EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 4, Article Number 6307
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Fuzzy Geodetic and Detour Spectra:
Geodetic-Laplacian Energy in Fuzzy Graphs

R. Rajeshkumar1, A. M. Anto2,∗, V. Mary Mettilda Rose3

1 Department of Mathematics, Malankara Catholic College, Affiliated to Manonmaniam
Sundaranar University, Tirunelveli, 627012, Tamil Nadu, India
2 Department of Mathematics, St. Albert’s College (Autonomous), Ernakulam, 682018,
Kerala, India
3 Department of Mathematics, Christ Nagar College, Affiliated to University of Kerala,
Maranalloor 695512, Kerala, India

Abstract. This research study delves into the spectral characteristics and energy measures associ-
ated with fuzzy graphs. We present the fuzzy geodetic spectrum and fuzzy detour spectrum, which
capture the spectral properties of fuzzy graphs under geodetic and detour constraints. In addition,
we propose and derive novel expressions for the fuzzy geodetic-Laplacian and detour-Laplacian
energies, incorporating both upper and lower bounds. Towards the end, a real-world application
of fuzzy geodetic-Laplacian energy pertaining to illegal immigration and human trafficking is de-
scribed. It emphasizes its potential to improve decision-making measures, allowing more effective
and coordinated crime prevention and resolution initiatives.

2020 Mathematics Subject Classifications: 05C72, 05C50, 05C20

Key Words and Phrases: Fuzzy graphs, geodetic spectrum, detour spectrum, geodetic-Laplacian
energy

1. Introduction

Lotfi A. Zadeh set forth the idea of fuzzy sets [1], which handle vagueness and unpre-
dictability in set theory. Zadeh [2] explored fuzzy relations and their underlying mech-
anisms, aiming to address the inherent vagueness in human reasoning. His work laid
the groundwork for applying fuzzy logic to diverse fields such as abstraction, information
technology, and communication. Inspired by his notion of fuzzy sets, Azriel Rosenfeld
[3] instilled fuzzy set theory into fuzzy graph theory in 1975. This state of the art of
graph theory allowed us to convey the strength of relationships within the interval [0, 1].
Different mathematicians were given the freedom to experiment after being inspired by

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6307

Email addresses: rajeshrajendrakumar09@gmail.com (R. Rajeshkumar),
antoalexam@gmail.com (A. M. Anto), marymettilda@cnc.ac.in (V. Mary Mettilda Rose)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 2 of 25

Rosenfeld’s papers. In fuzzy graphs, he developed fuzzy relations on fuzzy sets and veri-
fied several fundamental concepts. Recent works such as Platil and Tanaka [4] developed
multi-criteria evaluation frameworks for intuitionistic fuzzy sets using set-relations, which
may serve as a useful theoretical backdrop for the spectral approaches. It was Gary Char-
trand, Gamy L. Johns, and Songlin Tian who presented the notion of detour distance in
graphs [5]. Later, Rosenfeld, who developed a metric called the µ distance in fuzzy graphs.
Several other researchers, including P. S. Nair [6], Mathew and Mordeson [7, 8] presented
geodetic distance and other properties in fuzzy graphs, Vijayakumar and Sunitha [9], and
Rajeshkumar and Anto [10, 11], have conducted significant studies within the frameworks
of fuzzy and intuitionistic fuzzy graph theory. Nagoorgani and Umamaheswari presented
the notion of fuzzy detour µ-distance and some of its properties in [12].

The concept of graph energy was introduced by I. Gutman in 1978 [13] in a concise
and accessible form. As discussed in [14–18], several energy-related concepts have been
proposed by various researchers, expanding the scope of graph energy studies. However,
most of the existing studies on graph energy have focused primarily on crisp graphs,
with limited extensions into fuzzy graphs. Even within the fuzzy graph domain, energy-
related measures have largely overlooked the roles of alternative distance metrics such as
geodetic and detour distances. The concept of fuzzy graph energy, in particular, introduces
additional complexity, as the fuzziness of both vertices and edges must be integrated into
spectral computations. Anjali and Mathew [19] were among the first to explore this
area, proposing a method to derive spectral properties from fuzzy adjacency matrices
and generalizing the classical notion of graph energy to accommodate fuzzy structures.
Gutman and Zhou explored the Laplacian energy in classical graph theory [20], while S.
Rahimi Sharbaf and F. Fayazi introduced the concept of Laplacian energy for fuzzy graphs
[21]. M. Nath and S. Paul have investigated the distance Laplacian spectra of graphs [22].
Furthermore, there has been little exploration of how these distance-based energies interact
with structural properties of fuzzy graphs, especially in terms of spectral measures like the
Laplacian. The theoretical foundation of the present study draws inspiration from these
key works.

The application-oriented motivation for this study stems from the foundational work
of J. N. Mordeson and S. Mathew on distance measures and energy concepts in fuzzy
graphs [8], as well as the study by Binu, Mathew, and Mordeson, which applied the fuzzy
Wiener index to model illegal immigration networks [23]. Furthermore, a recent investiga-
tion into the intuitionistic fuzzy geodetic Wiener index for global human trading analysis
[24] has highlighted the relevance of fuzzy distance-based measures in addressing complex
real-world problems. These works collectively inspired the present study to explore novel
parameters for evaluating uncertain and dynamic network structures. Additional insights
from related literature [25–27] have also enriched the conceptual and methodological de-
velopment of this work.

This study addresses these gaps by extending the concept of fuzzy graph energy to
include fuzzy geodetic energy, fuzzy detour energy, and, most notably, fuzzy geodetic-
Laplacian energy and fuzzy detour-Laplacian energy. These new energy constructs provide
a more comprehensive framework for analyzing fuzzy graphs, as they integrate distance-



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 3 of 25

based information and spectral properties. The novelty of the proposed fuzzy geodetic-
Laplacian and detour-Laplacian energies lies in their ability to reflect both the topological
variation and fuzzy connectivity of the graph—features that classical energy measures
do not fully capture. Compared to traditional graph energies, these fuzzy spectral en-
ergies offer greater flexibility and descriptive power, especially for applications involving
uncertainty and imprecision.

