Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 31 https://internationalpubls.com Elegant Fuzzy Labeling of Graphs Nithya Sree A1,*, Mahavir B2 and Paulraj M S3 1,2,3PG and Research Department of Mathematics, Agurchand Manmull Jain College, University of Madras, Chennai, India. E-mail: *ananthanarayanan.nithya@gmail.com (corresponding author); mahavirb@gmail.com; mspaulraj65@gmail.com Article History: Received: 05-08-2024 Revised: 25-09-2024 Accepted: 07-10-2024 Abstract: Fuzzy graphs and fuzzy labelings are crucial in modeling real-time systems by accommodating varying degrees of information precision. In this paper, we introduce a novel fuzzy labeling called Elegant Fuzzy Labeling. We prove that a simple graph admits Elegant Fuzzy Labeling if and only if it admits elegant labeling. Furthermore, we prove that while a simple graph that admits Elegant Fuzzy Labeling also admits fuzzy labeling, but not conversely. We identify specific classes of simple graphs that admit Elegant Fuzzy Labeling and explore practical application of this new labeling approach. Keywords: Elegant labeling, Fuzzy graph, Fuzzy labeling, Elegant Fuzzy Labeling. Mathematics Subject Classification: 05C78, 05C72. 1. Introduction Graph labeling refers to assigning labels to vertices, edges, or both, of a graph while adhering to specific conditions. Graph labeling trace their origin back to the one introduced by A. Rosa [20] in 1967 called β-labeling. S. W. Golomb [9] renamed β-labeling as graceful labeling. Gallian [6] in his survey paper has given an extensive account of various types of graph labelings. The graph labeling has wide range of applications such as x-ray crystallography, coding theory, radar astronomy, network design, circuit design, etc. In real life, we encounter many problems that are imprecise and ambiguous. To deal with such problems Zadeh [30] introduced the concept, fuzzy sets. Since the graphs are used to model many real world problems, fuzzy graph models are required to represent the vagueness in the objects and the vagueness in the relationship between them. The first definition of Fuzzy graphs was given by Kaufman [11] that was based on Zadeh’s fuzzy relations. Later, Rosenfield [21] introduced the basic graph theoretic concepts such as bridges, paths, cycles, trees and connectedness in the fuzzy setting and established some of their properties. Fuzzy graphs are highly effective in modeling real- time systems, where varying levels of imprecision are inherently present. Gani et al. [7,15] introduced the concept of fuzzy labeling graphs and studied their properties. Fuzzy labeling is more appropriate than the classical labeling for many real-world problems. Several studies have explored different aspects of fuzzy graphs, highlighting their versatility and utility. For instance, Fathalian et al. [5] studied fuzzy magic labeling on simple graphs. Giri et al. [8] analyzed fermatean fuzzy graphs. Kosari et al. [13] studied perfectly regular fuzzy graphs. Selvarasu and Murugan [23] studied fuzzy anti- magic labeling in graphs. Shanmugapriya and Hemalatha [24] examined fuzzy vertex magic labeling. Borzooei and Rashmanlou [1] investigated cayley interval-valued fuzzy graphs. Shoaib et al. [29] studied pythagorean fuzzy graphs. Further extending the field, Shi et al. [28] expanded