0.5 0.7 0.5 0.5 Fµ 0.6 0.4 0.4 0.3 0.4

The Laplacian matrix of a FQNFG is given by   (0.6, 1.0, 1.2, 0.9) (−0.3, −0.5, −0.5, −0.6) (−0.3, −0.5, −0.7, −0.4) (0.0, 0.0, 0.0, 0.0) (−0.3, −0.5, −0.5, −0.6) (1.1, 1.6, 1.5, 1.3) (−0.2, −0.5, −0.5, −0.4) (−0.4, −0.6, −0.5, −0.4) L(G ) =   F Q   (−0.3, −0.5, −0.7, −0.4) (−0.2, −0.5, −0.5, −0.4) (0.9, 1.5, 1.7, 0.9) (−0.4, −0.5, −0.5, −0.3) (0.0, 0.0, 0.0, 0.0) (−0.4, −0.6, −0.5, −0.4) (−0.4, −0.5, −0.5, −0.3) (0.8, 1.1, 1.0, 0.7) The Laplacian spectrum of a FQNFG is Laplacian spec(Tµ (pipj)) = {0.0442, 0.6740, 1.4952, 1.1866} Laplacian spec(Cµ (pipj)) = {0, 1.0432, 2.1568, 2} Laplacian spec(Iµ (pipj)) = {0, 1.0755, 2.3245, 2} Laplacian spec(Fµ (pipj)) = {−0.1044, 0.7650, 1.8304, 1.3090} Laplacian spec(GF Q(pipj)) = {(0.0442, 0, 0, −0.1044), (0.6740, 1.0432, 1.0755, 0.7650), (1.4952, 2.1568, 2.3245, 1.8304), (1.1866, 2, 2, 1.3090)}

Theorem 4.3. Let GF Q = (σF Q, µF Q) be a FQNFG and L(GF Q) = (L(Tµ (pipj)), L(Cµ (pipj)), L(Iµ (pipj)), L(Fµ (pipj))) be the LM of GF Q. If γ1 ≥ γ2 ≥ · · · ≥ γn, ϕ1 ≥ ϕ2 ≥ · · · ≥ ϕn, τ1 ≥ τ2 ≥ · · · ≥ τn, and ρ1 ≥ ρ2 ≥ · · · ≥ ρn were eigen values of L(Tµ (pipj)), L(Cµ (pipj)), L(Iµ (pipj)) and L(Fµ (pipj)) correspondingly. Therefore, P n P P n P P n P (1) γi = 2 Tµ (pipj) , ϕi = 2 Cµ (pipj) , τi = 2 Iµ (pipj) ,

i=1 1≤i≤j≤n i=1 1≤i≤j≤n i=1 1≤i≤j≤n γi∈QL ϕi∈RL τi∈SL

P n P ρi = 2 Fµ (pipj)

i=1 1≤i≤j≤n ρi∈TL

P n P P n

2 2 2

(2) γ = 2 Tµ (pipj) + d (pi),

i F Q Tµ (pipj ) i=1 1≤i≤j≤n i=1 γi∈QL

P n P P n

2 2 2

ϕ = 2 Cµ (pipj) + d (pi),

i F Q Cµ (pipj ) i=1 1≤i≤j≤n i=1 ϕi∈RL

P n P P n

2 2 2

τ = 2 Iµ (pipj) + d (pi),

i F Q Iµ (pipj ) i=1 1≤i≤j≤n i=1 τi∈SL

P n P P n

2 2 2

ρ = 2 Fµ (pipj) + d (pi).

i F Q Fµ (pipj ) i=1 1≤i≤j≤n i=1 ρi∈TL

Proof : (1) We know that L(GF Q) denotes a symmetric matrix of positive values. Then,

γi = tr(L(GF Q))

γi∈QL

= dT (p p )(pi)

µF Q i j

= 2 Tµ (pipj)