This paper systematically extends spectral concepts to the domain of fuzzy graphs by
introducing novel distance-based energies and exploring their structural implications. The
core contributions are organized across seven sections. Section 2 introduces foundational
definitions and notations relevant to fuzzy graph theory. Section 3 presents the concept
of fuzzy geodetic energy, extending classical spectral ideas by defining the geodetic spec-
trum and computing energy based on fuzzy distances. In addition, the study also outlines
an algorithm for computing geodetic paths in fuzzy graphs, thereby complementing the
theoretical developments with computational techniques that ultimately lead to the deter-
mination of the fuzzy geodetic energy. Section 4 investigates fuzzy detour energy derived
from the spectrum of the fuzzy detour distance matrix, offering new structural insights
using long-path distance measures. Section 5 defines fuzzy geodetic-Laplacian energy, a
novel spectral invariant based on the fuzzy geodetic-distance matrix, representing a new in-
novation within the framework of Laplacian energy concepts.. Section 6 introduces fuzzy
detour-Laplacian energy by applying spectral analysis to the detour-distance Laplacian
matrix, bridging classical Laplacian energy and detour-based metrics. Finally, Section 7
explores the application of fuzzy geodetic-Laplacian energy in optimizing decision-making
processes, particularly in areas such as modern-day slavery [28] prevention and law en-
forcement modeling.

2. Preliminaries

This section presents the foundational definitions necessary for the study. Fuzzy graphs
and fuzzy relations were first introduced by Rosenfeld [3]. The algebraic underpinnings
of fuzzy graph structures may be further contextualized by fuzzy Γ-semimodule theory,
as discussed by Platil and Petalcorin [29], which enriches the mathematical modeling of
generalized fuzzy relationships. In this work, we adopt the notion of fuzzy graphs (FGs)
as discussed in [7, 8], along with other related concepts from existing literature.

Definition 1. [7] A fuzzy graph G : (V, σ, µ) consists of a non-empty set V along with a pair
of functions σ : V→ [0, 1] and µ : E → [0, 1] such that ∀u, v ∈ V, with µ(u, v) ≤ σ(u)∧σ(v).
Where the elements in V are the vertices and the elements in E are edges of the fuzzy
graph.

Definition 2. [8] Let G : (V, σ, µ) is a fuzzy graph, then H : (V, τ, ϑ ) is called a partial
fuzzy subgraph of G if τ ⊆ σ and ϑ ⊆ µ . Similarly, a fuzzy subgraph H : (V ′

, τ, ϑ) of G
is induced by V ′

if V ′ ⊆ V, τ (u) = σ(u) ∀ u ϵV ′
and ϑ (u, v) = µ(u, v) ∀u, v ϵ P.

Definition 3. [8] The support of σ is described as, Supp (σ) = u ϵ V: σ (u) > 0 and
is denoted as σ∗. The definitions and additional findings that we utilize later in our



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 4 of 25

discussion are listed below.

Definition 4. [12] In a fuzzy graph G : (V, σ, µ) a path P is a chain of distinct vertices
v0, v1, . . . . . . vn such that (vi−1, vi) > 0, 1 ≤ i ≤ n, where n denotes the length of the
path.

Definition 5. [8] Every path has a strength and is defined to be
∧n

i=1{µ (vi−1, vi)}. The
symbol

∧
denotes the minimum.

Definition 6. [7] The strength of connectedness between two vertices is the maximum
strength among all paths connecting them and is represented as µ ∞ (vi−1, vi).

Definition 7. [8] A fuzzy graph G : (V, σ, µ) is defined to be a complete fuzzy graph if
µ (vi, vj) = σ (vi) ∧ σ(vj) ∀ vi , vj ϵ σ∗.

Definition 8. [7] A connected fuzzy graph is called a fuzzy tree if it has a fuzzy spanning
subgraph H:(V , σ, τ), which is a tree where for all vivj not in H, µ (vi, vj) < τ∞(vi, vj).

Definition 9. [8] In a fuzzy graph (FG), the fuzzy distance between any two nodes is
defined and denoted as df (u, v) =

∧
P{l(P)∗S(P)}, where the length of the path is denoted

as l(P) and the strength of the path is denoted as S(P), while the symbol
∧

denotes the
minimum over all such paths. Moreover, a vi− vj path is called a fuzzy vi− vj geodesic if
df (vi, vj) = l(P) ∗ S(P).

Definition 10. [12] A u – v path is called a fuzzy detour if its length is ∆(u, v), where
∆(u, v) is the fuzzy detour µ – distance between pair of vertices u and v and is defined by
the maximum µ-length of any u – v path, where the µ- length of a path P : u = v0, v1,
. . . . . . , vn = v is l (P ) =

∑n
i=1

1
µ(vi−1, vi)

.

Definition 11. [20] An m×m matrix FDG = [aij ], where aij = dFG(si), if i = j and 0
otherwise, is called the degree matrix of fuzzy graph G : (V, σ, µ).

Definition 12. [20] Let G : (V, σ, µ) be an FG with |V| = n and µ1 ≥ µ2 ≥ ...... ≥ µn be

the Laplacian eigenvalues. Then LE(G) = |µi −
2
∑

1≤i<j≤n µ(vi, vj)

n |.

Theorem 1. [18] Let G be a weighted graph of order n each of whose edges has nonzero
weight and e1, . . . , em be all the edges of G. Then E(G) ≤ 2

∑m
i=1 |w(ei)|, where equality

holds if and only if each connected component of G has at most two vertices.

Theorem 2. [8] Let G : (V, σ, µ) be FG with |V| = n and µ∗ = {e1, ..., em}. Then,
E(G) ≤ 2

∑m
i=1 µ(ei).

Theorem 3. [8] Let H : (V, σ, µ) be a complete fuzzy subgraph of G. Then, H is a fuzzy
geodetic block (FGB) if and only if {vivj} is geodetic convex for any two vertices.

