EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6621 ISSN 1307-5543 – ejpam.com Published by New York Business Global Neutrosophic Soft n-Super-HyperGraphs with Real-World Applications Takaaki Fujita1,∗, Florentin Smarandache2 1 Independent Researcher, Shinjuku, Shinjuku-ku, Tokyo, Japan 2 University of New Mexico, Gallup Campus, NM 87301, USA Abstract. Graph theory provides a fundamental framework for modeling relationships using ver- tices and edges [1, 2]. Hypergraphs extend this framework by allowing hyperedges that can simulta- neously connect multiple vertices [3], while n-Super-HyperGraphs further generalize hypergraphs via iterated power-set constructions to capture hierarchical relationships [4, 5]. In parallel, various uncertainty modeling paradigms—such as fuzzy sets [6], soft sets [7], intuitionistic fuzzy sets [8–10], neutrosophic sets, and plithogenic sets—have been developed to handle imprecise or indeterminate information. In this paper, we propose a novel framework called the Neutrosophic Soft n-Super-HyperGraph, which integrates neutrosophic logic, soft set theory, and n-Super-HyperGraph structures. This model has the potential to facilitate effective decision-making in complex and uncertain networked environments. 2020 Mathematics Subject Classifications: 05C65 Key Words and Phrases: Super-HyperGraph, Hypergraph, Fuzzy Graph, Soft Graph, Neutro- sophic Graph 1. Introduction and Literature Review Traditional graphs capture binary relationships by representing entities as vertices and their connections as edges [1, 11]. Hypergraphs extend this notion by allowing each hyperedge to join any nonempty subset of vertices, thereby modeling higher-order inter- actions [12–14]. The concept is taken further by n-Super-HyperGraphs, which iteratively apply the power-set operator to encode nested, hierarchical connectivity patterns within a single unified structure [15–18]. These graph-based formalisms have proven invaluable across domains such as net- work science, data mining, chemistry, and physics, providing both intuitive visualization and rigorous analytical frameworks [2, 5]. The additional abstraction levels in n-Super- HyperGraphs are particularly well-suited for disentangling multi-tiered relationships in complex systems. Here, n is assumed to be a positive integer. https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 2 of 35 Structure Notation Edge Definition Description Graph G = (V,E) E ⊆ { {u, v} | u, v ∈ V, u ̸= v } Simple pairwise con- nections. Hypergraph H = (V,E) E ⊆ P(V ) \ {∅} Hyperedges connect arbitrary subsets of V . n-Super-HyperGraph SHG(n) = (V,E) V,E ⊆ Pn(V0) Hierarchical, it- erated–powerset construction. Table 1: Comparison of graphs, hypergraphs, and n-Super-HyperGraphs. Table 1 summarizes the principal distinctions among standard graphs, hypergraphs, and n-Super-HyperGraphs. Real-world networks often exhibit imprecision or missing information. To address these challenges, classical set theory has been extended with various uncertainty models: fuzzy sets [6], intuitionistic fuzzy sets [9], soft sets [7, 19], vague sets [20, 21], hyperfuzzy sets [22], neutrosophic sets [23, 24], and plithogenic sets [25, 26]. Embedding these into graph theory yields fuzzy graphs [27], intuitionistic fuzzy graphs [28], neutrosophic graphs [29, 30], plithogenic graphs [31], and soft graphs [32], each enriching the topology with graded uncertainty semantics. Soft graph models form a parameterized family of subgraphs, where each parameter selects a specific subgraph within a fixed vertex–edge universe [7, 32, 33]. This mirrors soft set theory, which collects subsets indexed by parameters to capture ambiguity or prefer- ence without numeric membership values [19, 34]. Extending this concept to hypergraphs, a soft hypergraph assigns to each parameter both a subset of vertices and its induced hyperedges [35–38]. The notion of a soft n-Super-HyperGraph carries this idea into the layered framework of an n-Super-HyperGraph, producing a hierarchical, parameter-driven model [39]. Table 2 presents an overview of soft graph models and their generalizations to hypergraphs. Moreover, a single-valued neutrosophic hypergraph endows every ver- tex and hyperedge with three membership degrees—truth, indeterminacy, and falsity—in [0, 1], subject to neutrosophic constraints [40]. These graph-based frameworks have seen extensive applications in decision-making and related domains. We describe the motivation and contributions of this paper. Neutrosophic Soft Hy- perGraphs and related studies on Neutrosophic Graphs, Soft Graphs, and HyperGraphs are well established and of great importance. However, as noted above, HyperGraphs alone struggle to model deeply hierarchical graph and network concepts—a limitation that naturally extends to Neutrosophic Soft HyperGraphs. To address this gap, this pa- per introduces the Neutrosophic Soft n-Super-HyperGraph, a novel framework that fuses neutrosophic logic, soft set theory, and n-Super-HyperGraph architecture. Our model pro- vides a flexible yet robust foundation for decision-making over networks characterized by ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6621 Email addresses: Takaaki.fujita060@gmail.com (T.Fujita), smarand@unm.edu (F. Smarandache) T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 3 of 35 Model Definition Parameter Mapping Soft graph A parameterized family of subgraphs of G = (V,E), where each parameter c ∈ C selects one subgraph (A(c), B(c)). A : C → P(V ), B : C → P(E), B(c) ⊆ {{u, v} | u, v ∈ A(c)}. Soft hypergraph A soft hypergraph over H = (V,E) in which each c ∈ C induces the subhypergraph (A(c), B(c)). A : C → P(V ), B : C → P(E), B(c) ⊆ {e ∈ E | e ⊆ A(c)}. Soft n-Super-HyperGraph A hierarchical, parameter-driven model built on SuHG(n) = (V,E), where each c ∈ C yields the substructure (A(c), B(c)). A : C → P(V ), B : C → P(E), B(c) ⊆ {e ∈ E | e ⊆ A(c)}. Table 2: Overview of soft graph models and their hypergraph generalizations both layered connectivity and degree-based uncertainty. We present its formal definition, examine its key properties, and illustrate its applicability through examples in complex, uncertain environments. We present a concise overview of neutrosophic soft graph models in Table 3. Model Description Neutrosophic Soft Graph A graph in which, for each context, every vertex and edge is assigned degrees of truth, indeterminacy, and falsity. Neutrosophic Soft Hypergraph A hypergraph where each parameter selects a substruc- ture, and vertices and hyperedges carry neutrosophic degrees per parameter. Neutrosophic Soft n-Super-HyperGraph A multi-level generalization: supervertices and su- peredges at n levels are chosen per context and endowed with neutrosophic degrees. Table 3: Concise overview of neutrosophic soft graph models. The structure of this paper is as follows. Section 2 provides an overview of n-Super- HyperGraphs and introduces the concept of Neutrosophic Soft HyperGraphs. In Section 3, we formally define the Neutrosophic Soft n-Super-HyperGraph. Section 4 discusses the algorithm for constructing this type of graph. Finally, Section 5 presents the conclusion and outlines potential directions for future research. 2. Preliminaries In this section, we introduce the fundamental concepts and notation used throughout this paper. We assume all graphs are finite, simple, and undirected unless noted otherwise. For more extensive treatments, see the cited literature. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 4 of 35 2.1. n-Super-HyperGraphs A hypergraph enhances graphs by allowing edges (called hyperedges) to join any subset of vertices, making it suitable for modeling multi-way relationships [3, 12, 41–44]. A Super- HyperGraph extends this idea by applying the power-set operation repeatedly, capturing hierarchical or recursive connectivity patterns [5, 15, 16, 45–47]. Note that in general, n is assumed to be a positive integer. Definition 1 (Base Set). Let S be a base set, serving as the ground domain for all higher-order constructions: S = {x : x belongs to the universe of discourse}. Definition 2 (power-set). The power-set of S, denoted PS(S), is the set of all subsets of S, including the empty set: PS(S) = {A : A ⊆ S}. Definition 3 (Hypergraph). [3, 41] A hypergraph is a pair H = (V,E) where • V is a finite set of vertices, • E is a collection of nonempty subsets of V , each called a hyperedge. Example 1 (Online Social Network as a Hypergraph). Consider a small social network with five users: V = {Hiroko, Tae, Yusuke, Dave, Eve}. We model four interest-based groups as hyperedges: e1 = {Hiroko, Tae, Yusuke} (Photography Club), e2 = {Tae, Dave, Eve} (Music Enthusiasts), e3 = {Hiroko, Dave, Eve} (Travel Group), e4 = {Yusuke, Eve} (Book Discussion). Thus the hypergraph H = ( V, {e1, e2, e3, e4} ) captures the overlapping community structure of the social network, where each hyperedge represents a multi-user group or interest. Definition 4 (n-th Iterated power-set). [48, 49] Define the iterated power-set of a set X by PS1(X) = PS(X), PSk+1(X) = PS ( PSk(X) ) , k ≥ 1. The corresponding nonempty iterated power-set is PS∗1(X) = PS(X) \ {∅}, PS∗k+1(X) = PS ( PS∗k(X) ) \ {∅}. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 5 of 35 Example 2 (Corporate Hierarchy as a 2-Iterated Power-Set). Let X = {AlphaCorp, BetaInc, GammaLLC} be three affiliated companies. The first power-set is PS1(X) = {∅, {AlphaCorp}, {BetaInc}, {GammaLLC}, {AlphaCorp,BetaInc}, {AlphaCorp,GammaLLC}, {BetaInc,GammaLLC}, {AlphaCorp,BetaInc,GammaLLC}}. Excluding the empty set, we may identify two holding companies: H1 = {AlphaCorp,BetaInc}, H2 = {GammaLLC}, H1, H2 ∈ PS∗1(X). The second iterated power-set (nonempty) is PS∗2(X) = PS ( PS∗1(X) ) \ {∅}. In particular, the set C = {H1, H2} ∈ PS∗2(X) represents a parent company C that directly controls two subsidiaries H1 and H2, each of which in turn holds its constituent firms—a clear two-level corporate hierarchy. Definition 5 (n-Super-HyperGraph). [47, 50, 51] Let V0 be a finite base set and define PSk(V0) by iterating the power-set k times. An n-Super-HyperGraph is a pair SuHG(n) = (V,E), where V ⊆ PSn(V0), E ⊆ PSn(V0). Members of V are called n-supervertices and members of E are n-superedges. Example 3 (Autonomous Robot Operating Modes as a 2-Super-HyperGraph). Consider the core software modules of an autonomous robot: V0 = {Perception, Planning, Actuation}. At the first level (n = 1), we form two functional clusters (1-supervertices): c1 = {Perception, Planning}, c2 = {Planning, Actuation}, V1 = {c1, c2}. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 6 of 35 At the second level (n = 2), we combine these clusters into two operational modes (2- supervertices): sfull = {c1, c2}, (full operation mode) snav = {c1}, (navigation-only mode) V2 = { sfull, snav }. We then introduce two 2-superedges to represent available mode selections: e1 = { sfull }, e2 = { snav }, E2 = {e1, e2}. Thus the 2-Super-HyperGraph is SuHG(2) = ( V2, E2 ) . Here: • SuHG(2) captures two levels of grouping: first into functional clusters, then into high-level operating modes. • The superedge e1 allows selection of the full operation mode, while e2 restricts the robot to navigation only. This concrete 2-Super-HyperGraph models how an autonomous robot’s low-level modules combine into clusters, which in turn form distinct modes of operation in real-world mis- sions. 