Likewise, it follows that, P n P P n P P n P ϕi = 2 Cµ (pipj) , τi = 2 Iµ (pipj) , ρi = 2 Fµ (pipj)

i=1 1≤i≤j≤n i=1 1≤i≤j≤n i=1 1≤i≤j≤n ϕi∈RL τi∈SL ρi∈TL (2) By the definition of LM, dT (p p )(p1) −Tµ (p1p2) · · · −Tµ (p1pn)

µF Q i j F Q F Q

 −Tµ (p2p1) dT (p p )(p2) · · · −Tµ (p2pn) 

F Q µF Q i j F Q

L(T (p p )) =   By matrix trace theory, µF Q i j  . . . .   . . .. .  . . .

−Tµ (pnp1) −Tµ (pnp2) · · · dT (p p )(pn)

F Q F Q µF Q i j

tr((L(Tµ (pipj))) ) = γi

γi∈QL

where

2 2 2 2

tr((L(Tµ (pipj))) ) = d (p1) + Tµ (p1p2) + · · · + Tµ (p1pn) +

F Q Tµ (pipj ) F Q F Q 2 2 2

Tµ (p2p1) + dT (p p )(p2) + · · · + Tµ (p2pn) + · · ·

F Q µF Q i j F Q

2 2 2

+ Tµ (pnp1) + Tµ (pnp2) + · · · + dT (p p )(pn)

F Q F Q µF Q i j

= 2 Tµ (pipj) + d (pi)

F Q Tµ (pipj ) 1≤i≤j≤n i=1

X X X

2 2 2

∴ γi = 2 Tµ (pipj) + d (pi)

F Q Tµ (pipj ) i=1 1≤i≤j≤n i=1 γi∈QL

Likewise, we get P n P P n

2 2 2

ϕ = 2 Cµ (pipj) + d (pi),

i F Q Cµ (pipj ) i=1 1≤i≤j≤n i=1 ϕi∈RL

P n P P n

2 2 2

τ = 2 Iµ (pipj) + d (pi),

i F Q Iµ (pipj ) i=1 1≤i≤j≤n i=1 τi∈SL

P n P P n

2 2 2

ρ = 2 Fµ (pipj) + d (pi). □

i F Q Fµ (pipj ) i=1 1≤i≤j≤n i=1 ρi∈TL

The LE of FQNFG GF Q = (σF Q, µF Q) is denoted by LE(GF Q) = (LE(Tµ (pipj)), LE(Cµ (pipj)), LE(Iµ (pipj)), LE(Fµ (pipj)))

F Q F Q F Q F Q

n n n n

X X X X = |δi|, |∆i|, |θi|, |ζi|

i=1 i=1 i=1 i=1

where, P P 2 Tµ (pipj) 2 Cµ (pipj)

1≤i≤j≤n 1≤i≤j≤n

δi = γi − ; ∆i = ϕi − ; P n P n 2 Iµ (pipj) 2 Fµ (pipj)

1≤i≤j≤n 1≤i≤j≤n

θi = τi − ; ζi = ρi − .

Theorem 4.4. Let GF Q = (σF Q, µF Q) be a FQNFG and L(GF Q) be the LM of GF Q. If γ1 ≥ γ2 ≥ · · · ≥ γn, ϕ1 ≥ ϕ2 ≥ · · · ≥ ϕn, τ1 ≥ τ2 ≥ · · · ≥ τn, and ρ1 ≥ ρ2 ≥ · · · ≥ ρn were eigen values of L(Tµ (pipj)), P F Q 2 Tµ (pipj) L(Cµ (pipj)), L(Iµ (pipj)) and L(Fµ (pipj)) respectively and δi = γi − ; ∆i = P P P n 2 Cµ (pipj) 2 Iµ (pipj) 2 Fµ (pipj)