The key symbols, notations and abbreviations employed throughout this paper are
summarized in Table 1.



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 5 of 25

Table 1: Symbols, notations and abbreviations.

Symbols/notations Definition

G : (V, σ, µ) A fuzzy graph (FG) with membership functions σ and µ.
H : (V, τ, ϑ ) Partial fuzzy subgraph of an FG.

µ fuzzy edge membership function.
σ fuzzy vertex membership function.
P A path in an FG.∧

Symbol for minimum.
min Minimum.
max Maximum.
FGB Fuzzy geodetic block.

df (vi, vj) Fuzzy vi − vj geodesic
∆ (u, v) Fuzzy detour µ – distance.
E(G) Energy of an FG.
LE(G) Laplacian energy of an FG.
FgE(G)) Fuzzy geodetic energy (Fg-energy) .
FdE(G)) Fuzzy detour energy (Fd-energy) .
GLE(G) Geodetic Laplacian energy.
DLE(G) Detour Laplacian energy.
MFgd

Fuzzy geodetic distance matrix (Fg-matrix).

TFgd
Fuzzy geodetic transmission matrix.

MFgld
Fuzzy geodetic-Laplacian distance matrix.

MFdd
Fuzzy detour distance matrix (Fd-matrix).

TFdd
Fuzzy detour transmission matrix.

MFdld
Fuzzy detour Laplacian-distance matrix .

FgSr Fuzzy geodetic spectral radius.
FdSr Fuzzy detour spectral radius.

3. Fuzzy Geodetic Spectrum

In classical graph theory, the distance spectrum focuses on the lengths of paths between
pairs of vertices. In recent years, researchers have extended spectral concepts to fuzzy
graphs, where uncertainty in connections must be taken into account. The concept of fuzzy
graph energy by Anjali and Mathew in [19], involves deriving spectral properties from fuzzy
adjacency matrices to extend classical graph energy to fuzzy structures. Motivated by the
mathematical richness of graph energy and its variants, we take the helm in this context
by introducing the fuzzy geodetic spectrum and investigating its mathematical properties.
Furthermore, we define and compute the fuzzy geodetic energy of fuzzy graphs, thereby
extending classical spectral concepts into the fuzzy domain.

Definition 13. Let G be a fuzzy graph with vertex set σ∗ = v1, v2, . . . , vm. The fuzzy
geodetic distance matrix (Fg-matrix) of G is the n × n matrix MFgd

= [bij], where bij =
df (vi, vj) denotes the fuzzy geodetic distance between vi and vj. The set of eigenvalues of



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 6 of 25

MFgd
is referred to as the fuzzy geodetic spectrum (Fg-spectrum) of G.

Definition 14. The fuzzy geodetic spectral radius (FgSr) of the matrixMFg is defined as
the maximum eigenvalue in the Fg-spectrum of G.

Definition 15. The fuzzy geodetic energy (Fg-energy or FgE(G)) is defined as the sum
of the absolute values of the Fg-eigenvalues of the matrixMFgd

.

Example 1. Consider the FG G = (V, σ, µ) as shown in Figure 1. Let the Fg-matrix of
G, corresponding to the vertex set σ∗ = {v1, v2, v3, v4}, be given as follows:

0.0 0.2 0.1 0.2
0.2 0.0 0.2 0.3
0.1 0.2 0.0 0.2
0.2 0.3 0.2 0.0

.

Figure 1: A fuzzy graph with Fg-spectrum {0.6123, −0.3, −0.2123, −0.1}.

To compute the Fg-spectrum of G, we employ the classical matrix theory approach: the
eigenvalues of the Fg-matrix MFgd

are obtained by solving the characteristic polynomial
equation

det(λI −MFgd
) = 0, (1)

where I denotes the identity matrix of order |σ∗|.
For the present case, solving (1) yields the eigenvalues,

λ1 = 0.6123, λ2 = −0.3, λ3 = −0.2123, λ4 = −0.1.

Thus, the Fg-spectrum of G is {0.6123, −0.3, −0.2123, −0.1}. Accordingly, the Fg-energy
of G is evaluated as,

FgE(G) =
4∑

i=1

|λi| = 0.6123 + 0.3 + 0.2123 + 0.1 = 1.2246.



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Theorem 4. Let G be an FG with |V| = m and |E| = n. Suppose the fuzzy geodetic
distance between vertices vi and vj is given by df (vi, vj) = l(P) × S(P) = dij, and let
MFgd

denote the fuzzy geodetic distance matrix. Then, the fuzzy geodetic energy FgE(G)
satisfies the following bounds:

√
2
∑
i<j

(dij)2 +m(m− 1)|MFgd
|
2
m ≤ FgE(G) ≤

√√√√√2

∑
i<j

(dij)2

m.

Proof. By applying the Cauchy–Schwarz inequality to the number of vertices and the
set of absolute values of the eigenvalues,

m∑
i=1

|θi| ≤
√
m

√√√√ m∑
i=1

|θi|2 . (2)

(
m∑
i=1

θi

)2

=
m∑
i=1

|θi|2 + 2
∑
i<j

θiθj . (3)

Next, we analyze the coefficients of
∏m

i=1(θ − θi) = |MFgd
− θI| implies∑

i<j

θiθj = −
∑
i<j

d 2
ij . (4)

Now, in light of Equation (3)

m∑
i=1

|θi|2 = 2
∑
i<j

(dij)
2 . (5)

Substituting Equation (5) into Equation (2) yields

m∑
i=1

|θi| ≤
√
m

√
2
∑
i<j

d 2
ij =

√√√√√2

∑
i<j

d 2
ij

m

FgE(G) ≤

√√√√√2

∑
i<j

(dij)2

m.



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 8 of 25

Regarding the second inequality, we have

[FgE(G)]2 =

(
m∑
i=1

θi

)2

=

m∑
i=1

|θi|2 + 2
∑
i<j

θiθj

≥ 2
∑
i<j

(dij)
2 +m(m− 1)GM{|θiθj |}

=⇒ FgE(G) ≥
√

2
∑
i<j

(dij)2 +m(m− 1)GM{|θiθj |} .