2.2. Neutrosophic n-Super-HyperGraph A single-valued neutrosophic set assigns to each element three membership degrees—truth, indeterminacy, and falsity—in [0, 1], with their sum at most three [23, 52]. A single-valued neutrosophic graph extends this concept to graphs, equipping both vertices and edges with neutrosophic memberships to model uncertain relationships [29, 53, 54]. A single-valued neutrosophic hypergraph further generalizes to hypergraphs, where each vertex–hyperedge incidence is endowed with its own triple of membership degrees under the same constraint [55–58]. We begin by recalling the definition of a single-valued neutrosophic hypergraph and then extend it to the framework of n-Super-HyperGraphs (cf. [55–58]). Definition 6 (Neutrosophic Set). [23] Let X be a non-empty set. A Neutrosophic Set (NS) A on X is characterized by three membership functions: TA : X → [0, 1], IA : X → [0, 1], FA : X → [0, 1], where for each x ∈ X, the values TA(x), IA(x), and FA(x) represent the degrees of truth, indeterminacy, and falsity, respectively. These values satisfy the following condition: 0 ≤ TA(x) + IA(x) + FA(x) ≤ 3. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 7 of 35 Definition 7 (Single-Valued Neutrosophic Graph). A single-valued neutrosophic graph (SVNG) is an ordered septuple GN = ( V, E, TV , IV , FV , TE , IE , FE ) , where • V is a nonempty set of vertices, • E ⊆ V × V is the set of edges, • TV , IV , FV : V → [0, 1] are the truth-, indeterminacy-, and falsity-membership func- tions on vertices, • TE , IE , FE : E → [0, 1] are the truth-, indeterminacy-, and falsity-membership func- tions on edges, satisfying the following conditions: (i) For every vertex v ∈ V , 0 ≤ TV (v) + IV (v) + FV (v) ≤ 3. (ii) For every edge e ∈ E, 0 ≤ TE(e) + IE(e) + FE(e) ≤ 3. (iii) For each edge e = (u, v) ∈ E, the edge-vertex consistency constraints hold: TE(e) ≤ min{TV (u), TV (v)}, IE(e) ≤ min{IV (u), IV (v)}, FE(e) ≥ max{FV (u), FV (v)}. Definition 8 (Single-Valued Neutrosophic Hypergraph). (cf.[57, 58]) Let V = {v1, . . . , vN} be a finite vertex set, and let {Ei}Mi=1 be a collection of non-empty neutrosophic subsets of V such that V = M⋃ i=1 supp(Ei). Each hyperedge Ei is specified by three membership functions TEi , IEi , FEi : V → [0, 1], assigning to each vertex v ∈ V its truth, indeterminacy, and falsity degrees, respectively, and satisfying 0 ≤ TEi(v) + IEi(v) + FEi(v) ≤ 3 ∀ v ∈ V. We represent Ei as the set Ei = { (v, TEi(v), IEi(v), FEi(v)) : v ∈ V } . The pair H = (V, {Ei}) is called a single-valued neutrosophic hypergraph. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 8 of 35 Example 4 (Air Pollution Events as a Single-Valued Neutrosophic Hypergraph). Con- sider a monitoring network for four common air pollutants: V = {NO2, CO, PM2.5, O3}. We model three types of pollution episodes as neutrosophic hyperedges: E1 = {NO2,CO,PM2.5,O3} (Smog event), E2 = {NO2,CO} (Traffic pollution), E3 = {PM2.5,O3} (Industrial emission). For each Ei, define membership functions TEi , IEi , FEi : V → [0, 1], satisfying 0 ≤ TEi(v) + IEi(v) + FEi(v) ≤ 3. A plausible assignment is: Smog event (E1): TE1(NO2) = 0.80, IE1(NO2) = 0.10, FE1(NO2) = 0.10, TE1(CO) = 0.75, IE1(CO) = 0.15, FE1(CO) = 0.10, TE1(PM2.5) = 0.85, IE1(PM2.5) = 0.10, FE1(PM2.5) = 0.05, TE1(O3) = 0.70, IE1(O3) = 0.20, FE1(O3) = 0.10; Traffic pollution (E2): TE2(NO2) = 0.90, IE2(NO2) = 0.05, FE2(NO2) = 0.05, TE2(CO) = 0.88, IE2(CO) = 0.07, FE2(CO) = 0.05; Industrial emission (E3): TE3(PM2.5) = 0.95, IE3(PM2.5) = 0.03, FE3(PM2.5) = 0.02, TE3(O3) = 0.82, IE3(O3) = 0.10, FE3(O3) = 0.08. Then Ei = { (v, TEi(v), IEi(v), FEi(v)) : v ∈ Ei } , i = 1, 2, 3, and the pair H = ( V, {E1, E2, E3} ) is a single-valued neutrosophic hypergraph capturing both the membership degrees and the uncertainty of each pollutant in different pollution events. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 9 of 35 Definition 9 (Neutrosophic n-Super-HyperGraph). Let V0 be a finite ground set. Define iterated power-sets by PS0(V0) = V0, PSk+1(V0) = PS ( PSk(V0) ) , k ≥ 0. An n-Super-HyperGraph is a pair SuHG(n) = (V,E) with V ⊆ PSn(V0), E ⊆ PSn(V0). A neutrosophic n-Super-HyperGraph enhances this structure by equipping both superver- tices and superedges with neutrosophic memberships:( V, E, TV , IV , FV , TE , IE , FE ) , where • TV , IV , FV : V → [0, 1] assign to each supervertex v its truth, indeterminacy, and falsity degrees, subject to 0 ≤ TV (v) + IV (v) + FV (v) ≤ 3 ∀ v ∈ V. • TE , IE , FE : E × V → [0, 1] assign to each pair (e, v) the corresponding neutrosophic membership values, satisfying 0 ≤ TE(e, v) + IE(e, v) + FE(e, v) ≤ 3 ∀ e ∈ E, v ∈ V. These must obey the containment constraints: TE(e, v) ≤ TV (v), IE(e, v) ≤ IV (v), FE(e, v) ≤ FV (v), ∀ e ∈ E, v ∈ V. Example 5 (Organizational Workflow as a Neutrosophic 2-Super-HyperGraph). Consider a small organization with four employees: V0 = {Hiroko, Yutaka, Haruko, Shinya}. They form two departments: DeptA = {Hiroko,Yutaka}, DeptB = {Haruko,Shinya}, so that PS1(V0) ⊇ V1 = {DeptA, DeptB}. At the next level, these two departments collaborate on a joint division: PS2(V0) ⊇ V2 = {Div1}, Div1 = {DeptA,DeptB}. A single superedge models their shared project: E2 = {ProjectX}, ProjectX = {DeptA, DeptB}. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 10 of 35 We assign neutrosophic membership degrees to each supervertex v ∈ V2: TV (Div1) = 0.9, IV (Div1) = 0.05, FV (Div1) = 0.05, indicating strong confidence that Div1 is active, with slight uncertainty or doubt. Likewise, for the superedge ProjectX and each constituent department v: TE(ProjectX,DeptA) = 0.85, IE(ProjectX,DeptA) = 0.10, FE(ProjectX,DeptA) = 0.05, TE(ProjectX,DeptB) = 0.80, IE(ProjectX,DeptB) = 0.15, FE(ProjectX,DeptB) = 0.05. These values satisfy 0 ≤ TV (v) + IV (v) + FV (v) ≤ 3, 0 ≤ TE(e, v) + IE(e, v) + FE(e, v) ≤ 3, and the containment constraints TE(e, v) ≤ TV (v), IE(e, v) ≤ IV (v), FE(e, v) ≤ FV (v) hold for all v ∈ V2, e ∈ E2. Thus the tuple( V2, E2, TV , IV , FV , TE , IE , FE ) constitutes a neutrosophic 2-Super-HyperGraph capturing the organization’s hierarchical structure and uncertainty in collaboration. Example 6 (Corporate Divisions as a Neutrosophic 3-Super-HyperGraph). Consider a small company with four employees: V0 = {Hiroko, Yutaka, Haruko, Shinya}. These employees form teams: T1 = {Hiroko,Yutaka}, T2 = {Haruko}, T3 = {Shinya}, so that PS1(V0) ⊇ V1 = {T1, T2, T3}. At the next level, teams assemble into departments: DA = {T1, T2}, DB = {T3}, giving PS2(V0) ⊇ V2 = {DA, DB}. Finally, departments are grouped into divisions (the third power-set): PS3(V0) ⊇ V3 = { {DA, DB}, {DA}, {DB} } . Choose the set of supervertices and superedges as V = { S1, S2, S3 } , E = {S1}, where S1 = {DA, DB}, S2 = {DA}, S3 = {DB}. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 11 of 35 Assign neutrosophic membership degrees to each supervertex Si: TV (S1) = 0.90, IV (S1) = 0.05, FV (S1) = 0.05, TV (S2) = 0.75, IV (S2) = 0.15, FV (S2) = 0.10, TV (S3) = 0.80, IV (S3) = 0.10, FV (S3) = 0.10, all satisfying 0 ≤ TV + IV + FV ≤ 3. Next, for the single superedge S1 and each supervertex Si ∈ V , define: TE(S1, S1) = 0.95, IE(S1, S1) = 0.03, FE(S1, S1) = 0.02, TE(S1, S2) = 0.85, IE(S1, S2) = 0.10, FE(S1, S2) = 0.05, TE(S1, S3) = 0.88, IE(S1, S3) = 0.07, FE(S1, S3) = 0.05. These values obey 0 ≤ TE + IE + FE ≤ 3, TE(e, v) ≤ TV (v), IE(e, v) ≤ IV (v), FE(e, v) ≤ FV (v) for all (e, v). Consequently, the tuple( V, E, TV , IV , FV , TE , IE , FE ) is a valid neutrosophic 3-Super-HyperGraph, modeling the company’s four-level hierarchy under uncertainty. 2.3. Soft Super-HyperGraphs Soft graph models represent a family of subgraphs indexed by a parameter set, where each parameter selects a particular subgraph on a fixed vertex–edge universe [7, 32, 33]. This notion coincides with the theory of soft sets—a parameterized collection of subsets that encodes uncertainty or preference without numerical membership values [19, 34]. Extending this to hypergraphs, a soft hypergraph assigns to each parameter a vertex subset and its induced hyperedges [35–38]. A soft n-Super-HyperGraph carries this idea into the layered setting of an n-Super-HyperGraph, producing a hierarchical, parameter-driven model [39]. Definition 10 (Soft Set). [7, 19] Let U be a universal set and A be a set of attributes. A soft set over U is a pair (F , S), where S ⊆ A and F : S → P(U). Here, P(U) denotes the power set of U . Mathematically, a soft set is represented as: (F , S) = {(α,F(α)) | α ∈ S,F(α) ∈ P(U)}. Each α ∈ S is called a parameter, and F(α) is the set of elements in U associated with α. Definition 11 (Soft Hypergraph). [35–38] Let H = (V,E) be a hypergraph and let C be a nonempty set of parameters. A soft hypergraph over H with parameter set C is a quadruple ( H, C, A, B ) , T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 12 of 35 where A : C → PS(V ), B : C → PS(E), and for each c ∈ C, B(c) ⊆ { e ∈ E : e ⊆ A(c)}. The induced pair ( A(c), B(c) ) is called the soft subhypergraph at parameter c. Example 7 (Monthly Training Programs as a Soft Hypergraph). Consider a training center offering the following courses: V = {Intro to Python, Data Analysis, Machine Learning, Deep Learning}. Three certificate programs are defined by their required courses: E1 = {Intro to Python, Data Analysis}, E2 = {Data Analysis, Machine Learning}, E3 = {Machine Learning, Deep Learning}. Let the parameter set denote the monthly sessions: C = {May, June}. Define the mappings A and B by A(May) = {Intro to Python, Data Analysis, Machine Learning}, B(May) = {E1, E2}, A(June) = {Data Analysis, Deep Learning}, B(June) = ∅. Here B(c) ⊆ {e ∈ E : e ⊆ A(c)} for each c ∈ C, so ( H = (V,E), C, A, B ) is a soft hypergraph showing which programs can run in each month. Definition 12 (Soft n-Super-HyperGraph). Let SuHG(n) = (V,E) be an n-Super-HyperGraph and C a nonempty parameter set. A soft n-Super-HyperGraph is the structure( V, E, C, A, B ) , with A : C → PS(V ), B : C → PS(E), such that for every c ∈ C, A(c) ⊆ V, B(c) ⊆ { e ∈ E : e ⊆ A(c)}. Each ( A(c), B(c) ) forms a sub-Super-HyperGraph of SuHG(n) corresponding to parameter c. Example 8 (Consulting Engagements as a Soft 2-Super-HyperGraph). Consider a small consultancy with four consultants: V0 = {Hiroko, Yutaka, Haruko, Shinya}. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 13 of 35 They form two teams: T1 = {Hiroko,Yutaka}, T2 = {Haruko,Shinya}, and these teams jointly run a division: D = {T1, T2}. Thus the 2-supervertices are V = {T1, T2, D}, and we define superedges that model collaborations: E = { E1, E2, E3 } , where E1 = {T1}, E2 = {T2}, E3 = {T1, T2}. This yields the 2-Super-HyperGraph SuHG(2) = (V,E). Next, let the parameter set represent two ongoing projects: C = {Alpha, Beta}. Define A : C → PS(V ), B : C → PS(E), by A(Alpha) = {T1, D}, B(Alpha) = {E1, E3}, A(Beta) = {T2}, B(Beta) = {E2}. Observe that B(c) ⊆ { e ∈ E : e ⊆ A(c)} for each c ∈ C, so (A(c), B(c)) is indeed a sub-Super-HyperGraph for each project. Concretely: • Project Alpha: involves Team 1 and the Division D. A(Alpha) = {T1, D}, B(Alpha) = {{T1}, {T1, T2}}. • Project Beta: involves only Team 2. A(Beta) = {T2}, B(Beta) = {{T2}}. Hence the tuple ( V, E, C, A, B ) is a soft 2-Super-HyperGraph modeling how different projects select subsets of the com- pany’s hierarchical structure. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 14 of 35 Example 9 (Strategic Initiatives as a Soft 3-Super-HyperGraph). Consider a consultancy with four consultants: V0 = {Hiroko, Yutaka, Haruko, Shinya}. They form two teams: T1 = {Hiroko,Yutaka}, T2 = {Haruko,Shinya}, and these teams assemble into departments: D1 = {T1}, D2 = {T2}. At the third level, departments are grouped into divisions: V3 ⊆ PS3(V0) = { {D1}, {D2}, {D1, D2} } . Select as our supervertices and superedges: V = {S1, S2, S3}, E = {E1, E2}, with S1 = {D1, D2}, S2 = {D1}, S3 = {D2}, E1 = {S1, S2}, E2 = {S1, S3}. Thus SuHG(3) = (V,E) models the firm’s four-level hierarchy. Suppose the firm runs two concurrent strategic initiatives: C = {Alpha, Beta}. Define soft-selection maps A : C → PS(V ), B : C → PS(E), by A(Alpha) = {S1, S2}, B(Alpha) = {E1}, A(Beta) = {S1, S3}, B(Beta) = {E2}. One checks easily that for each c ∈ C, B(c) ⊆ { e ∈ E : e ⊆ A(c)}, so (A(c), B(c)) is a valid sub-Super-HyperGraph of SuHG(3). Concretely: • Initiative Alpha: involves Division S1 (both departments) and S2 (Dept. 1 only), modeling the kick-off phase: A(Alpha) = {S1, S2}, B(Alpha) = {{S1, S2}}. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 15 of 35 • Initiative Beta: involves Division S1 and S3 (Dept. 2), representing a spin-off project: A(Beta) = {S1, S3}, B(Beta) = {{S1, S3}}. Hence the tuple ( V, E, C, A, B ) is a soft 3-Super-HyperGraph, capturing how different initiatives select subsets of the organization’s three-level superstructure. 2.4. Neutrosophic Soft HyperGraph A neutrosophic soft set is a parameterized collection of single-valued neutrosophic sub- sets, modeling uncertainty across distinct elements under each parameter (cf.[59–61]). A neutrosophic soft graph is a set of parameterized neutrosophic assignments to graph ver- tices and edges, capturing uncertain structural preferences (cf.[62–64]). We now introduce the notion of a neutrosophic soft hypergraph, which combines the ideas of single-valued neutrosophic sets and soft sets on the underlying hypergraph (cf.[37, 65]). Definition 13 (Neutrosophic Soft Set). (cf.[59–61]) Let U be a nonempty universe of discourse and let E be a nonempty set of parameters. Fix a subset A ⊆ E. A neutrosophic soft set over U with parameter set A is a pair (F,A), where F : A −→ SNS(U), and for each c ∈ A, F (c) = { ( u, TF (c;u), IF (c;u), FF (c;u) ) : u ∈ U } . Here TF , IF , FF : A× U → [0, 1] are the truth, indeterminacy, and falsity membership functions, satisfying the neutro- sophic constraint 0 ≤ TF (c;u) + IF (c;u) + FF (c;u) ≤ 3 ∀ c ∈ A, ∀u ∈ U. Definition 14 (Neutrosophic Soft Graph). [62–64] Let G∗ = (V,E, TV , IV , FV , TE , IE , FE) be a single-valued neutrosophic graph, where TV , IV , FV : V → [0, 1], TE , IE , FE : E × V → [0, 1], satisfy the usual neutrosophic constraints (see Definition 3). Let C be a nonempty set of parameters. A neutrosophic soft graph over G∗ with parameter set C is a 4-tuple( G∗, C, R, S ) , T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 16 of 35 where R : C −→ SNS(V ), S : C −→ SNS(E), and for each c ∈ C: R(c) = { (v, TR(c; v), IR(c; v), FR(c; v)) : v ∈ V } , S(c) = { (e, TS(c; e), IS(c; e), FS(c; e)) : e ∈ E } , with 0 ≤ TR(c; v) + IR(c; v) + FR(c; v) ≤ 3, 0 ≤ TS(c; e) + IS(c; e) + FS(c; e) ≤ 3, and the following consistency conditions for every e = {u, v} ∈ E: TS(c; e) ≤ min{TR(c;u), TR(c; v)}, IS(c; e) ≤ min{IR(c;u), IR(c; v)}, FS(c; e) ≥ max{FR(c;u), FR(c; v)}. Finally, we require the coverage condition⋃ e∈E supp ( S(c; e) ) = V for each c ∈ C, so that every vertex appears in at least one neutrosophic soft edge under each parameter. Example 10 (Contextual Trust Network as a Neutrosophic Soft Graph). Let V = {Hiroko, Tae, Yusuke} and E = { e1 = {Hiroko,Tae}, e2 = {Tae,Yusuke}}. Define two contexts C = {Work, Friends}. For each context c ∈ C, we assign neutrosophic soft vertex- and edge-memberships R(c) and S(c) as follows: Work context (c = Work): R(Work) = { (Hiroko, 0.90, 0.05, 0.05), (Tae, 0.85, 0.10, 0.05), (Yusuke, 0.80, 0.15, 0.05) } , S(Work) = { (e1, 0.80, 0.10, 0.10), (e2, 0.60, 0.20, 0.20) } . Friends context (c = Friends): R(Friends) = { (Hiroko, 0.70, 0.20, 0.10), (Tae, 0.90, 0.05, 0.05), (Yusuke, 0.95, 0.03, 0.02) } , T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 17 of 35 S(Friends) = { (e1, 0.75, 0.15, 0.10), (e2, 0.85, 0.10, 0.05) } . One verifies for each c ∈ C: 0 ≤ TR(c; v) + IR(c; v) + FR(c; v) ≤ 3, 0 ≤ TS(c; e) + IS(c; e) + FS(c; e) ≤ 3, and for each edge e = {u, v}: TS(c; e) ≤ min{TR(c;u), TR(c; v)}, IS(c; e) ≤ min{IR(c;u), IR(c; v)}, FS(c; e) ≥ max{FR(c;u), FR(c; v)}. Finally, every vertex appears in some soft edge under each context, so ( G∗, C,R, S ) is a valid neutrosophic soft graph modeling context-dependent trust relationships. Definition 15 (Neutrosophic Soft Hypergraph). (cf.