1≤i≤j≤n 1≤i≤j≤n 1≤i≤j≤n

ϕi − ; θi = τi − ; ζi = ρi − . Then δi = n n n i=1 P n P n P n P n P n P n P n

2 2 2 2

0, ∆i = 0, θi = 0, ζi = 0, δi = 2NT , ∆i = 2NC, θi = 2NI, ζi = 2NF , where

i=1 i=1 i=1 i=1 i=1 i=1 i=1  P 2 2 Tµ (pipj) P 1 P n 2  1≤i≤j≤n  NT = Tµ (pipj) + dT (p p )(pi) − 

F Q µF Q i j

1≤i≤j≤n 2 i=1 n  P 2 2 Cµ (pipj) P 1 P n 2  1≤i≤j≤n  NC = Cµ (pipj) + dC (p p )(pi) − 

F Q µF Q i j

1≤i≤j≤n 2 i=1 n  P 2 2 Iµ (pipj) P 1 P n 2  1≤i≤j≤n  NI = Iµ (pipj) + dI (p p )(pi) − 

F Q µF Q i j

1≤i≤j≤n 2 i=1 n  P 2 2 Fµ (pipj) P 1 P n 2  1≤i≤j≤n  NF = Fµ (pipj) + dF (p p )(pi) − 

F Q µF Q i j

1≤i≤j≤n 2 i=1 n

Theorem 4.5. Consider FQNFG, GF Q = (σF Q, µF Q) with n vertices and L(GF Q) = (L(Tµ (pipj)), L(Cµ (pipj)), L(Iµ (pipj)), L(Fµ (pipj))) be the LM of GF Q. Hence,

u  P 2 u 2 Tµ (pipj) u P P n u 2  1≤i≤j≤n  (1) LE(Tµ (pipj)) ≤ t2n Tµ (pipj) + n dT (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n i=1 n

u  P 2 u 2 Cµ (pipj) u P P n u 2  1≤i≤j≤n  (2) LE(Cµ (pipj)) ≤ t2n Cµ (pipj) + n dC (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n i=1 n

u  P 2 u 2 Iµ (pipj) u P P n u 2  1≤i≤j≤n  (3) LE(Iµ (pipj)) ≤ t2n Iµ (pipj) + n dI (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n i=1 n

u  P 2 u 2 Fµ (pipj) u P P n u 2  1≤i≤j≤n  (4) LE(Fµ (pipj)) ≤ t2n Fµ (pipj) + n dF (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n i=1 n

Proof : Applying Cauchy-Schwarz inequality to the n numbers 1, 1, · · · , 1 and |δ1|, |δ2|, · · · , |δn| results in,

n u n X √ uX |δi| ≤ n |δi|

i=1 i=1

√ p p LE(Tµ (pipj)) ≤ n 2NT = 2nNT  P 2 2 Tµ (pipj) P 1 P n 2  1≤i≤j≤n  Since NT = Tµ (pipj) + dT (p p )(pi) − 

F Q µF Q i j

1≤i≤j≤n 2 i=1 n

u  P 2 u 2 Tµ (pipj) u P P n u 2  1≤i≤j≤n  ∴ LE(Tµ (pipj)) ≤ t2n Tµ (pipj) + n dT (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n i=1 n

Likewise, we get u  P 2 u 2 Cµ (pipj) u P P n u 2  1≤i≤j≤n  LE(Cµ (pipj)) ≤ t2n Cµ (pipj) + n dC (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n i=1 n

u  P 2 u 2 Iµ (pipj) u P P n u 2  1≤i≤j≤n  LE(Iµ (pipj)) ≤ t2n Iµ (pipj) + n dI (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n i=1 n

u  P 2 u 2 Fµ (pipj) u P P n u 2  1≤i≤j≤n  LE(Fµ (pipj)) ≤ t2n Fµ (pipj) + n dF (p p )(pi) −  □