But,

GM{|θiθj |} =

∏
i<j

|θiθj |

 2
m(m−1)

=

∏
i<j

|θi|

 2
m

= |Mdf |
2
m

.

Now,

FgE(G) ≥
√
2
∑
i<j

(dij)2 +m(m− 1)|MFgd
|
2
m .

This completes the proof.

Upper bounds for the Fg-energy of an FG can be derived using the spectral radius.

Theorem 5. Let G be an FG with m vertices, and let ri denote the sum of the entries in
the ith row of the Fg-matrix. Then, FgE(G) ≤ m max{ri}, 1 ≤ i ≤ m.

Proof. The Fg-matrix is a non-negative square matrix with zero diagonal entries. By
the properties of non-negative matrices, the spectral radius FgSr satisfies

min
1≤i≤m

ri ≤ FgSr ≤ max
1≤i≤m

ri,

where ri is the sum of the ith row of the Fg-matrix.
Thus,

FgSr ≤ max
1≤i≤m

ri.

Since FgE(G) is the sum of the absolute values of the eigenvalues, and the spectral
radius bounds the largest absolute eigenvalue, it follows that

FgE(G) ≤ m max
1≤i≤m

ri,

completing the proof.

To establish a lower bound for FgE(G), a similar approach can be applied. The
following theorem presents the equivalence of adjacency energy and geodetic energy within
fuzzy geodetic blocks.



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 9 of 25

Theorem 6. Let G be a fuzzy geodetic block graph (FGB). Then, the spectra of the Fg-
matrix and the adjacency matrix A of G coincide, and consequently, their energies are
equal.

Proof. Assume G is an FGB. By definition, the pair {vi, vj} is fuzzy geodetic convex
for any two vertices vi and vj . Thus, every edge corresponds to a unique fuzzy geodesic,
implying that all vertex pairs are adjacent. Therefore, the adjacency matrix A and the
fuzzy geodetic distance matrix MFgd

coincide. This establishes the equivalence of their
spectra and hence their energies.

3.1. Geodesic Path Identification and Fuzzy Geodetic Energy Algorithm

The pseudocode presented below offers a clear and systematic description of the geodesic
path identification process, along with its application in constructing the fuzzy geodetic
distance matrix, spectrum, and energy.

Input: A fuzzy graph G = (V, σ, µ) with vertex set V = {v1, v2, . . . , vn} and fuzzy edge
weights µ
Output: Fuzzy geodesic paths, fuzzy geodetic distance matrix MFgd

, fuzzy geodetic
spectrum, and fuzzy geodetic energy

(i) Initialize an n× n matrixMFgd
with zeros.

(ii) For each unique unordered vertex pair (vi, vj):

(a) Generate all possible paths P between vi and vj .

(b) For each path P:
i. Compute the path length l(P) = number of edges in P.
ii. Compute the path strength S(P) = min{µ(e) : e ∈ P}.

(c) Determine the fuzzy geodetic distance:

df (vi, vj) = min
P
{l(P)× S(P)}.

(d) AssignMFgd
[i][j]← df (vi, vj).

(iii) After processing every pair,MFgd
represents the fuzzy geodetic distance matrix.

(iv) Evaluate the eigenvalues {λ1, λ2, . . . , λn} ofMFgd
.

(v) Define the fuzzy geodetic spectrum as:

Fg-spectrum = {λ1, λ2, . . . , λn}.



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 10 of 25

(vi) Determine the fuzzy geodetic energy:

FgE(G) =
n∑

i=1

|λi|.

(vii) Display the identified geodesic paths, the constructed Fg-matrix, its spectrum, and
the computed energy.

4. Fuzzy Detour Spectrum

In fuzzy graph theory, Nagoorgani and Umamaheswari introduced the fuzzy detour
µ-distance and examined its fundamental properties. Later, Rajesh and Anto explored
detour convexity properties. Extending this framework, we investigate the fuzzy detour
energy derived from the spectrum of the fuzzy detour distance matrix. This development
enriches the spectral analysis of fuzzy graphs by incorporating long-path distance measures
under fuzziness, leading to new bounds and structural insights.

Definition 16. The fuzzy detour distance matrix (Fd - matrix) of G with σ∗ = {v1, v2, ...
.. vm} is an m×m matrix MFdd

= [cij ] where cij = ∆(vi, vj). The fuzzy detour energy
(Fd-Energy) is the sum of the absolute values of its Fd- eigenvalues of MFdd

. The set
of Fd- eigenvalues is called the fuzzy detour-spectrum (Fd − spectrum) of G. The fuzzy
detour-spectral radius (FdSr) ofM is the maximum of its Fd- eigenvalues.

Example 2. For instance, consider the Fd-matrix (distance has been corrected to two
decimal places) of G in Figure 2 with σ∗ = {v1, v2, v3, v4}.

Figure 2: A fuzzy graph with fuzzy detour-spectral radius 36.9236.



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 11 of 25
0.00 11.66 10.00 13.33
11.66 0.00 12.00 14.66
10.00 12.00 0.00 12.50
13.33 14.16 12.50 0.00

.

Here, the Fd-spectrum is {36.9236, −14.6074, −12.3964, −9.91979}, and thus the FdE(G) =
73.84719. The FdSr = 36.9236.

Theorem 7. Let G be an FG. Then,

FdE(G) ≤ 2
∑
i,j

∆(vi, vj), ∀ i, j.

Proof. Consider the fuzzy detour distance matrix MFdd
, which is a non-negative,

symmetric square matrix. The trace ofM2
Fdd

is given by

Tr(M2
Fdd

) =
m∑
i=1

m∑
j=1

∆(vi, vj)∆(vj , vi).

SinceMFdd
is symmetric, we have ∆(vi, vj) = ∆(vj , vi), and thus

Tr(M2
Fdd

) =
m∑
i=1

m∑
j=1

∆(vi, vj)
2.