the concept of energy on the picture fuzzy graphs. Rao et al. [22] introduced intuitionistic fuzzy trees. Moreover, studies on vague graphs, an extension of fuzzy graphs, have been conducted by Kosari et al. [12], Rashmanlou et al. [19], and Shao et al. [25,26]. mailto:ananthanarayanan.nithya@gmail.com mailto:mahavirb@gmail.com mailto:mspaulraj65@gmail.com Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 32 https://internationalpubls.com In 1980, harmonious graphs were introduced by Graham and Sloane [10]. They defined a graph 𝐺 with 𝑞 edges to be harmonious if there is an injection 𝑓 from the vertices of 𝐺 to the group of integers modulo 𝑞 such that when each edge 𝑢𝑣 is assigned the label (𝑓(𝑢) + 𝑓(𝑣)) 𝑚𝑜𝑑 𝑞, the resulting edge labels are distinct. Elegant labeling, a variation of harmonious labeling was defined by Chang et al. [3] in 1981. They defined, elegant labeling 𝑓 of a graph 𝐺 with 𝑞 edges as an injective function from the vertices of 𝐺 to the set {0,1,… , 𝑞} such that when each edge 𝑢𝑣 is assigned the label (𝑓(𝑢) + 𝑓(𝑣)) 𝑚𝑜𝑑 (𝑞 + 1), the resulting edge labels are distinct and nonzero. Cahit [2], Elumalai and Sethuraman [4], Maulidia and Purwanto [14], Parwati and Purwanto [16], Prihandini et al. [17,18] and, Sherly and Purwanto [27] have produced several results on elegant graphs. Motivated by the need to address uncertainties and impreciseness in problems more effectively, in this paper, we introduce a new type of fuzzy labeling called the Elegant Fuzzy Labeling. We prove that if a simple graph admits Elegant Fuzzy Labeling, then it admits elegant labeling and fuzzy labeling, and also prove that if a simple graph admits elegant labeling, then it admits Elegant Fuzzy Labeling. We prove that the path graphs 𝑃𝑛, 𝑛 ≠ 4 and cycles 𝐶𝑛, 𝑛 ≡ 0,3 𝑚𝑜𝑑 4, admit Elegant Fuzzy Labeling. The line graph 𝐿(𝐺) of a simple graph 𝐺 is the graph with edges of 𝐺 as its vertices, where two vertices are adjacent in 𝐿(𝐺) if and only if the corresponding edges are incident in 𝐺. We prove that 𝐿(𝑃𝑛), where 𝑛 ≠ 5 and 𝐿(𝐶𝑛), where 𝑛 ≡ 0,3 𝑚𝑜𝑑 4, admit Elegant Fuzzy Labeling. We also provide an application of Elegant Fuzzy Labeling. If 𝐺 is a graph, we denote the vertex set of 𝐺 by 𝑉(𝐺) and the edge set of 𝐺 by 𝐸(𝐺). 2. Preliminaries Definition 2.1. [3] Elegant labeling 𝑓 of a graph 𝐺 with 𝑞 edges is an injective function from 𝑉(𝐺) into the set {0, 1,… , 𝑞} such that the function 𝑔 from 𝐸(𝐺) into the set {1, 2,… , 𝑞} defined as 𝑔(𝑢𝑣) = (𝑓(𝑢) + 𝑓(𝑣)) 𝑚𝑜𝑑 (𝑞 + 1) for every 𝑢𝑣 in 𝐸(𝐺) is injective. Example 2.2. Fig. 1 illustrates an elegant graph 𝐺 with 𝑉(𝐺) = {𝑣1, 𝑣2, 𝑣3} and 𝐸(𝐺) = {𝑣1𝑣2, 𝑣2𝑣3, 𝑣1𝑣3}. An elegant labeling 𝑓: 𝑉(𝐺) → {0, 1, 2, 3} of 𝐺 is defined as 𝑓(𝑣1) = 0, 𝑓(𝑣2) = 1, 𝑓(𝑣3) = 2. Hence the edge labels are 𝑔(𝑣1𝑣2) = 1, 𝑔(𝑣2𝑣3) = 3, 𝑔(𝑣1𝑣3) = 2. Definition 2.3. [21] A graph 𝐺 = (𝜇, 𝜌) with vertex set 𝑉(𝐺), edge set 𝐸(𝐺) and a pair of functions 𝜇: 𝑉(𝐺) → [0,1] and 𝜌: 𝐸(𝐺) → [0,1] is called fuzzy graph if for every 𝑢𝑣 in 𝐸(𝐺), 𝜌(𝑢𝑣) ≤ 𝜇(𝑢) ∧ 𝜇(𝑣) (= 𝑚𝑖𝑛{𝜇(𝑢), 𝜇(𝑣)}). Example 2.4. Fig. 2 illustrates a fuzzy graph G with 𝑉(𝐺) = {𝑣1, 𝑣2, 𝑣3, 𝑣4} and 𝐸(𝐺) = {𝑣1𝑣2, 𝑣2𝑣3, 𝑣3𝑣4, 𝑣1𝑣4}. A function 𝜇: 𝑉(𝐺) → [0,1] is defined as 𝜇(𝑣1) = 0.5, 𝜇(𝑣2) = 0.3, 𝜇(𝑣3) = 0.4, 𝜇(𝑣4) = 0.6, and 𝜌: 𝐸(𝐺) → [0,1] is defined as 𝜌(𝑣1𝑣2) = 0.2, 𝜌(𝑣2𝑣3) = 0.3, 𝜌(𝑣3𝑣4) = 0.4, 𝜌(𝑣1𝑣4) = 0.1. Definition 2.5. [7] A fuzzy graph 𝐺 = (𝜇, 𝜌) is said to be a fuzzy labeling graph, if 𝜇: 𝑉(𝐺) → [0,1] and 𝜌: 𝐸(𝐺) → [0,1] are injective and for every 𝑢𝑣 in 𝐸(𝐺), 𝜌(𝑢𝑣) < 𝜇(𝑢) ∧ 𝜇(𝑣). Example 2.6. Fig. 3 illustrates a fuzzy labeling graph 𝐺 with 𝑉(𝐺) = {𝑣1, 𝑣2, 𝑣3, 𝑣4} and 𝐸(𝐺) = {𝑣1𝑣2, 𝑣2𝑣3, 𝑣3𝑣4, 𝑣2𝑣4}. A fuzzy labeling 𝜇: 𝑉(𝐺) → [0,1] and 𝜌: 𝐸(𝐺) → [0,1] are defined as follows: 𝜇(𝑣1) = 0.5, 𝜇(𝑣2) = 0.6, 𝜇(𝑣3) = 0.7, 𝜇(𝑣4) = 0.8, 𝜌(𝑣1𝑣2) = 0.1, 𝜌(𝑣2𝑣3) = 0.2, 𝜌(𝑣3𝑣4) = 0.3, 𝜌(𝑣2𝑣4) = 0.4. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 33 https://internationalpubls.com Definition 2.7. Let 𝐺 be a (𝑝, 𝑞) graph. Let 𝑓 be an injective function from 𝑉(𝐺) into the set {ℎ′, ℎ′ + 1, . . . , ℎ′ + 𝑞 − 1, ℎ′ + 𝑞} where ℎ′ is a suitable constant, and 𝑔 be an injective function from 𝐸(𝐺) into the set {0,1,… , 𝑞} defined as, 𝑔(𝑢𝑣) = (𝑓(𝑢) + 𝑓(𝑣)) 𝑚𝑜𝑑 (𝑞 + 1) for every 𝑢𝑣 in 𝐸(𝐺). Define vertex labelling 𝜇: 𝑉(𝐺) → [0,1] as 𝜇(𝑣) = 𝑓(𝑣). ℎ and the edge labeling 𝜌: 𝐸(𝐺) → [0,1] as 𝜌(𝑢𝑣) = 𝑔(𝑢𝑣). ℎ, where ℎ ≤ 1 ℎ′+𝑞 . If the edge labels are distinct and nonzero, and if the condition 𝜌(𝑢𝑣) < 𝜇(𝑢) ∧ 𝜇(𝑣) is satisfied for all 𝑢𝑣 ∈ 𝐸(𝐺), then 𝐺 is said to be an Elegant Fuzzy Labeling Graph and (𝜇, 𝜌) is an Elegant Fuzzy Labeling of 𝐺. In the Definition 2.7, if ℎ′ = 0, then 𝑓 is an elegant labeling. Suppose 𝑓(𝑣) = 0 for some 𝑣 ∈ 𝑉(𝐺) and 𝑢𝑣 ∈ 𝐸(𝐺) then, 𝑔(𝑢𝑣) = (𝑓(𝑢) + 𝑓(𝑣)) 𝑚𝑜𝑑 (𝑞 + 1), where 𝑓(𝑣) = 0 and 𝑓(𝑢) ≠ 0, simplifying to 𝑔(𝑢𝑣) = 𝑓(𝑢) 𝑚𝑜𝑑 (𝑞 + 1). According to Definition 2.7, 0 < 𝑔(𝑢𝑣) < 𝑞 + 1, hence we have 𝑔(𝑢𝑣) > 𝑓(𝑢) ∧ 𝑓(𝑣). Similarly, suppose 𝑓(𝑣′) = 1 for some 𝑣′ ∈ 𝑉(𝐺) and 𝑢′𝑣′ ∈ 𝐸(𝐺), we get 𝑔(𝑢′𝑣′) ≥ 𝑓(𝑢′) ∧ 𝑓(𝑣′). Since 𝑔: 𝐸(𝐺) → {1, 2, … , 𝑞} is injective, by the Definition 2.7, for some 𝑢′′𝑣′′ ∈ 𝐺, 𝑔(𝑢′′𝑣′′) = 𝑞 which implies, 𝑔(𝑢′′𝑣′′) > 𝑓(𝑢′′) ∧ 𝑓(𝑣′′). However, we must have 𝑔(𝑢𝑣) < 𝑓(𝑢) ∧ 𝑓(𝑣) for all 𝑢𝑣 ∈ 𝐸(𝐺) to attain 𝜌(𝑢𝑣) < μ(𝑢) ∧ μ(𝑣) for all 𝑢𝑣 ∈ 𝐸(𝐺). Thus, we have translated the set {0,1,… , 𝑞} to {ℎ′, ℎ′ + 1,… , ℎ′ + 𝑞}, where ℎ′ is the suitable translation constant. To define labeling functions of 𝑉(𝐺) and 𝐸(𝐺) into [0,1], we have performed contraction with scaling factor ℎ. 3. Elegant Fuzzy Labeling In the sequel we take ℎ′ = 𝑞 + 1 and ℎ = 1 2𝑞+1 . So we consider the following definition of Elegant Fuzzy Labeling of graphs, and obtain the results and prove that certain classes of graphs admit Elegant Fuzzy Labeling. Definition 3.1. Let 𝐺 be a (𝑝, 𝑞) graph. Let 𝑓 be an injective function from 𝑉(𝐺) into the set {𝑞 + 1, 𝑞 + 2, … , 2𝑞 + 1} and 𝑔 be an injective function from 𝐸(𝐺) into the set {0,1,… , 𝑞} defined as, 𝑔(𝑢𝑣) = (𝑓(𝑢) + 𝑓(𝑣)) 𝑚𝑜𝑑 (𝑞 + 1) for every 𝑢𝑣 in 𝐸(𝐺). Define vertex labeling 𝜇: 𝑉(𝐺) → [0,1] as 𝜇(𝑣) = 𝑓(𝑣). ℎ and the edge labeling 𝜌: 𝐸(𝐺) → [0,1] as 𝜌(𝑢𝑣) = 𝑔(𝑢𝑣). ℎ, where ℎ = 1 2𝑞+1 . If the edge labels are distinct and nonzero, and if the condition 𝜌(𝑢𝑣) < 𝜇(𝑢) ∧ 𝜇(𝑣) is satisfied for all 𝑢𝑣 ∈ 𝐸(𝐺), then 𝐺 is said to be an Elegant Fuzzy Labeling Graph and (𝜇, 𝜌) is an Elegant Fuzzy Labeling of 𝐺. Example 3.2. Fig. 4 illustrates an Elegant Fuzzy Labeling Graph 𝐺, where 𝑉(𝐺) = {𝑣1, 𝑣2, 𝑣3, 𝑣4} and 𝐸(𝐺) = {𝑣1𝑣2,  𝑣1𝑣3,  𝑣1𝑣4,  𝑣2𝑣3,  𝑣2𝑣4,  𝑣3𝑣4}. The function 𝑓 is defined as follows: 𝑓(𝑣1) = 7, 𝑓(𝑣2) = 8, 𝑓(𝑣3) = 9, 𝑓(𝑣4) = 11. Consequently, 𝑔(𝑣1𝑣2) = 1, 𝑔(𝑣1𝑣3) = 2, 𝑔(𝑣1𝑣4) = 4, 𝑔(𝑣2𝑣3) = 3, 𝑔(𝑣2𝑣4) = 5, 𝑔(𝑣3𝑣4) = 6. Therefore, an Elegant Fuzzy Labeling (𝜇, 𝜌) is given as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 34 https://internationalpubls.com follows: 𝜇(𝑣1) = 0.54, 𝜇(𝑣2) = 0.62, 𝜇(𝑣3) = 0.69, 𝜇(𝑣4) = 0.85, 𝜌(𝑣1𝑣2) = 0.077, 𝜌(𝑣1𝑣3) = 0.15, 𝜌(𝑣1𝑣4) = 0.31, 𝜌(𝑣2𝑣3) = 0.23, 𝜌(𝑣2𝑣4) = 0.38, 𝜌(𝑣3𝑣4) = 0.46. 