F Q F Q µF Q i j

1≤i≤j≤n i=1 n

Theorem 4.6. Consider FQNFG, GF Q = (σF Q, µF Q) with n vertices and L(GF Q) = (L(Tµ (pipj)), L(Cµ (pipj)), L(Iµ (pipj)), L(Fµ (pipj))) are LM of GF Q. Thus,

u  P 2 u 2 Tµ (pipj) u P P n u 2 1  1≤i≤j≤n  (1) LE(Tµ (pipj)) ≥ 2t Tµ (pipj) + dT (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n 2 i=1 n

u  P 2 u 2 Cµ (pipj) u P P n u 2 1  1≤i≤j≤n  (2) LE(Cµ (pipj)) ≥ 2t Cµ (pipj) + dC (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n 2 i=1 n

u  P 2 u 2 Iµ (pipj) u P P n u 2 1  1≤i≤j≤n  (3) LE(Iµ (pipj)) ≥ 2t Iµ (pipj) + dI (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n 2 i=1 n

u  P 2 u 2 Fµ (pipj) u P P n u 2 1  1≤i≤j≤n  (4) LE(Fµ (pipj)) ≥ 2t Fµ (pipj) + dF (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n 2 i=1 n

Proof :

X X X |δi| = |δi| + 2 |δiδj| ≥ 4NT

i=1 i=1 1≤i≤j≤n

LE(Tµ (pipj)) ≥ 2NT We know that  P  2 Tµ (pipj) P 1 P n 2  1≤i≤j≤n  NT = Tµ (pipj) + dT (p p )(pi) − 

F Q µF Q i j

1≤i≤j≤n 2 i=1 n

u  P 2 u 2 Tµ (pipj) u P P n u 2 1  1≤i≤j≤n  ∴ LE(Tµ (pipj)) ≥ 2t Tµ (pipj) + dT (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n 2 i=1 n

Similary, it follows that v u  P 2 u 2 Cµ (pipj) u P P n u 2 1  1≤i≤j≤n  LE(Cµ (pipj)) ≥ 2t Cµ (pipj) + dC (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n 2 i=1 n u  P 2 u 2 Iµ (pipj) u P P n u 2 1  1≤i≤j≤n  LE(Iµ (pipj)) ≥ 2t Iµ (pipj) + dI (p p )(pi) − 

F Q F Q µF Q i j

1≤i≤j≤n 2 i=1 n

u  P 2 u 2 Fµ (pipj) u P P n u 2 1  1≤i≤j≤n  LE(Fµ (pipj)) ≥ 2t Fµ (pipj) + dF (p p )(pi) −  □

F Q F Q µF Q i j

1≤i≤j≤n 2 i=1 n

Theorem 4.7. Consider FQNFG, GF Q = (σF Q, µF Q) with n vertices and L(GF Q) = (L(Tµ (pipj)), L(Cµ (pipj)), L(Iµ (pipj)), L(Fµ (pipj))) are LM of GF Q. Thus, v   u  P  u 2 Tµ (pipj) u P P n  2 1≤i≤j≤n  u   2 (1) LE(Tµ (pipj)) ≤ |δ1|+ (n − 1) 2 Tµ (pipj) + dT (p p )(pi) − − δ  F Q t F Q µF Q i j 1

1≤i≤j≤n i=1

v   u  P  u 2 Cµ (pipj) u P P n  2 1≤i≤j≤n  u   2 (2) LE(Cµ (pipj)) ≤ |∆1|+ (n − 1) 2 Cµ (pipj) + dC (p p )(pi) − − ∆  F Q t F Q µF Q i j 1

1≤i≤j≤n i=1

v   u  P  u 2 Iµ (pipj) u P P n  2 1≤i≤j≤n  u   2 (3) LE(Iµ (pipj)) ≤ |θ1|+ (n − 1) 2 Iµ (pipj) + dI (p p )(pi) − − θ  F Q t F Q µF Q i j 1

1≤i≤j≤n i=1

v   u  P  u 2 Fµ (pipj) u P P n  2 1≤i≤j≤n  u   2 (4) LE(Fµ (pipj)) ≤ |ζ1|+ (n − 1) 2 Fµ (pipj) + dF (p p )(pi) − − ζ  F Q t F Q µF Q i j 1