Using Theorem 1 and properties of symmetric matrices, it follows that

FdE(G) ≤ 2
∑
i,j

∆(vi, vj), ∀ i, j,

as required.

Theorem 8. Let G be a fuzzy graph, and let ri denote the sum of the entries in the ith

row of the Fd-matrix. Then,
FdE(G) ≤ n max

1≤i≤n
ri.

Proof. The Fd-matrix is a non-negative square matrix with zero diagonal entries.
Thus, the spectral radius FdSr satisfies

min
1≤i≤n

ri ≤ FdSr ≤ max
1≤i≤n

ri.

Hence,
FdSr ≤ max

1≤i≤n
ri.

Since the fuzzy detour energy FdE(G) is the sum of the absolute values of the eigenvalues,
we have

FdE(G) ≤ n · max
1≤i≤n

ri,

as required.

To derive the lower bound for FdE(G), a similar proof is possible.



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 12 of 25

Theorem 9. Let G be a FG with |V| = m and |E| = n. If cij = ∆(vi, vj) denotes the
fuzzy detour distance andMFdd

is the fuzzy detour distance matrix, then

FdE(G) ≤
√

(m− 1)
[
κ− θ2i

]
, where κ =

∑
i<j

c2ij .

Proof. By the Cauchy–Schwarz inequality, we have:

m∑
i=2

|θi| ≤

√√√√(m− 1)

m∑
i=2

θ2i . (6)

Suppose that the total sum of squared eigenvalues satisfies:

m∑
i=1

θ2i ≤ κ =
∑
i<j

c2ij .

Then,

θ21 +

m∑
i=2

θ2i ≤ κ ⇒
m∑
i=2

θ2i ≤ κ− θ21.

Substituting into (6), we get:

m∑
i=2

|θi| ≤
√
(m− 1)(κ− θ21).

Therefore, the fuzzy detour energy satisfies:

FdE(G) =
m∑
i=1

|θi| ≤ θ1 +
√

(m− 1)(κ− θ21),

as required.

Another, potentially more satisfying, upper and lower bound is established in the next
theorem.

Theorem 10. Let G be a FG with |V| = m and |E| = n. If cij = ∆(vi, vj) denotes the
fuzzy detour distance andMFdd

is the fuzzy detour distance matrix, then

√
2
∑
i<j

(cij)2 +m(m− 1)|MFdd
|
2
m ≤ FdE(G) ≤

√√√√√2

∑
i<j

(cij)2

m.

Proof. The proof follows similarly to that of Theorem 4.



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 13 of 25

5. Fuzzy Geodetic-Laplacian Energy

In classical graph theory, distance Laplacian spectra were introduced by Milan Nath
and Somnath Paul. In fuzzy graphs, the ordinary and Laplacian energies, studied in
[19] and [21], share several spectral similarities with their crisp counterparts. However,
to extend Laplacian-based energy to geodetic frameworks, we define the fuzzy geodetic-
Laplacian energy, derived from the spectrum of the fuzzy geodetic-distance matrix, offering
a novel spectral invariant for fuzzy graphs.

Definition 17. For a connected FG G = (V, σ, µ) with |V| = n, let MFgd
denote the

Fg-matrix. Then the fuzzy geodetic transmission matrix, TFgd
, is an n×n diagonal matrix,

where each diagonal entry tii is defined as

tii =
n∑

j=1

df (vi, vj).

Definition 18. LetMFgd
and TFgd

be the Fg-matrix and the fuzzy geodetic transmission
matrix, respectively, of an FG G = (V, σ, µ). Their difference is called the fuzzy geodetic-
Laplacian matrix, represented byMFgld

, and is defined as

MFgld
= TFgd

−MFgd
.

Definition 19. Let G = (V, σ, µ) be an FG with |V| = n, and let γgl1 ≥ γgl2 ≥ · · · ≥ γgln
be the fuzzy geodetic-Laplacian eigenvalues. Then, the geodetic-Laplacian energy is defined
as

GLE(G) =
n∑

i=1

|ψi|,

where

|ψi| =
∣∣∣∣γgli − 2

∑
1≤i<j≤n df (vi, vj)

n

∣∣∣∣ .
Example 3. Figure 3 illustrates the fuzzy geodetic-Laplacian matrix and its eigenvalues
for the FG G.

The associated fuzzy geodetic-Laplacian matrix is:
1.4 −0.6 −0.5 −0.3
−0.6 1.7 −0.5 −0.6
−0.5 −0.5 1.3 −0.3
−0.3 −0.6 −0.3 1.2

.

Here, γgl1 = 2.2842, γgl2 = 1.8384, γgl3 = 1.4774, and γgl4 = 0.0, then GLE(G) = 2.8.

The following theorems establish the upper and lower bounds for the geodetic-Laplacian
energy.



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 14 of 25

Figure 3: Illustration of the fuzzy geodetic–Laplacian matrix

Theorem 11. Let G = (V, σ, µ) be an FG with fuzzy geodetic-Laplacian matrix MFgld

whose diagonal entries are denoted as dMFgld
(ti). Then, the geodetic-Laplacian energy

satisfies:

GLE(G) ≤

√√√√√n

2
∑

1≤i<j≤n

d2ij +

n∑
i=1

(
dMFgld

(ti)−
2
∑
dij

n

)2
.

Proof. Applying the Cauchy–Schwarz inequality to the vector (|ψ1|, |ψ2|, . . . , |ψn|), we
have ∣∣∣∣∣

n∑
i=1

ψi

∣∣∣∣∣
2

≤ n
n∑

i=1

|ψi|2,

where

ψi = γgli −
2
∑

1≤i<j≤n df (vi, vj)

n
.

By Definition 19,

GLE(G) =
n∑

i=1

|ψi| ≤

√√√√n
n∑

i=1

|ψi|2 =
√
2Mn,

where

M =
∑

1≤i<j≤n

d2ij +
1

2

n∑
i=1

(
dMFgld

(ti)−
2
∑
dij

n

)2

.



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 15 of 25

Thus,

GLE(G) ≤

√√√√√n

2
∑

1≤i<j≤n

d2ij +

n∑
i=1

(
dMFgld

(ti)−
2
∑
dij

n

)2
.