3.1. General results on Elegant Fuzzy Labeling of simple graphs Proposition 3.1.1. A simple graph 𝐺 admits Elegant Fuzzy Labeling if and only if it admits elegant labeling. Proof. Let 𝐺 be a simple graph with 𝑝 vertices and 𝑞 edges. Let 𝐺 admit Elegant Fuzzy Labeling. By Definition 3.1, 𝐺 admits elegant labeling. Conversely, let 𝐺 admit elegant labeling. By Definition 2.1, there exist an injective function 𝑓: 𝑉(𝐺) → {0,1, … , 𝑞} and an injective function 𝑔: 𝐸(𝐺) → {1,… , 𝑞} defined as 𝑔(𝑢𝑣) = (𝑓(𝑢) + 𝑓(𝑣)) 𝑚𝑜𝑑 (𝑞 + 1). Now translate the function 𝑓 by (𝑞 + 1), So 𝑓(𝑢) increases by 𝑞 + 1 and 𝑔(𝑢𝑣) remains unchanged. Now define vertex labeling 𝜇: 𝑉(𝐺) → [0,1] as 𝜇(𝑢) = 𝑓(𝑢). ℎ and edge labeling 𝜌: 𝐸(𝐺) → [0,1] as 𝜌(𝑢𝑣) = 𝑔(𝑢𝑣). ℎ, where ℎ = 1 2𝑞+1 . The functions f and g are injective. Hence, the vertex labels and edge labels are distinct and nonzero. Since 𝑚𝑖𝑛(𝑓(𝑢)) = 𝑞 + 1 and 𝑚𝑎𝑥(𝑔(𝑢𝑣)) = 𝑞, edge labels are less than the minimum of their respective endpoints labels. Therefore (𝜇, 𝜌) is an Elegant Fuzzy Labeling of 𝐺. □ Proposition 3.1.2. If a simple graph 𝐺 admits Elegant Fuzzy Labeling then it admits fuzzy labeling, but not conversely. Proof. Let 𝐺 be a simple graph with 𝑝 vertices and 𝑞 edges. Let 𝐺 admit Elegant Fuzzy Labeling. By Definition 3.1, 𝐺 admits fuzzy labeling. Conversely, if 𝐺 admits fuzzy labeling then it is not necessary that 𝐺 admits Elegant Fuzzy Labeling. For instance consider the graph 𝑃4 with vertex set 𝑉(𝑃4) = {𝑣1, 𝑣2, 𝑣3, 𝑣4} and edge set 𝐸(𝑃4) = {𝑣𝑖𝑣𝑖+1, 1 ≤ 𝑖 ≤ 3}. Define vertex labeling 𝜇: 𝑉(𝑃4) → [0,1] as 𝜇(𝑣𝑖) = 𝑖+3 10 and edge labeling 𝜌: 𝐸(𝑃4) → [0,1] as 𝜌(𝑣𝑖𝑣𝑖+1) = 𝑖 10 . By Definition 2.5, 𝑃4 admits fuzzy labeling. But 𝑃4 is not elegant [2]. □ 3.2. Elegant Fuzzy Labeling of certain classes of simple graphs Theorem 3.2.1. Path graphs 𝑃𝑛 admit Elegant Fuzzy Labeling except for 𝑛 = 4. Proof. Let 𝑃𝑛 be a path with vertex set 𝑉(𝑃𝑛) = {𝑣1, 𝑣2, … , 𝑣𝑛} and edge set 𝐸(𝑃𝑛) = {𝑣1𝑣2, 𝑣2𝑣3, … , 𝑣𝑛−1𝑣𝑛}. Let 𝑓 be an injective function from 𝑉(𝑃𝑛) into the set { 𝑛 , 𝑛 + 1 ,… , 2𝑛 − 1} and 𝑔 be a function from 𝐸(𝑃𝑛) into the set {0,1,… , 𝑛 − 1 } defined as, 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣𝑖) + 𝑓(𝑣𝑖+1)) 𝑚𝑜𝑑 𝑛 for 1 ≤ 𝑖 ≤ 𝑛 − 1. Case 1: For 𝑃2𝑚+1, 𝑚 = 1, 2, 3,… , 𝑓: 𝑉(𝑃2𝑚+1) → {2𝑚 + 1, 2𝑚 + 2, 2𝑚 + 3,… , 4𝑚 + 1} is defined as, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 35 https://internationalpubls.com 𝑓(𝑣𝑖) = { 3𝑚 + 1 + 𝑖 1 ≤ 𝑖 ≤ 𝑚 2𝑚 + 1 𝑖 = 𝑚 + 1 𝑓(𝑣𝑖−1) + 1 𝑚 + 2 ≤ 𝑖 ≤ 2𝑚 + 1. For 1 ≤ 𝑖 ≤ 𝑚 − 1, 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣𝑖) + 𝑓(𝑣𝑖+1)) 𝑚𝑜𝑑 (2𝑚 + 1) = (3𝑚 + 1 + 𝑖 + 3𝑚 + 1+ 𝑖 + 1) 𝑚𝑜𝑑 (2𝑚 + 1) = (3(2𝑚 + 1) + 2𝑖) 𝑚𝑜𝑑 (2𝑚 + 1) = 2𝑖. For 𝑖 = 𝑚, 𝑔(𝑣𝑚𝑣𝑚+1) = (𝑓(𝑣𝑚) + 𝑓(𝑣𝑚+1)) 𝑚𝑜𝑑 (2𝑚 + 1) = (3𝑚 + 1 +𝑚 + 2𝑚+ 1) 𝑚𝑜𝑑 (2𝑚 + 1) = 2𝑚. For 𝑚 + 1 ≤ 𝑖 ≤ 2𝑚, 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣𝑖) + 𝑓(𝑣𝑖+1)) 𝑚𝑜𝑑 (2𝑚 + 1) = (2𝑓(𝑣𝑖) + 1) 𝑚𝑜𝑑 (2𝑚 + 1) = (2𝑓(𝑣𝑖−1) + 3) 𝑚𝑜𝑑 (2𝑚 + 1) ⋮ = (2𝑓(𝑣𝑖−𝑗) + 2𝑗 + 1) 𝑚𝑜𝑑 (2𝑚 + 1), 𝑗 = 0, 1, 2,… , 𝑖 − 𝑚 − 1, and when 𝑗 = 𝑖 −𝑚 − 1, 𝑔(𝑣𝑖𝑣𝑖+1) = (2𝑓(𝑣𝑚+1) + 2(𝑖 − 𝑚 − 1) + 1) 𝑚𝑜𝑑 (2𝑚 + 1) = (2(2𝑚 + 1) + 2(𝑖 − 𝑚 − 1) + 1)𝑚𝑜𝑑(2𝑚 + 1) = 2(𝑖 − 𝑚 − 1) + 1. Hence, 𝑔(𝑣𝑖𝑣𝑖+1) = { 2𝑖 1 ≤ 𝑖 ≤ 𝑚 2(𝑖 − 𝑚 − 1) + 1 𝑚 + 1 ≤ 𝑖 ≤ 2𝑚. Case 2: For 𝑃2𝑚, 𝑚 = 3, 5, 7,… , 𝑓: 𝑉(𝑃2𝑚) → {2𝑚, 2𝑚 + 1, 2𝑚 + 2,… , 4𝑚 − 1} is defined as, 𝑓(𝑣𝑖) = { 2𝑚 + 1 𝑖 = 1 𝑓(𝑣𝑖−1) + 2 2 ≤ 𝑖 ≤ ⌊ 𝑚 2 ⌋ 3𝑚 𝑖 = ⌊ 𝑚 2 ⌋ + 1 𝑓(𝑣𝑖−1) + 1 ⌊ 𝑚 2 ⌋ + 2 ≤ 𝑖 ≤ 𝑚 + ⌊ 𝑚 2 ⌋ 2𝑚 𝑖 = 𝑚 + ⌊ 𝑚 2 ⌋ + 1 𝑓(𝑣𝑖−1) + 2 𝑚 + ⌊ 𝑚 2 ⌋ + 2 ≤ 𝑖 ≤ 2𝑚. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 36 https://internationalpubls.com Following a similar approach as in Case 1, we obtain, 𝑔(𝑣𝑖𝑣𝑖+1) = { 4𝑖 1 ≤ 𝑖 ≤ ⌊ 𝑚 2 ⌋ 2 (𝑖 − ⌊ 𝑚 2 ⌋ − 1) + 1 ⌊ 𝑚 2 ⌋ + 1 ≤ 𝑖 ≤ 𝑚 + ⌊ 𝑚 2 ⌋ − 1 2𝑚 − 1 𝑖 = 𝑚 + ⌊ 𝑚 2 ⌋ 4 (𝑖 − 𝑚 − ⌊ 𝑚 2 ⌋ − 1) + 2 𝑚 + ⌊ 𝑚 2 ⌋ + 1 ≤ 𝑖 ≤ 2𝑚 − 1. Case 3: For 𝑃4𝑚, 𝑚 = 3, 5, 7,… , 𝑓: 𝑉(𝑃4𝑚) → {4𝑚, 4𝑚 + 1, 4𝑚 + 2,… , 8𝑚 − 1} is defined as, 𝑓(𝑣𝑖) = { 4𝑚 𝑖 = 1 𝑓(𝑣𝑖−1) + 2 2 ≤ 𝑖 ≤ 𝑚 4𝑚 + 1 𝑖 = 𝑚 + 1 𝑓(𝑣𝑖−1) + 2 𝑚 + 2 ≤ 𝑖 ≤ 𝑚 + ⌊ 𝑚 2 ⌋ 7𝑚 𝑖 = 𝑚 + ⌊ 𝑚 2 ⌋ + 1 𝑓(𝑣𝑖−1) + 1 𝑚 + ⌊ 𝑚 2 ⌋ + 2 ≤ 𝑖 ≤ 2𝑚 + ⌊ 𝑚 2 ⌋ 6𝑚 − 1 𝑖 = 2𝑚 + ⌊ 𝑚 2 ⌋ + 1 𝑓(𝑣𝑖−1) − 2 2𝑚 + ⌊ 𝑚 2 ⌋ + 2 ≤ 𝑖 ≤ 3𝑚 7𝑚 − 1 𝑖 = 3𝑚 + 1 𝑓(𝑣𝑖−1) − 1 3𝑚 + 2 ≤ 𝑖 ≤ 4𝑚. Following a similar approach as in Case 1, we obtain, 𝑔(𝑣𝑖𝑣𝑖+1) = { 4𝑖 − 2 1 ≤ 𝑖 ≤ 𝑚 − 1 2𝑚 − 1 𝑖 = 𝑚 4𝑖 − 4𝑚 𝑚 + 1 ≤ 𝑖 ≤ 𝑚 + ⌊ 𝑚 2 ⌋ − 1 4𝑚 − 2 𝑖 = 𝑚 + ⌊ 𝑚 2 ⌋ 2𝑖 − 𝑚 𝑚 + ⌊ 𝑚 2 ⌋ + 1 ≤ 𝑖 ≤ 2𝑚 + ⌊ 𝑚 2 ⌋ − 1 2𝑚 − 2 𝑖 = 2𝑚 + ⌊ 𝑚 2 ⌋ 14𝑚 − 4𝑖 − 2 2𝑚 + ⌊ 𝑚 2 ⌋ + 1 ≤ 𝑖 ≤ 3𝑚 − 1 4𝑚 − 1 𝑖 = 3𝑚 8𝑚 − 2𝑖 − 1 3𝑚 + 1 ≤ 𝑖 ≤ 4𝑚 − 1. Case 4: For 𝑃8𝑚, 𝑚 = 1, 2, 3,… , 𝑓: 𝑉(𝑃8𝑚) → {8𝑚, 8𝑚 + 1, 8𝑚 + 2,… , 16𝑚 − 1} is defined as, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 37 https://internationalpubls.com 𝑓(𝑣𝑖) = { 14𝑚 − 1 𝑖 = 1 10𝑚 − 2 𝑖 = 2 𝑓(𝑣𝑖−2) − 2 3 ≤ 𝑖 ≤ 2𝑚 12𝑚 − 1 𝑖 = 2𝑚 + 1 𝑓(𝑣𝑖−1) − 2 2𝑚 + 2 ≤ 𝑖 ≤ 4𝑚 16𝑚 − 2 𝑖 = 4𝑚 + 1 𝑓(𝑣𝑖−1) − 2 4𝑚 + 2 ≤ 𝑖 ≤ 6𝑚 + 1 16𝑚 − 1 𝑖 = 6𝑚 + 2 𝑓(𝑣𝑖−2) − 2 6𝑚 + 3 ≤ 𝑖 ≤ 8𝑚. Following a similar approach as in Case 1, we obtain, 𝑔(𝑣𝑖𝑣𝑖+1) = { 8𝑚 − 3 𝑖 = 1 8𝑚 − 2𝑖 − 1 2 ≤ 𝑖 ≤ 2𝑚 − 1 4𝑚 − 1 𝑖 = 2𝑚 16𝑚 − 4𝑖 2𝑚 + 1 ≤ 𝑖 ≤ 4𝑚 − 1 8𝑚 − 1 𝑖 = 4𝑚 24𝑚 − (4𝑖 + 2) 4𝑚 + 1 ≤ 𝑖 ≤ 6𝑚 − 1 32𝑚 − (4𝑖 + 2) 𝑖 = 6𝑚 4𝑚 − 3 𝑖 = 6𝑚 + 1 16𝑚 − 2𝑖 − 1 6𝑚 + 2 ≤ 𝑖 ≤ 8𝑚 − 1. Now we have ℎ = 1 2𝑛−1 . The vertex labeling 𝜇: 𝑉(𝑃𝑛) → [0,1] is defined as, 𝜇(𝑣𝑖) = 𝑓(𝑣𝑖). 1 2𝑛−1 for every 1 ≤ 𝑖 ≤ 𝑛 and edge labeling 𝜌: 𝐸(𝑃𝑛) → [0,1] is defined as, 𝜌(𝑣𝑖𝑣𝑖+1) = 𝑔(𝑣𝑖𝑣𝑖+1). 1 2𝑛−1 for every 1 ≤ 𝑖 ≤ 𝑛 − 1. It can be verified that in all the cases 𝑔(𝑣𝑖𝑣𝑖+1) are distinct and nonzero, therefore 𝜌(𝑣𝑖𝑣𝑖+1) are distinct and nonzero. Further max{𝑔(𝑣𝑖𝑣𝑖+1)} = 𝑛 − 1 and min{ 𝑓(𝑣𝑖)} = 𝑛. Therefore, 𝑔(𝑣𝑖𝑣𝑖+1) < 𝑓(𝑣𝑖) ∧ 𝑓(𝑣𝑖+1) for every 1 ≤ 𝑖 ≤ 𝑛 − 1. Hence for every 1 ≤ 𝑖 ≤ 𝑛 − 1, 𝜌(𝑣𝑖𝑣𝑖+1) = 𝑔(𝑣𝑖𝑣𝑖+1). 1 2𝑛−1 < (𝑓(𝑣𝑖) ∧ 𝑓(𝑣𝑖+1)). 1 2𝑛−1 < 𝑓(𝑣𝑖). 1 2𝑛−1 ∧ 𝑓(𝑣𝑖+1). 1 2𝑛−1 < 𝜇(𝑣𝑖) ∧ 𝜇(𝑣𝑖+1). This implies that, the edge labels are less than the minimum of their respective endpoints labels. Consequently, 𝑃𝑛, 𝑛 ≠ 4 are Elegant Fuzzy Labeling Graphs and (𝜇, 𝜌) is an Elegant Fuzzy Labeling of 𝑃𝑛, 𝑛 ≠ 4. □ Theorem 3.2.2. Line graph of path graphs 𝐿(𝑃𝑛) admit Elegant Fuzzy Labeling except for 𝑛 = 5. Proof. By Theorem 3.2.1, 𝑃𝑛 admit Elegant Fuzzy Labeling except for 𝑛 = 4. 𝐿(𝑃𝑛) ≡ 𝑃𝑛−1. Therefore 𝐿(𝑃𝑛) admit Elegant Fuzzy Labeling except for 𝑛 = 5. □ Example 3.2.3. Consider a path graph 𝑃5 in Fig. 5. By Theorem 3.2.1, a function 𝑓: 𝑉(𝑃5) → {5, 6, 7, 8, 9} is defined as follows: 𝑓(𝑣1) = 8, 𝑓(𝑣2) = 9, 𝑓(𝑣3) = 5, 𝑓(𝑣4) = 6, 𝑓(𝑣5) = 7. Consequently, 𝑔(𝑣1𝑣2) = 2, 𝑔(𝑣2𝑣3) = 4, 𝑔(𝑣3𝑣4) = 1, 𝑔(𝑣4𝑣5) = 3. Hence an Elegant Fuzzy Labeling (𝜇, 𝜌) of 𝑃5 is given as follows: 𝜇(𝑣1) = 0.89, 𝜇(𝑣2) = 1, 𝜇(𝑣3) = 0.56, 𝜇(𝑣4) = 0.67, 𝜇(𝑣5) = 0.78, 𝜌(𝑣1𝑣2) = 0.22, 𝜌(𝑣2𝑣3) = 0.44, 𝜌(𝑣3𝑣4) = 0.11, 𝜌(𝑣4𝑣5) = 0.33. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 38 https://internationalpubls.com Theorem 3.2.4. Cycles 𝐶𝑛 admit Elegant Fuzzy Labeling for 𝑛 ≡ 0, 3 𝑚𝑜𝑑 4. Proof. Let 𝐶𝑛 be a cycle with vertex set 𝑉(𝐶𝑛) = {𝑣1, 𝑣2, … , 𝑣𝑛} and edge set 𝐸(𝐶𝑛) = {𝑣1𝑣2, 𝑣2𝑣3, … , 𝑣𝑛−1𝑣𝑛 , 𝑣𝑛𝑣1}. Let 𝑓 be an injective function from 𝑉(𝐶𝑛) into the set { 𝑛 + 1 , 𝑛 + 2 ,… , 2𝑛 + 1} and 𝑔 be a function from 𝐸(𝐶𝑛) into the set {0,1,… , 𝑛 } defined as, 𝑔(𝑣𝑛𝑣1) = (𝑓(𝑣𝑛) + 𝑓(𝑣1)) 𝑚𝑜𝑑 (𝑛 + 1) and 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣𝑖) + 𝑓(𝑣𝑖+1)) 𝑚𝑜𝑑 (𝑛 + 1) for 1 ≤ 𝑖 ≤ 𝑛 − 1. Case 1: For 𝐶2𝑚+1, 𝑚 = 1, 3, 5, … , 𝑓: 𝑉(𝐶2𝑚+1) → {2𝑚 + 2, 2𝑚 + 3, 2𝑚 + 4,… , 4𝑚 + 3} is defined as, 𝑓(𝑣𝑖) = { 2𝑚 + 2 𝑖 = 1 3𝑚 + 1 + 𝑖 2 ≤ 𝑖 ≤ 3 2𝑚 + 3 𝑖 = 4 𝑓(𝑣𝑖−2) + 1 5 ≤ 𝑖 ≤ 𝑚 + 2 𝑓(𝑣𝑖−1) + 1 𝑖 = 𝑚 + 3 𝑓(𝑣𝑖−3) + 2 𝑖 = 𝑚 + 4 𝑓(𝑣𝑖−2) + 1 𝑚 + 5 ≤ 𝑖 ≤ 2𝑚 + 1. For 𝑖 = 1, 𝑔(𝑣1𝑣2) = (2𝑚 + 2 + 3𝑚 + 3) 𝑚𝑜𝑑 (2𝑚 + 2) = (5𝑚 + 5) 𝑚𝑜𝑑 (2𝑚 + 2) = 𝑚 + 1. For 𝑖 = 2, 𝑔(𝑣2𝑣3) = (3𝑚 + 3 + 3𝑚 + 4) 𝑚𝑜𝑑 (2𝑚 + 2) = (6𝑚 + 7) 𝑚𝑜𝑑 (2𝑚 + 2) = 1. For 𝑖 = 3, 𝑔(𝑣3𝑣4) = (3𝑚 + 4 + 2𝑚 + 3) 𝑚𝑜𝑑 (2𝑚 + 2) = (5𝑚 + 7) 𝑚𝑜𝑑 (2𝑚 + 2) = 𝑚 + 3. For 4 ≤ 𝑖 ≤ 𝑚 + 1, 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣𝑖) + 𝑓(𝑣𝑖+1)) 𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣𝑖−2) + 1 + 𝑓(𝑣𝑖+1−2) + 1) 𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣𝑖−4) + 2 + 𝑓(𝑣𝑖+1−4) + 2) 𝑚𝑜𝑑 (2𝑚 + 2) ⋮ Subcase 1: When 4 ≤ 𝑖 ≤ 𝑚 + 1 and odd, 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣𝑖−𝑗) + 𝑗 2 + 𝑓(𝑣𝑖+1−𝑗) + 𝑗 2 ) 𝑚𝑜𝑑 (2𝑚 + 2), 𝑗 = 2, 4, … , 𝑖 − 3, and when 𝑗 = 𝑖 − 3, Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 39 https://internationalpubls.com 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣3) + 𝑓(𝑣4) + 2 ( 𝑖 − 3 2 )) 𝑚𝑜𝑑 (2𝑚 + 2) = (3𝑚 + 4+ 2𝑚 + 3 + 𝑖 − 3) 𝑚𝑜𝑑 (2𝑚 + 2) = (2(2𝑚 + 2) +𝑚 + 𝑖) 𝑚𝑜𝑑 (2𝑚 + 2) = 𝑚 + 𝑖. Subcase 2: When 4 ≤ 𝑖 ≤ 𝑚 + 1 and even, 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣𝑖−𝑗) + 𝑗 2 + 𝑓(𝑣𝑖+1−𝑗) + 𝑗 2 ) 𝑚𝑜𝑑 (2𝑚 + 2), 𝑗 = 0, 2, 4,… , 𝑖 − 4, and when 𝑗 = 𝑖 − 4, 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣4) + 𝑓(𝑣5) + 2 ( 𝑖 − 4 2 )) 𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣4) + 𝑓(𝑣3) + 1 + 2 ( 𝑖−4 2 )) 𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣3) + 𝑓(𝑣4) + 2 ( 𝑖−3 2 )) 𝑚𝑜𝑑 (2𝑚 + 2) = (3𝑚 + 4+ 2𝑚 + 3 + 𝑖 − 3) 𝑚𝑜𝑑 (2𝑚 + 2) = (2(2𝑚 + 2) +𝑚 + 𝑖) 𝑚𝑜𝑑 (2𝑚 + 2) = 𝑚 + 𝑖 . For 𝑖 = 𝑚 + 2, 𝑔(𝑣𝑚+2𝑣𝑚+3) = (𝑓(𝑣𝑚+2) + 𝑓(𝑣𝑚+3)) 𝑚𝑜𝑑 (2𝑚 + 2) = (2𝑓(𝑣𝑚+2) + 1) 𝑚𝑜𝑑 (2𝑚 + 2) = (2𝑓(𝑣𝑚) + 2 + 1) 𝑚𝑜𝑑 (2𝑚 + 2) = (2𝑓(𝑣𝑚−2) + 4 + 1) 𝑚𝑜𝑑 (2𝑚 + 2) ⋮ = (2𝑓(𝑣(𝑚+2)−(𝑚+2−3)) + 𝑚 + 2 − 3 + 1) 𝑚𝑜𝑑 (2𝑚 + 2) = (2𝑓(𝑣3) + 𝑚 + 2 − 2) 𝑚𝑜𝑑 (2𝑚 + 2) = (6𝑚 + 6 + 2 + 𝑚) 𝑚𝑜𝑑 (2𝑚 + 2) = 𝑚 + 2. For 𝑖 = 𝑚 + 3, 𝑔(𝑣𝑚+3𝑣𝑚+4) = (𝑓(𝑣𝑚+3) + 𝑓(𝑣𝑚+4)) 𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣𝑚+2) + 1 + 𝑓(𝑣𝑚+1) + 2) 𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣𝑚) + 1 + 𝑓(𝑣𝑚−1) + 1 + 3) 𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣𝑚−2) + 2 + 𝑓(𝑣𝑚−3) + 2 + 3) 𝑚𝑜𝑑 (2𝑚 + 2) ⋮ = (𝑓(𝑣3) + 𝑚−1 2 + 𝑓(𝑣4) + 𝑚−3 2 + 3) 𝑚𝑜𝑑 (2𝑚 + 2) = (3𝑚 + 4 + 2𝑚 + 3 + 𝑚−1 2 + 𝑚−3 2 + 3)𝑚𝑜𝑑 (2𝑚 + 2) = (5𝑚 + 10 +𝑚 − 2) 𝑚𝑜𝑑 (2𝑚 + 2) = 2. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 40 https://internationalpubls.com For 𝑚 + 4 ≤ 𝑖 ≤ 2𝑚, 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣𝑖) + 𝑓(𝑣𝑖+1)) 𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣𝑖−2) + 1 + 𝑓(𝑣𝑖+1−2) + 1) 𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣𝑖−4) + 2 + 𝑓(𝑣𝑖+1−4) + 2) 𝑚𝑜𝑑 (2𝑚 + 2) ⋮ Subcase 1: When 𝑚 + 4 ≤ 𝑖 ≤ 2𝑚 and even, 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣𝑖−𝑗) + 𝑗 2 + 𝑓(𝑣𝑖+1−𝑗) + 𝑗 2 )𝑚𝑜𝑑(2𝑚 + 2), 𝑗 = 2, 4, … , 𝑖 − 𝑚 − 3, and when 𝑗 = 𝑖 −𝑚 − 3, 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣𝑚+3) + 𝑓(𝑣𝑚+4) + (𝑖 − 𝑚 − 3)) 𝑚𝑜𝑑 (2𝑚 + 2) = (6𝑚 + 6 + 2 + 𝑖 −𝑚 − 3) 𝑚𝑜𝑑 (2𝑚 + 2) = 𝑖 − 𝑚 − 1. Subcase 2: When 𝑚 + 4 ≤ 𝑖 ≤ 2𝑚 and odd, 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣𝑖−𝑗) + 𝑗 2 + 𝑓(𝑣𝑖+1−𝑗) + 𝑗 2 ) 𝑚𝑜𝑑 (2𝑚 + 2), 𝑗 = 0, 2, 4,… , 𝑖 − 𝑚 − 4, and when 𝑗 = 𝑖 −𝑚 − 4, 𝑔(𝑣𝑖𝑣𝑖+1) = (𝑓(𝑣𝑚+4) + 𝑓(𝑣𝑚+5) + (𝑖 − 𝑚 − 4)) 𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣𝑚+4) + 𝑓(𝑣𝑚+3) + 1 + (𝑖 −𝑚 − 4)) 𝑚𝑜𝑑(2𝑚 + 2) = (𝑓(𝑣𝑚+3) + 𝑓(𝑣𝑚+4) + (𝑖 − 𝑚 − 3)) 𝑚𝑜𝑑(2𝑚 + 2) = (6𝑚 + 6 + 2 + 𝑖 −𝑚 − 3)𝑚𝑜𝑑(2𝑚 + 2) = 𝑖 − 𝑚 − 1. Now, 𝑔(𝑣2𝑚+1𝑣1) = (𝑓(𝑣2𝑚+1) + 𝑓(𝑣1)) 𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣2𝑚+1−2) + 1 + 2𝑚 + 2) 𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣2𝑚+1−4) + 2 + 2𝑚 + 2) 𝑚𝑜𝑑 (2𝑚 + 2) ⋮ = (𝑓(𝑣𝑚+4) + 2𝑚+1−𝑚−4 2 + 2𝑚+ 2) 𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣𝑚+4−3) + 2 + 𝑚−3 2 + 2𝑚 + 2)𝑚𝑜𝑑 (2𝑚 + 2) = (𝑓(𝑣𝑚+1) + 𝑚−3 2 + 2𝑚+ 4)𝑚𝑜𝑑(2𝑚 + 2) = (𝑓(𝑣𝑚+1−2) + 1 + 𝑚−3 2 + 2𝑚 + 4)𝑚𝑜𝑑(2𝑚 + 2) ⋮ = (𝑓(𝑣𝑚+1−𝑚−1+4) + 𝑚−3 2 + 𝑚−3 2 + 2𝑚 + 4)𝑚𝑜𝑑(2𝑚 + 2) = (𝑓(𝑣4) + 𝑚−3 2 + 𝑚−3 2 + 2𝑚+ 4) 𝑚𝑜𝑑(2𝑚 + 2) = (2𝑚 + 3 + 𝑚 − 3 + 2𝑚 + 4) 𝑚𝑜𝑑(2𝑚 + 2) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 41 https://internationalpubls.com = (4𝑚 + 4 + 𝑚) 𝑚𝑜𝑑(2𝑚 + 2) = 𝑚. Hence, 𝑔(𝑣𝑖𝑣𝑖+1) = { 𝑚 + 1 𝑖 = 1 1 𝑖 = 2 𝑚 + 3 𝑖 = 3 𝑚 + 𝑖 4 ≤ 𝑖 ≤ 𝑚 + 1 𝑚 + 2 𝑖 = 𝑚 + 2 2 𝑖 = 𝑚 + 3 𝑖 −𝑚 − 1 𝑚 + 4 ≤ 𝑖 ≤ 2𝑚. 𝑔(𝑣2𝑚+1𝑣1) = 𝑚 Case 2: For 𝐶4𝑚, 𝑚 = 1 ,2, 3,… , 𝑓: 𝑉(𝐶4𝑚) → {4𝑚 + 1, 4𝑚 + 2, 4𝑚 + 3,… , 8𝑚 + 1} is defined as, 𝑓(𝑣𝑖) = { 4𝑚 + 2 𝑖 = 1 8𝑚 + 𝑖 − 2 2 ≤ 𝑖 ≤ 3 𝑓(𝑣𝑖−2) − 2 4 ≤ 𝑖 ≤ 2𝑚 + 1 𝑓(𝑣𝑖−1) − 2 𝑖 = 2𝑚 + 2 𝑓(𝑣𝑖−1) − 1 2𝑚 + 3 ≤ 𝑖 ≤ 4𝑚. Following a similar approach as in Case 1, we obtain, 𝑔(𝑣𝑖𝑣𝑖+1) = { 4𝑚 𝑖 = 1 4𝑚 − 2 𝑖 = 2 4𝑚 + 2 − 2𝑖 3 ≤ 𝑖 ≤ 2𝑚 1 𝑖 = 2𝑚 + 1 8𝑚 + 3 − 2𝑖 2𝑚 + 2 ≤ 𝑖 ≤ 4𝑚 − 1. 𝑔(𝑣4𝑚𝑣1) = 3. Now we have ℎ = 1 2𝑛+1 . The vertex labeling 𝜇: 𝑉(𝐶𝑛) → [0,1] is defined as, 𝜇(𝑣𝑖) = 𝑓(𝑣𝑖). 1 2𝑛+1 for every 1 ≤ 𝑖 ≤ 𝑛 and the edge labeling 𝜌: 𝐸(𝐶𝑛) → [0,1] is defined as, 𝜌(𝑣𝑖𝑣𝑖+1) = 𝑔(𝑣𝑖𝑣𝑖+1). 1 2𝑛+1 for every 1 ≤ 𝑖 ≤ 𝑛 − 1 and 𝜌(𝑣𝑛𝑣1) = 𝑔(𝑣𝑛𝑣1). 1 2𝑛+1 . It can be verified that in all the cases the elements of the set {𝑔(𝑣𝑛𝑣1), 𝑔(𝑣𝑖𝑣𝑖+1), 1 ≤ 𝑖 ≤ 𝑛 − 1} are distinct and nonzero. Therefore, the edge labels are distinct and non zero. Further max{ 𝑔(𝑣𝑛𝑣1), 𝑔(𝑣𝑖𝑣𝑖+1), 1 ≤ 𝑖 ≤ 𝑛 − 1} = 𝑛 and min{ 𝑓(𝑣𝑖)} = 𝑛 + 1. Therefore, 𝑔(𝑣𝑛𝑣1) < 𝑓(𝑣𝑛) ∧ 𝑓(𝑣1) and 𝑔(𝑣𝑖𝑣𝑖+1) < 𝑓(𝑣𝑖) ∧ 𝑓(𝑣𝑖+1) for every 1 ≤ 𝑖 ≤ 𝑛 − 1. Hence the edge labels are less than the minimum of their respective endpoints labels. Consequently, 𝐶𝑛 for 𝑛 ≡ 0,3 𝑚𝑜𝑑 4 are Elegant Fuzzy Labeling Graphs and (𝜇, 𝜌) is an Elegant Fuzzy Labeling of 𝐶𝑛 for 𝑛 ≡ 0,3 𝑚𝑜𝑑 4. □ Remark 3.2.5. Cycles 𝐶𝑛 where 𝑛 ≡ 1 𝑚𝑜𝑑 4 are not Elegant [3], therefore it does not admit Elegant Fuzzy Labeling. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 42 https://internationalpubls.com Note. The case for cycles 𝐶𝑛 where 𝑛 ≡ 2 𝑚𝑜𝑑 4 remains an open question. Future work could investigate whether this class of cycles can admit Elegant Fuzzy Labeling or if modifications to the labeling scheme are required to accommodate these structures. Theorem 3.2.6. Line graph of cycles 𝐿(𝐶𝑛) admit Elegant Fuzzy Labeling when 𝑛 ≡ 0,3 𝑚𝑜𝑑 4. Proof. By Theorem 3.2.4, 𝐶𝑛 admit Elegant Fuzzy Labeling when 𝑛 ≡ 0,3 𝑚𝑜𝑑 4. 𝐿(𝐶𝑛) ≡ 𝐶𝑛. Therefore 𝐿(𝐶𝑛) admit Elegant Fuzzy Labeling when 𝑛 ≡ 0,3 𝑚𝑜𝑑 4. □ Example 3.2.7. Consider a cycle 𝐶8 in Fig.6. By Theorem 3.2.4, the function 𝑓: 𝑉(𝐶8) → {9, 10, 11, 12, 13,14, 15, 16, 17} is defined as follows: 𝑓(𝑣1) = 10, 𝑓(𝑣2) = 16, 𝑓(𝑣3) = 17, 𝑓(𝑣4) = 14, 𝑓(𝑣5) = 15, 𝑓(𝑣6) = 13, 𝑓(𝑣7) = 12, 𝑓(𝑣8) = 11. Consequently, 𝑔(𝑣1𝑣2) = 8, 𝑔(𝑣2𝑣3) = 6, 𝑔(𝑣3𝑣4) = 4, 𝑔(𝑣4𝑣5) = 2, 𝑔(𝑣5𝑣6) = 1, 𝑔(𝑣6𝑣7) = 7, 𝑔(𝑣7𝑣8) = 5, 𝑔(𝑣8𝑣1) = 3. Hence, an Elegant Fuzzy Labeling (𝜇, 𝜌) of 𝐶8 is given as follows: 𝜇(𝑣1) = 0.59, 𝜇(𝑣2) = 0.94, 𝜇(𝑣3) = 1, 𝜇(𝑣4) = 0.82, 𝜇(𝑣5) = 0.88, 𝜇(𝑣6) = 0.76, 𝜇(𝑣7) = 0.71, 𝜇(𝑣8) = 0.65, 𝜌(𝑣1𝑣2) = 0.47, 𝜌(𝑣2𝑣3) = 0.35, 𝜌(𝑣3𝑣4) = 0.24, 𝜌(𝑣4𝑣5) = 0.12, 𝜌(𝑣5𝑣6) = 0.06, 𝜌(𝑣6𝑣7) = 0.41, 𝜌(𝑣7𝑣8) = 0.29, 𝜌(𝑣8𝑣1) = 0.18. 4. Application Consider a transit network as an Elegant Fuzzy Labeling Graph, where vertices represent stations and edges represent routes. Vertex labels denote passenger capacity at each station, while edge labels represent the passenger flow along the routes. The condition of Elegant Fuzzy Labeling, that labels are distinct and non-zero, guarantees that every station and route is utilized effectively. The condition that edge labels are less than their endpoints labels ensures that the passenger flow along the route does not exceed the station capacity, thus preventing congestion at stations and along the routes. This approach enables judicious allocation of resources, improves routing and scheduling, and ensures smooth movement of passengers. 5. Conclusion In this paper, we have introduced a new type of fuzzy labeling called Elegant Fuzzy Labeling. We proved that a simple graph admits Elegant Fuzzy Labeling if and only if it admits elegant labeling. Additionally, we showed that while a simple graph admitting Elegant Fuzzy Labeling will also admit fuzzy labeling, the converse is not necessarily true. We have investigated certain classes of simple graphs to determine if they admit Elegant Fuzzy Labeling and provided an application using this labeling method. We employed translation and contraction of mapping to achieve the labeling. For future work, we plan to explore Elegant Fuzzy Labeling on other graph families, investigate whether Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 1s (2025) 43 https://internationalpubls.com different translation constants and contraction scales can be used to define Elegant Fuzzy Labeling and extend the concept to various extensions of fuzzy graphs. Declarations Conflicts of Interest: None of the authors have any conflict of interest. References [1] Borzooei, R. A., Rashmanlou, H.: Cayley interval-valued fuzzy graphs. UPB Scientific Bulletin, Series A: Applied Mathematics and Physics 78, no. 3: 83-94 (2016). [2] Cahit, Ibrahim.: Elegant valuation of the paths. Ars Combinatoria 16 (1983). [3] Chang, G. J., Hue, D. 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