1≤i≤j≤n i=1

P n P n P n P n Proof : The Cauchy-Schwarz Inequality yields, |δi| ≤ n |δi| , |δi| ≤ (n − 1) |δi|

i=1 i=1 i=2 i=2

LE(Tµ (pipj)) − |δ1| ≤ (n − 1)(2NT − δ )

F Q 1

LE(Tµ (pipj)) ≤ |δ1| + (n − 1)(2NT − δ )

F Q 1

We know that,  P 2 2 Tµ (pipj) P 1 P n 2  1≤i≤j≤n  NT = Tµ (pipj) + dT (p p )(pi) − 

F Q µF Q i j

1≤i≤j≤n 2 i=1 n

u   P 2  u 2 Tµ (pipj) u P P n u  2  1≤i≤j≤n   ∴ LE(Tµ (pipj)) ≤ |δ1| + t(n − 1) 2 Tµ (pipj) + dT (p p )(pi) −  − δ 

F Q F Q µF Q i j 1

1≤i≤j≤n i=1 n

Similarly, we can prove v u   P   u 2 Cµ (pipj) u P P n  2 1≤i≤j≤n  u   2 LE(Cµ (pipj)) ≤ |∆1|+ (n − 1) 2 Cµ (pipj) + dC (p p )(pi) − − ∆  F Q t F Q µF Q i j 1

1≤i≤j≤n i=1 v   u  P  u 2 Iµ (pipj) u P P n  2 1≤i≤j≤n  u   2 LE(Iµ (pipj)) ≤ |θ1| + (n − 1) 2 Iµ (pipj) + dI (p p )(pi) − − θ  F Q t F Q µF Q i j 1

1≤i≤j≤n i=1

v   u  P  u 2 Fµ (pipj) u P P n  2 1≤i≤j≤n  u   2 LE(Fµ (pipj)) ≤ |ζ1|+ (n − 1) 2 Fµ (pipj) + dF (p p )(pi) − − ζ  F Q t F Q µF Q i j 1

1≤i≤j≤n i=1

To purchase a new laptop using MCDM, the criteria are

(1) Processor

(2) Memory & Storage

(3) Battery life

(4) Display quality

Embarking on the New Laptop Hunt requires careful consideration of several key criteria. Processing Power is paramount, demanding fast performance and the ability to handle seamless multitasking. Equally important are Memory & Storage, requiring ample RAM for smooth operation and sufficient storage to accommodate all essential files and programs. Battery Life is a crucial factor for uninterrupted productivity, ideally lasting for the entire workday. Finally, Display Quality plays a significant role in both work and entertainment, necessitating vibrant visuals and crisp details for an optimal viewing experience.

Now, using a decision-making method with 4 criteria, we will consider 4 laptops to find the best one to buy. Laptop 1:

(1) Processor – (0.3, 0.4, 0.3, 0.1)

(2) Memory & Storage – (0.6, 0.5, 0.3, 0.2)

(3) Battery life – (0.3, 0.5, 0.2, 0.1)

(4) Display quality – (0.2, 0.1, 0.6, 0.3)

Laptop 2:

(1) Processor – (0.8, 0.2, 0.1, 0.1)

(2) Memory & Storage – (0.7, 0.4, 0.3, 0.2)

(3) Battery life – (0.7, 0.6, 0.3, 0.4)

(4) Display quality – (0.5, 0.8, 0.2, 0.3)

Laptop 3:

(1) Processor – (0.6, 0.6, 0.5, 0.3)

(2) Memory & Storage – (0.5, 0.4, 0.6, 0.2)

(3) Battery life – (0.8, 0.2, 0.6, 0.2)

(4) Display quality – (0.2, 0.3, 0.5, 0.3)

Laptop 4:

(1) Processor – (0.6, 0.5, 0.3, 0.1)

(2) Memory & Storage – (0.4, 0.4, 0.5, 0.1) (3) Battery life – (0.6, 0.3, 0.1, 0.2)

(4) Display quality – (0.7, 0.6, 0.4, 0.2)