Theorem 12. Let G = (V, σ, µ) be an FG with fuzzy geodetic-Laplacian matrix MFgld
,

and let its diagonal entries be represented by dMFgld
(ti). Then, the geodetic-Laplacian

energy satisfies the inequality:

GLE(G) ≥ 2

√(∑
1≤i<j≤n d

2
ij +

1
2

∑n
i=1

(
dMFgld

(ti)− 2
∑

dij
n

)2)
.

Proof. By Definition 19, we have

(GLE(G))2 = (
∑n

i=1 |ψi|)2 =
∑n

i=1 |ψi|2 + 2
∑

1≤i<j≤n |ψi||ψj | ≥ 4M .

Thus we get,

GLE(G) ≥ 2

√(∑
1≤i<j≤n d

2
ij +

1
2

∑n
i=1

(
dMFgld

(ti)− 2
∑

dij
n

)2)
.

Theorem 13. Let G = (V, σ, µ) be an FG. If G is a geodetic block, then LE(G) = GLE(G).
Proof. Since G is a geodetic block, every edge corresponds to a distinct fuzzy geodesic.

Consequently, the fuzzy geodetic-Laplacian matrix MFgld
is identical to the standard

Laplacian matrix. Hence, the spectra of both matrices are the same, and it follows that
LE(G) = GLE(G).

6. Fuzzy Detour-Laplacian Energy

In fuzzy graph theory, detour distance has been studied for its role in path analy-
sis, although detour-based spectral properties have been explored in earlier sections, the
extension of Laplacian energy concepts to detour-based frameworks remains largely un-
addressed. In this section, we define the fuzzy detour-Laplacian energy by employing the
spectrum of the fuzzy detour-distance Laplacian matrix. This formulation offers a new
spectral invariant and bridges the gap between classical Laplacian energy and detour-based
distance measures in fuzzy graphs.

Definition 20. Let G = (V, σ, µ) be a connected FG with |V| = n, and let MFdd
denote

the Fd-matrix. Then, the detour transmission matrix TFdd
is an n × n diagonal matrix

whose diagonal entries are given by

tii =
n∑

j=1

∆(vi, vj),

where ∆(vi, vj) denotes the fuzzy detour distance between vertices vi and vj.



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 16 of 25

Definition 21. Let G = (V, σ, µ) be an FG. IfMFdd
and TFdd

denote the Fd-matrix and
the detour transmission matrix of G, respectively, then their difference is defined as the
fuzzy detour-Laplacian matrixMFdld

, given by

MFdld
= TFdd

−MFdd
.

Definition 22. Let G = (V, σ, µ) be an FG with |V| = n, and let γdl1 ≥ γdl2 ≥ · · · ≥
γdln denote the fuzzy detour-Laplacian eigenvalues of G. Then the fuzzy detour-Laplacian
energy is defined as

DLE(G) =
n∑

i=1

|χi|,

where

χi = γdli −
2

n

∑
1≤i<j≤n

∆(vi, vj).

Example 4. To comprehend the fuzzy detour-Laplacian matrix and its eigenvalues, con-
sider the graph G shown in Figure 2. The associated fuzzy detour-Laplacian matrixMFdld

is given by: 
34.99 −11.66 −10.00 −13.33
−11.66 38.32 −12.00 −14.66
−10.00 −12.00 34.5 −12.50
−13.33 −14.16 −12.50 39.99

.

Here, γdl1 = 54.0166, γdl2 = 49.0913, γdl3 = 44.6921 and γdl4 = 0.0000, DLE(G) = 148.8.

The following theorems establish upper and lower bounds for the detour-Laplacian
energy.

Theorem 14. Let G : (V, σ, µ) be an FG with fuzzy detour-Laplacian matrix MFdld
, and

let its diagonal entries be denoted as dMFdld
(ti). Then,

DLE(G) ≤

√√√√√
2

∑
1≤i<j≤n

c2ij +

n∑
i=1

(
dMFdld

(ti)−
2
∑
cij

n

)2
n.

Proof. Applying the Cauchy–Schwarz inequality to the vector (|χ1|, |χ2|, . . . , |χn|)
yields the result. The argument proceeds analogously to Theorem 11 and is thus omitted
for brevity.

Theorem 15. Let G : (V, σ, µ) be an FG with fuzzy detour-Laplacian matrix MFdld
, and

let its diagonal entries be denoted as dMFdld
(ti). Then,

DLE(G) ≥ 2

√√√√√
 ∑

1≤i<j≤n

c2ij +
1

2

n∑
i=1

(
dMFdld

(ti)−
2
∑
cij

n

)2
,

where cij = ∆(vi, vj).



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 17 of 25

Proof. By Definition 22, we have

(DLE(G))2 =

(
n∑

i=1

|χi|

)2

=

n∑
i=1

|χi|2 + 2
∑

1≤i<j≤n

|χi||χj |.

Since both terms are non-negative, it follows that

(DLE(G))2 ≥ 4M,

where

M =
∑

1≤i<j≤n

c2ij +
1

2

n∑
i=1

(
dMFdld

(ti)−
2
∑
cij

n

)2

.

Taking square roots on both sides yields the desired inequality.

7. Applications of Fuzzy Geodetic-Laplacian Energy

This section explores the use of geodetic-Laplacian energy in decision-making processes
aimed at addressing modern-day slavery. Additionally, the study highlights an effective
mathematical model for its application in modern law enforcement, particularly in the
areas of efficient crime detection and combating illegal activities.

7.1. Analyzing the Illicit Routes of Human Trading Networks

Human trafficking often operates through complex networks of individuals and organi-
zations. Given the inherent ambiguity and uncertainty in these connections, Fuzzy Graph
(FG) theory provides a valuable framework for examining and visualizing these networks.
This strategy could be helpful to law enforcement and governments in identifying the struc-
ture and dynamics of trafficking routes. In this study, we focus on the major illicit human
trafficking routes into the United States, as outlined in studies [23]. Our primary aim is
to evaluate the fuzzy geodetic Laplacian energy as a comparative measure, in contrast to
the various fuzzy graph metrics discussed in previous researches [23–25].

The Global Slavery Index, as referenced in [28], offers a detailed assessment of the
susceptibility to modern slavery across 150+ countries. Over the past two decades, global
institutions like the United Nations Office on Drugs and Crime (UNODC) and the Inter-
national Labour Organization (ILO) have significantly advanced efforts to quantify the
scale of modern slavery worldwide. Various methods have been applied to fuzzy graphs in
order to assess the likelihood of illegal immigration along different routes from countries
of origin to the United States.

Notably, Binu et al. [23] explored the use of the Wiener index on fuzzy graphs to
examine pathways of illegal immigration, providing a novel approach to studying human
trafficking networks. Additionally, studies referenced in [24] have introduced the use of



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 18 of 25

the Intuitionistic Fuzzy Geodetic Wiener Index, a method that proves effective in mod-
eling illicit trafficking by capturing migration flows and network centralities within fuzzy
relational frameworks.

Building upon these earlier works, this paper adopts the fuzzy graph structure em-
ployed by Binu et al.[23] to test the applicability and accuracy of the fuzzy geodetic
Laplacian energy as a metric for analyzing human trafficking networks. By comparing
this approach with existing fuzzy graph measures, we aim to enhance our understanding
of the dynamics and complexities of trafficking networks, ultimately contributing to more
effective measures for combating human exploitation. Let df (Pu→v) represent the fuzzy
geodetic distance relation matrix for various path relations Pu→v, and let FDG(Pu→v)
denote the relation degree matrix.

Figure 4: Main routes of illicit trade in humans to the U.S.

In Figure 4, consider the different paths to the United States. First, we evaluate
df (Pu→v) for any one of the paths—specifically, say df (PChina→U. S.). Now we have

df (PChina→U. S.) =


0.0 0.19 0.38 0.38 0.57
0.19 0.0 0.3 0.3 0.6
0.38 0.3 0.0 0.32 0.64
0.38 0.3 0.32 0.0 0.47
0.57 0.6 0.64 0.47 0.0

.

Also, we have



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 19 of 25

TF gd(PChina→U. S.) =


1.52 0.0 0.0 0.0 0.0
0.0 1.39 0.0 0.0 0.0
0.0 0.0 1.64 0.0 0.0
0.0 0.0 0.0 1.47 0.0
0.0 0.0 0.0 0.0 2.28

.

Thus, the corresponding fuzzy geodetic-Laplacian matrix, MFgld
(PChina→U. S.) is as fol-

lows:

MFgld
(PChina→U. S.) =


1.52 −0.19 −0.38 −0.38 −0.57
−0.19 1.39 −0.3 −0.3 −0.6
−0.38 −0.3 1.64 −0.32 −0.64
−0.38 −0.3 −0.32 1.47 −0.47
−0.57 −0.6 −0.64 −0.47 2.28

.

The eigenvalues are γgl1 = 2.8692, γgl2 = 1.9900, γgl3 = 1.8317, γgl4 = 1.6090, and
γgl5 = 0.0000. Then,

GLE(PChina→U. S.) =
∣∣∣γgli − 2

∑
1≤i<j≤n df (PChina→U. S.)

n

∣∣∣.
GLE(PChina→U. S.) = |2.8692− 1.66|+ |1.9900− 1.66|

+ |1.8317− 1.66|+ |1.6090− 1.66|
+ |0.0000− 1.66|

= 3.4219.

Likewise, along the PIndia→U. S. route, we have:

df (PIndia→U. S.) =


0.0 0.26 0.52 0.78
0.26 0.0 0.32 0.64
0.52 0.32 0.0 0.47
0.78 0.64 0.47 0.0


and

TF gd(PIndia→U. S.) =


1.56 0.0 0.0 0.0
0.0 1.22 0.0 0.0
0.0 0.0 1.31 0.0
0.0 0.0 0.0 1.89

.

Thus

MFgld
(PIndia→U. S.) =


1.56 −0.26 −0.52 −0.78
−0.26 1.22 −0.32 −0.64
−0.52 −0.32 1.31 −0.47
−0.78 −0.64 −0.47 1.89

.



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 20 of 25

Then we obtain, GLE(PIndia→U. S.) = 2.9789. Using similar computations for the
PEthiopia→U. S., we obtainGLE(PEthiopia→U. S.) = 5.026. Along PSomalia→U. S. route, GLE(PSomalia→U. S.)
= 5.855. In the route PNigeria→U. S., GLE(PNigeria→U. S.) = 5.518.

Now we analyze the human trafficking from the continents of Africa and Asia to enter
the U.S. illegally over the Mexican border. Let’s first evaluateMFgld

(PAsia→U. S.). Thus

MFgld
(PAsia→U. S.) =


0.78 0.0 0.0 0.0 −0.78
0.0 0..57 0.0 0.0 −0.57
0.0 0.0 0.47 −0.17 −0.3
0.0 0.0 −0.17 0.41 −0.24
−0.78 −0.57 −0.3 −0.24 1.89

.

The eigenvalues are γgl1 = 1.545, γgl2 = 1.281, γgl3 = 0.612, γgl4 = 0.491, and γgl5 = 0.191.
Therefore, we have

GLE(PAsia→U. S.) = 2.356.

Similarly, along the PAfrica→U. S. route, we have

MFgld
(PAfrica→U. S.) =


1.05 0.00 0.00 −1.05
0.00 0.36 0.00 −0.36
0.00 0.00 0.55 −0.55
−1.05 −0.36 −0.55 1.96

 .

Then we obtain:
GLE(PAfrica→U. S.) = 3.608.