The criteria have associated weights given by the vector W = (0.6,0.4,0.3,0.2). In the decision matrix P, row denotes laptop and column represents one of the four criteria.   (0.3, 0.4, 0.3, 0.1) (0.6, 0.5, 0.3, 0.2) (0.3, 0.5, 0.2, 0.1) (0.2, 0.1, 0.6, 0.3) (0.8, 0.2, 0.1, 0.1) (0.7, 0.4, 0.3, 0.2) (0.7, 0.6, 0.3, 0.4) (0.5, 0.8, 0.2, 0.3) P =   (0.6, 0.6, 0.5, 0.3) (0.5, 0.4, 0.6, 0.2) (0.8, 0.2, 0.6, 0.2) (0.2, 0.3, 0.5, 0.3) (0.6, 0.5, 0.3, 0.1) (0.4, 0.4, 0.5, 0.1) (0.6, 0.3, 0.1, 0.2) (0.7, 0.6, 0.4, 0.2) We will compute the Fermatean Quadripartitioned Neutrosophic arithmetic average across all laptops for each criterion, resulting in a single Fermatean Quadripartitioned Neutrosophic value per criterion. The aggregated Fermatean quadripartioned neutrosophic values are as follows.

0.3 + 0.6 + 0.3 + 0.2 0.4 + 0.5 + 0.5 + 0.1 0.3 + 0.3 + 0.2 + 0.6 0.1 + 0.2 + 0.1 + 0.3 L1 = , , , 4 4 4 4 = (0.35, 0.375, 0.35, 0.175) Similarly we Calculate for other Laptops as L2 = (0.675, 0.5, 0.225, 0.25); L3 = (0.525, 0.375, 0.55, 0.25); L4 = (0.575, 0.45, 0.325, 0.15).

Using these aggregated values for each criterion, we will now determine the overall score and accuracy for each laptop. We will use the aggregated Fermatean Quadripartitioned Neutrosophic values from the previous step to compute score and accuracy for all laptops seperately.

Score equation formula is

m X W T + W C + W I + W F

T i C i I i F i

S = × 2 WT + WC + WI + WF

After calculating We get, S(L1) = 0.67; S(L2) = 0.96; S(L3) = 0.9; S(L4) = 0.87 The accuracy function formula is

m 2 2 2 2

X (W T ) + (W C ) + (W I ) + (W F )

T i C i I i F i

After computation, A(L1) = 0.1985; A(L2) = 0.325; A(L3) = 0.2751; A(L4) = 0.2844.

We can now rank the laptops based on their calculated scores and accuracy values as follows:

L2 > L4 > L3 > L1. In conclusion Laptop L2 is identified as the best Laptop for purchasing.

Fermatean Quadripartitioned Neutrosophic Fuzzy Graph (FQNFG) provides a suitable modelling framework. These models generally offer improved precision, greater flexibility and enhanced compatibility compared to other fuzzy approaches. This manuscript dealt with the new idea of Energy and LE of FQNFG . The LB and UB of Energy and LE for FQNFG were derived. The energy of FQNFG holds potential for applications in Computer Science, Chemistry, Risk Assessment, Medical diagnosis etc. The proposed methodology offers the most effective method for laptop acquisition. The Future work will extend to energy of FQNFGs to Double Layered Fuzzy Graph, Hesitant Fuzzy Graph, Spherical Fuzzy Graph etc.

The authors deeply appreciate the editors and anonymous reviewers insightful comments and constructive feedback, which were instrumental in improving this work.

V. Divya: Software and editing. J. Jesintha Rosline: conceptualization and methodology. All authors have approved the final manuscript.

The authors affirm that no external financial assistance, administrative support, technical resources, or in-kind contributions were received for the work presented in this paper.Data Availability: The data supporting the findings of this study are available within the manuscript.

The authors declare no conflict of interest regarding the publication of this research. The funders had no involvement in the study design, data collection, analysis, interpretation, manuscript writing or publication decision.