Based on the methodology employed in [23] to assess the likelihood of illegal immi-
gration from countries of origin to the United States, as well as the approach used in
this study, Somalia emerges as the most vulnerable country of origin for migration routes
leading to the United States. Similar findings regarding Somalia’s high vulnerability are
reported in [23–25]. Moreover, our results indicate that the continent of Africa exhibits a
higher level of vulnerability compared to Asia. To quantify such vulnerability, this study
introduces the concept of fuzzy geodetic-Laplacian energy as a potential metric. This
work demonstrates that fuzzy geodetic-Laplacian energy can effectively analyze complex
relational structures, providing a valuable tool for identifying patterns of migratory risk.
The results validate the applicability of this approach, suggesting that it offers enhanced
insights into regional vulnerability assessments and supports informed decision-making.

7.2. Role of Different Agencies in Decision Making for Combating Crimes

In contemporary law enforcement, effective crime detection and prevention require co-
ordinated collaboration across multiple agencies. In this study, we propose a mathemat-
ical model that employs fuzzy geodetic-Laplacian energy to facilitate informed decision-
making. We consider a hypothetical team formation comprising four police departments—Alpha
(A), Bravo (B), Charlie (C), and Delta (D)—each authorized to detect, investigate, and



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 21 of 25

take remedial action against reported crimes. These departments interact across three key
relational dimensions: law enforcement coordination (Ψ1), joint investigative involvement
(Ψ2), and engagement with external agencies (Ψ3). As illustrated in Figure 5, the rela-
tionships among Alpha, Bravo, Charlie, and Delta are clearly depicted. To analyze these
interactions, we first compute the geodetic-Laplacian energies for each relational aspect
Ψi. Let df (Ψi) denote the fuzzy geodetic distance relation matrix corresponding to the
relations Ψi, and FDG(Ψi) represent the relation degree matrix.

Figure 5: Relational aspects of different agencies.

Now we evaluate GLE for all Ψi; from Figure 5 we have

df (Ψ1) =


0.0 0.1 0.0 0.2
0.1 0.0 0.0 0.2
0.0 0.0 0.0 0.0
0.2 0.2 0.0 0.0

 .

TFgd
(Ψ1) =


0.3 0.0 0.0 0.0
0.0 0.3 0.0 0.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.4

 .

Thus, the corresponding fuzzy geodetic-Laplacian matrix for Ψ1, denotedMFgld
(Ψ1),

is



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 22 of 25

MFgld
(Ψ1) =


0.3 −0.1 0.0 −0.2
−0.1 0.1 0.0 −0.2
0.0 0.0 0.0 0.0
−0.2 −0.2 0.0 0.2

 .

The eigenvalues are γgl1(Ψ1) = 0.6, γgl2(Ψ1) = 0.4, γgl3(Ψ1) = 0, and γgl4(Ψ1) = 0,
yielding

GLE(Ψ1) = 1.

Now, according to the relationship Ψ2, we have

df (Ψ2) =


0.0 0.1 0.2 0.2
0.1 0.0 0.1 0.2
0.2 0.1 0.0 0.2
0.2 0.2 0.2 0.0

 .

and

TFgd
(Ψ2) =


0.5 0.0 0.0 0.0
0.0 0.4 0.0 0.0
0.0 0.0 0.5 0.0
0.0 0.0 0.0 0.6

 .

Then,

MFgld
(Ψ2) =


0.5 −0.1 −0.2 −0.2
−0.1 0.4 −0.1 −0.2
−0.2 −0.1 0.5 −0.2
−0.2 −0.2 −0.2 0.6

 .

After computing the eigenvalues,

GLE(Ψ2) = 1.

In a similar fashion, for the relationship Ψ3,

MFgld
(Ψ3) =


0.0 0.0 0.0 0.0
0.0 0.7 −0.3 −0.4
0.0 −0.3 0.9 −0.6
0.0 −0.4 −0.6 1.0

 .



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 23 of 25

Thus,
GLE(Ψ3) = 2.599.

Therefore,
GLE(G) = {1.000, 1.000, 2.599}.

Key Findings:

• The highest fuzzy geodetic-Laplacian energy is associated with maintaining rela-
tionships with external agencies (Ψ3). This indicates that Bravo (B), Charlie (C),
and Delta (D) are primarily responsible for the complex task of coordinating and
leveraging support from outside organizations. The significant energy allocation un-
derscores the critical role these external collaborations play in enhancing the overall
effectiveness of crime prevention and control.

• The energy related to mutual involvement in investigations (Ψ2) ranks second. This
reflects a strong cooperative dynamic among Alpha (A), Bravo (B), Charlie (C), and
Delta (D) in investigative activities, demonstrating effective teamwork in pursuing
criminal cases.

• The energy devoted to enforcing law and order (Ψ1) is comparable to that of mutual
investigations (Ψ2). This suggests that the foundational collaboration between de-
partments is well established, facilitating smoother and more efficient investigative
processes.

Overall, to further enhance crime prevention and resolution outcomes, these depart-
ments should prioritize strengthening their partnerships with external agencies.

8. Conclusion

This study introduced and examined the fuzzy geodetic spectrum and fuzzy detour
spectrum as novel extensions of classical spectral graph theory to the fuzzy domain. We
defined the fuzzy geodetic-Laplacian energy and detour-Laplacian energy, derived their re-
spective expressions, and established upper and lower bounds. Notably, it was shown that
for geodetic blocks, the geodetic-Laplacian energy coincides with the standard Laplacian
energy, highlighting a structural consistency within fuzzy graph frameworks.

The practical applicability of fuzzy geodetic-Laplacian energy was illustrated through
a case analysis of trafficking channels, demonstrating its potential in decision-making
contexts such as crime detection and prevention. The findings are consistent with results
reported in the existing literature, supporting the validity and relevance of the proposed
spectral measures.

Future research may focus on extending these spectral invariants to directed or weighted
fuzzy graphs, enhancing robustness under data uncertainty, and applying the proposed
energies to infrastructure domains such as water distribution systems and wireless com-
munication networks. Algorithmic development for efficient spectral computation in large
fuzzy systems also represents a promising direction.



R. Rajeshkumar, A. M. Anto, V. Mary Mettilda Rose / Eur. J. Pure Appl. Math, 18 (4) (2025), 6307 24 of 25

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