[37, 65]) Let H = (V,E) be a fi- nite crisp hypergraph, and let C be a nonempty set of parameters. A neutrosophic soft hypergraph over H with parameters C is a triple( R, S, C ) , together with mappings R : C −→ SNS(V ), S : C −→ SNS(E), where SNS(X) denotes the family of single-valued neutrosophic subsets of X. Concretely, for each c ∈ C: • R(c) = { ( v, TR(c; v), IR(c; v), FR(c; v) ) : v ∈ V } satisfies 0 ≤ TR(c; v) + IR(c; v) + FR(c; v) ≤ 3, ∀ v ∈ V. • S(c) = { ( e, TS(c; e), IS(c; e), FS(c; e) ) : e ∈ E} satisfies 0 ≤ TS(c; e) + IS(c; e) + FS(c; e) ≤ 3, ∀ e ∈ E, and moreover TS(c; e) ≤ min v∈e TR(c; v), IS(c; e) ≤ min v∈e IR(c; v), FS(c; e) ≤ max v∈e FR(c; v). T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 18 of 35 Finally, we require the coverage condition⋃ e∈E supp ( S(c; e) ) = V for each c ∈ C, so that every vertex appears in some neutrosophic soft hyperedge under each parameter. Example 11 (Medical Diagnosis as a Neutrosophic Soft Hypergraph). Consider a medical- diagnosis hypergraph where V = {Fever, Cough, Headache} and two diseases are modeled as hyperedges: E = { Flu = {Fever,Cough,Headache}, Cold = {Cough,Headache} } . Let the parameter set C represent two patients: C = {Patient1, Patient2}. We define neutrosophic-soft mappings R : C → SNS(V ), S : C → SNS(E). For each c ∈ C, R(c) assigns to each symptom v ∈ V a triple ( TR(c; v), IR(c; v), FR(c; v) ) and S(c) assigns to each disease e ∈ E a triple ( TS(c; e), IS(c; e), FS(c; e) ) . Patient 1: R(Patient1) : (Fever, 0.90, 0.05, 0.05), (Cough, 0.80, 0.10, 0.10), (Headache, 0.70, 0.20, 0.10); S(Patient1) : { Flu : (0.85, 0.10, 0.05), Cold : (0.60, 0.25, 0.15). Patient 2: R(Patient2) : (Fever, 0.20, 0.30, 0.50), (Cough, 0.65, 0.20, 0.15), (Headache, 0.50, 0.30, 0.20); S(Patient2) : { Flu : (0.40, 0.30, 0.30), Cold : (0.55, 0.25, 0.20). One checks for each c, each v, 0 ≤ TR + IR + FR ≤ 3, and similarly for S. Moreover, TS(c; e) ≤ min v∈e TR(c; v), IS(c; e) ≤ min v∈e IR(c; v), FS(c; e) ≤ max v∈e FR(c; v). Finally, every symptom appears in at least one soft hyperedge: ⋃ e∈E supp(S(c; e)) = V . Hence ( R,S,C ) is a concrete neutrosophic soft hypergraph modeling uncertain symp- tom–disease relationships in two patient cases. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 19 of 35 3. Main Results This section presents the main results of the paper. 3.1. Neutrosophic Soft n-Super-HyperGraph We provide below the formal definition of a Neutrosophic Soft n-Super-HyperGraph. This concept applies the principles of neutrosophic sets and soft sets to the framework of n-Super-HyperGraphs. Definition 16 (Neutrosophic Soft n-Super-HyperGraph). Let SuHG(n) = (V,E) be an n- Super-HyperGraph on base set V0, and let C be a nonempty parameter set. A neutrosophic soft n-Super-HyperGraph is a septuple( V, E, C, A, B, TV , IV , FV , TE , IE , FE ) , where • A : C → PS(V ) and B : C → PS(E) satisfy A(c) ⊆ V, B(c) ⊆ { e ∈ E : e ⊆ A(c)}, ∀ c ∈ C. • TV , IV , FV : C × V → [0, 1] assign to each (c, v) its truth, indeterminacy, and falsity degrees, with 0 ≤ TV (c, v) + IV (c, v) + FV (c, v) ≤ 3 ∀ c ∈ C, v ∈ V. • TE , IE , FE : C × E × V → [0, 1] assign to each triple (c, e, v) its neutrosophic mem- bership, subject to 0 ≤ TE(c, e, v) + IE(c, e, v) + FE(c, e, v) ≤ 3 ∀ c ∈ C, e ∈ E, v ∈ V. Moreover, the following containment constraints must hold for all c ∈ C, e ∈ E, and v ∈ V : TE(c, e, v) ≤ TV (c, v), IE(c, e, v) ≤ IV (c, v), FE(c, e, v) ≤ FV (c, v). Example 12 (Disaster Response Network as a Neutrosophic Soft 2-Super-HyperGraph). Consider an emergency response system composed of four local units: V0 = {Unit A, Unit B, Unit C, Unit D}. They form two squads: Squad1 = {Unit A,Unit B}, Squad2 = {Unit C,Unit D}, T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 20 of 35 so that PS1(V0) ⊇ V1 = {Squad1,Squad2}. At the next level, these squads combine into a regional command: PS2(V0) ⊇ V2 = {Region}, Region = {Squad1,Squad2}. We take the single superedge to model a coordinated operation: E2 = {Operation}, Operation = {Squad1,Squad2}. Thus SuHG(2) = (V2, E2) is our 2-Super-HyperGraph. Next, let the parameter set represent two disaster scenarios: C = {Earthquake, Flood}. We define A : C → PS(V2), B : C → PS(E2), by A(Earthquake) = {Region, Squad1}, B(Earthquake) = {Operation}, A(Flood) = {Region, Squad2}, B(Flood) = {Operation}. Clearly A(c) ⊆ V2 and B(c) ⊆ {e : e ⊆ A(c)} for each c. Finally, assign neutrosophic-soft membership degrees: Vertex memberships (TV , IV , FV ): Earthquake Flood v TV IV FV TV IV FV Region 0.85 0.10 0.05 0.80 0.15 0.05 Squad1 0.90 0.08 0.02 0.60 0.30 0.10 Squad2 0.70 0.20 0.10 0.88 0.10 0.02 Each row sums to at most 3, satisfying 0 ≤ TV + IV + FV ≤ 3. Edge–vertex memberships (TE , IE , FE): Earthquake Flood v TE IE FE TE IE FE Squad1 0.88 0.10 0.02 0 0 0 Squad2 0.80 0.15 0.05 0.85 0.12 0.03 Entries for which v /∈ A(c) are zero. One checks for each (c, e, v): TE(c, e, v) ≤ TV (c, v), IE(c, e, v) ≤ IV (c, v), FE(c, e, v) ≤ FV (c, v). Therefore, the tuple ( V2, E2, C, A, B, TV , IV , FV , TE , IE , FE ) is a concrete neutrosophic soft 2-Super-HyperGraph modeling how different disaster sce- narios select and evaluate subsets of the response network under uncertainty. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 21 of 35 Example 13 (Autonomous Vehicle Sensor Fusion as a Neutrosophic Soft 1-Super-HyperGraph). In an autonomous driving system, four sensor types collect environmental data: V0 = {Camera, Lidar, Radar, Microphone}. We group them into three sensor clusters: Svision = {Camera, Lidar}, Sradar = {Radar}, Saudio = {Microphone}, so that our level-1 supervertices are V = {Svision, Sradar, Saudio }. We model the fusion module as a single hyperedge linking all clusters: E = { efusion}, efusion = {Svision, Sradar, Saudio }. Thus SuHG(1) = (V,E). Different driving scenarios induce uncertainty in sensor reliability. Let C = {Highway, Urban} be our parameter set. Define A : C → PS(V ), B : C → PS(E), by A(Highway) = {Svision, Sradar}, B(Highway) = {efusion}, A(Urban) = {Svision, Saudio}, B(Urban) = {efusion}. Clearly B(c) ⊆ {e ∈ E : e ⊆ A(c)} for each c. We now assign neutrosophic-soft memberships: Vertex memberships (TV , IV , FV ): Highway Urban Cluster TV IV FV TV IV FV Svision 0.95 0.03 0.02 0.90 0.07 0.03 Sradar 0.92 0.05 0.03 0 0 0 Saudio 0 0 0 0.85 0.10 0.05 Each row meets 0 ≤ TV + IV + FV ≤ 3. A zero entry reflects a cluster not used in that scenario. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 22 of 35 Edge–vertex memberships (TE , IE , FE): For the single fusion edge efusion, we record memberships only for clusters in A(c): Highway Urban Cluster TE IE FE TE IE FE Svision 0.93 0.05 0.02 0.88 0.08 0.04 Sradar 0.90 0.07 0.03 0 0 0 Saudio 0 0 0 0.82 0.12 0.06 One checks 0 ≤ TE + IE +FE ≤ 3 and the containment constraints TE(c, e, v) ≤ TV (c, v), IE(c, e, v) ≤ IV (c, v), and FE(c, e, v) ≤ FV (c, v) for each (c, efusion, v). Hence the tuple ( V, E, C, A, B, TV , IV , FV , TE , IE , FE ) constitutes a concrete neutrosophic soft 1-Super-HyperGraph capturing how sensor clus- ters are selected and weighted under uncertainty in different driving scenarios. Example 14 (Autonomous Robot Control as a Neutrosophic Soft 2-Super-HyperGraph). We model an autonomous robot’s software architecture under varying mission contexts. Base Modules (level 0): V0 = {Perception, Planning, Communication}. Level-1 supervertices (modules): V1 = { Perception, Planning, Communication}. Level-2 supervertices (functional clusters): Nav = {Perception, Planning}, CommOps = {Perception, Communication}. V2 = {Nav, CommOps}. Level-2 superedges (operation modes): efull = {Nav, CommOps}, (full system mode) enav = {Nav}, (navigation-only mode) E2 = { efull, enav }. Thus SuHG(2) = (V2, E2) is our 2-Super-HyperGraph. Parameters (missions): C = {Rescue, Survey}. Define the selection maps A : C → PS(V2), B : C → PS(E2), T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 23 of 35 by A(Rescue) = {Nav, CommOps}, B(Rescue) = {efull}, A(Survey) = {Nav}, B(Survey) = {enav}. One checks A(c) ⊆ V2 and B(c) ⊆ {e : e ⊆ A(c)}. Neutrosophic-soft memberships (TV , IV , FV ): Rescue Survey v TV IV FV TV IV FV Nav 0.90 0.05 0.05 0.80 0.15 0.05 CommOps 0.85 0.10 0.05 0 0 0 Each row satisfies 0 ≤ TV + IV + FV ≤ 3. zeros reflect unused clusters. Neutrosophic-soft memberships (TE , IE , FE): Rescue, efull Survey, enav v TE IE FE TE IE FE Nav 0.88 0.10 0.02 0.78 0.20 0.02 CommOps 0.82 0.15 0.03 0 0 0 One verifies for all (c, e, v): 0 ≤ TE+IE+FE ≤ 3, TE(c, e, v) ≤ TV (c, v), IE(c, e, v) ≤ IV (c, v), FE(c, e, v) ≤ FV (c, v). Hence the tuple ( V2, E2, C, A, B, TV , IV , FV , TE , IE , FE ) is a detailed neutrosophic soft 2-Super-HyperGraph representing how an autonomous robot’s functional clusters are selected and weighted under different mission scenarios. Theorem 1 (Generalization of Neutrosophic and Soft Models). Every neutrosophic n- Super-HyperGraph and every soft n-Super-HyperGraph can be realized as a special case of a neutrosophic soft n-Super-HyperGraph. Proof. We split the proof into two parts. (i) Embedding a neutrosophic n-Super-HyperGraph. Suppose we are given a neutrosophic n-Super-HyperGraph Hneut = (V, E, T̃V , ĨV , F̃V , T̃E , ĨE , F̃E). Define a neutrosophic soft structure by choosing C = {∗}, A(∗) = V, B(∗) = E, and let TV (∗, v) = T̃V (v), IV (∗, v) = ĨV (v), FV (∗, v) = F̃V (v), T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 24 of 35 TE(∗, e, v) = T̃E(e, v), IE(∗, e, v) = ĨE(e, v), FE(∗, e, v) = F̃E(e, v). All parameter-indexed constraints reduce to those of Hneut. In particular, since A(∗) = V and B(∗) = E, the containment constraints TE(∗, e, v) ≤ TV (∗, v), . . . are exactly those in the original neutrosophic model. Thus Hneut embeds as a neutrosophic soft n-Super-HyperGraph with a single parameter. (ii) Embedding a soft n-Super-HyperGraph. Now let Hsoft = (V, E, C, A, B) be a soft n-Super-HyperGraph. We define neutrosophic memberships by TV (c, v) = { 1, v ∈ A(c), 0, v /∈ A(c), IV (c, v) = FV (c, v) = 0, and TE(c, e, v) = { 1, e ∈ B(c) and v ∈ e, 0, otherwise, IE(c, e, v) = FE(c, e, v) = 0. Then 0 ≤ TV + IV + FV ≤ 3 and 0 ≤ TE + IE + FE ≤ 3 hold trivially. Moreover, if e ∈ B(c) and v ∈ e ⊆ A(c), then TE(c, e, v) = 1 ≤ TV (c, v) = 1; all other containment constraints hold since both sides vanish. Hence Hsoft arises as a neutrosophic soft n-Super- HyperGraph with binary membership values. Combining (i) and (ii), we see that the neutrosophic soft n-Super-HyperGraph frame- work simultaneously extends both the pure neutrosophic and the soft models. Theorem 2 (Neutrosophic Soft n-Super-HyperGraph Generalizes Neutrosophic Soft Hy- pergraph). Let ( H = (V,E), C, R, S ) be any neutrosophic soft hypergraph. Then it can be viewed as a neutrosophic soft 1-Super-HyperGraph by setting SuHG(1) = (V,E), A = C, and defining the neutrosophic soft membership functions on the level-1 supervertices and superedges by TV (c; v) = TR(c; v), IV (c; v) = IR(c; v), FV (c; v) = FR(c; v), TE(c; e, v) = { TS(c; e), v ∈ e, 0, v /∈ e, IE(c; e, v) = { IS(c; e), v ∈ e, 0, v /∈ e, T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 25 of 35 FE(c; e, v) = { FS(c; e), v ∈ e, 0, v /∈ e. Then the resulting tuple ( V, E, C, TV , IV , FV , TE , IE , FE ) satisfies all the axioms of a neutrosophic soft 1-Super-HyperGraph, and reproduces exactly the original neutrosophic soft hypergraph under the identification of level-1 superstructures with (V,E). Proof. Let (H,C,R, S) be given. We check each requirement in the definition of a neutrosophic soft 1-Super-HyperGraph: (i) Parameterized vertex sets. By construction, for every c ∈ C, A(c) = V and TV (c; v) = TR(c; v), IV (c; v) = IR(c; v), FV (c; v) = FR(c; v), so 0 ≤ TV (c; v) + IV (c; v) + FV (c; v) ≤ 3 ∀ v ∈ V. (ii) Parameterized edge–vertex memberships. For each triple (c, e, v), define TE(c; e, v), IE(c; e, v), FE(c; e, v) as above. If v /∈ e, all three are zero, and hence 0 ≤ TE + IE + FE ≤ 3. If v ∈ e, then TE(c; e, v) = TS(c; e) ≤ min u∈e TR(c;u) = min u∈e TV (c;u) ≤ TV (c; v), and similarly IE(c; e, v) ≤ IV (c; v), FE(c; e, v) ≤ FV (c; v). Thus the containment con- straints hold. (iii) Coverage condition. Since in the original neutrosophic soft hypergraph each pa- rameter c satisfies ⋃ e∈E supp ( S(c; e) ) = V , it follows immediately that in the level-1 Super-HyperGraph each vertex v has TV (c; v) > 0 for some edge–vertex pair (e, v). All remaining axioms (non-negativity, upper bound 3, etc.) follow at once from the corresponding properties of R and S. Hence the construction yields a valid neutrosophic soft 1-Super-HyperGraph whose projections onto the original vertex and edge sets recover exactly the given neutrosophic soft hypergraph. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 26 of 35 Algorithm 1 Constructing a Neutrosophic Soft n-Super-HyperGraph Require: Base set V0, integer n > 0, parameters C, soft–selection A,B : C → PSn(V0) with B(c) ⊆ {e ⊆ A(c)}, initial neutrosophic maps T 0 V , I0V , F 0 V : V0 → [0, 1], T 0 E , I 0 E , F 0 E : En × Vn → [0, 1]. Ensure: Neutrosophic soft n-Super-HyperGraph H∗ = (Vn, En, C,A,B, TV , IV , FV , TE , IE , FE). 1: // 1. build underlying n-super-hypergraph 2: V ′ ← V0 3: for k = 1 to n do 4: Vk ← PS(V ′), Ek ← Vk 5: V ′ ← Vk 6: end for 7: Vn ← Vn, En ← En 8: // 2. initialize neutrosophic memberships 9: for each c ∈ C do 10: for each v ∈ Vn do 11: ( TV , IV , FV ) (c, v)← { (T 0 V , I0V , F 0 V )(v), v ∈ A(c) (0, 0, 0), otherwise. 12: end for 13: for each e ∈ En do 14: for each v ∈ Vn do 15: (TE , IE , FE)(c, e, v)←  ( min(T 0 E(e, v), TV (c, v)), min(I0E(e, v), IV (c, v)), min(F 0 E(e, v), FV (c, v)) ) , v ∈ e, (0, 0, 0), v /∈ e. 16: end for 17: end for 18: end for 19: // 3. enforce neutrosophic constraints 20: for each (c, v) do 21: normalize if TV + IV + FV > 3 22: end for 23: for each (c, e, v) do 24: normalize if TE + IE + FE > 3 25: end for 26: return H∗ 4. Additional Result: Algorithm for Constructing a Neutrosophic Soft n-Super-HyperGraph In this section, we present the algorithm for constructing a Neutrosophic Soft n-Super- HyperGraph. The algorithm 1 is given below. Example 15 (Applying Algorithm 1 to a Disaster-Response Network). We illustrate each step of the construction algorithm in the context of a two–level emergency response system. 1. Problem data. V0 = {UnitA, UnitB, UnitC, UnitD}, n = 2, C = {Earthquake, Flood}. Define the soft-selection maps A(c), B(c) ⊆ PS2(V0) T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 27 of 35 by Earthquake : A = {Region, Squad1}, B = {Operation}, Flood : A = {Region, Squad2}, B = {Operation}, where Squad1 = {UnitA,UnitB}, Squad2 = {UnitC,UnitD}, Region = {Squad1, Squad2}, and Operation = Region. Initial neutrosophic maps on V0: T 0 V (UnitA) = 0.90, I0V (UnitA) = 0.05, F 0 V (UnitA) = 0.05, T 0 V (UnitB) = 0.85, I0V (UnitB) = 0.10, F 0 V (UnitB) = 0.05, T 0 V (UnitC) = 0.80, I0V (UnitC) = 0.15, F 0 V (UnitC) = 0.05, T 0 V (UnitD) = 0.75, I0V (UnitD) = 0.20, F 0 V (UnitD) = 0.05. On the unique 2-superedge Operation and each supervertex v ∈ {Squad1, Squad2}, set for example T 0 E(Operation, v) = T 0 V ( any member of v ) , I0E(Operation, v) = I0V ( any member of v ) , F 0 E(Operation, v) = F 0 V ( any member of v ) . 2. Build underlying 2-Super-HyperGraph. By iterating the power-set: V1 = PS(V0), E1 = V1, V2 = PS(V1), E2 = V2. We restrict to the chosen supervertices V ∗ 2 = {Region, Squad1 . . .} and superedges E∗ 2 = {Operation}. 3. Initialize neutrosophic memberships. For each scenario c ∈ C and each v ∈ V ∗ 2 : (TV , IV , FV )(c, v) = { (T 0 V , I 0 V , F 0 V )(v ′), v′ ∈ v ⊆ A(c), (0, 0, 0), otherwise. For the single superedge Operation and each supervertex v ∈ V ∗ 2 : (TE , IE , FE)(c,Operation, v) = {( min(T 0 E , I 0 E , F 0 E),min,min ) , v ∈ B(c), (0, 0, 0), otherwise. 4. Enforce neutrosophic constraints. Normalize any triple whose sum exceeds 3. In our numeric assignments above, all sums remain ≤ 3. Result. The algorithm yields H∗ = ( V ∗ 2 , E ∗ 2 , C, A, B, TV , IV , FV , TE , IE , FE ) , a fully specified neutrosophic soft 2-Super-HyperGraph modeling how different disaster sce- narios select and weight response clusters under uncertainty. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 28 of 35 Theorem 3 (Correctness of Algorithm 1). The output H∗ = (Vn, En, C,A,B, TV , IV , FV , TE , IE , FE) satisfies all the axioms of a neutrosophic soft n-Super-HyperGraph. Proof. By construction: (i) Underlying n-Super-HyperGraph: Step 1 computes Vk = PS(Vk−1), Ek = Vk (1 ≤ k ≤ n), so that Vn, En ⊆ PSn(V0), as required. (ii) Soft selection maps: We leave A : C → PSn(V0) and B : C → PSn(V0) unchanged, and the algorithm respects B(c) ⊆ {e : e ⊆ A(c)} by hypothesis. (iii) Vertex memberships: For each c ∈ C and v ∈ Vn, (TV , IV , FV )(c, v) = { (T 0 V , I 0 V , F 0 V )(v), v ∈ A(c), (0, 0, 0), v /∈ A(c). Hence 0 ≤ TV (c, v) + IV (c, v) + FV (c, v) ≤ 3. (iv) Edge–vertex memberships: For each c, e, v, (TE , IE , FE)(c, e, v) = ( min(T 0 E(e, v), TV (c, v)), min(I0E(e, v), IV (c, v)), min(F 0 E(e, v), FV (c, v)) ) , v ∈ e, (0, 0, 0), v /∈ e. By construction, TE(c, e, v) ≤ TV (c, v), IE(c, e, v) ≤ IV (c, v), and FE(c, e, v) ≤ FV (c, v), and each sum lies in [0, 3]. (v) Normalization: Step 3 rescales any triple whose sum exceeds 3, restoring the bound 0 ≤ T + I + F ≤ 3. No other axioms are violated. Thus H∗ is a valid neutrosophic soft n-Super-HyperGraph. Theorem 4 (Time Complexity of Algorithm 1). Let N0 = |V0| and define Nk = |PSk(V0)| for 1 ≤ k ≤ n. Then the algorithm runs in time O ( n∑ k=1 Nk−1Nk + |C| (Nn +N2 n) ) , which is exponential in n (and doubly exponential in N0). T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 29 of 35 Proof. • Step 1: Computing each power-set PS(V ′) of size Nk−1 requires O(Nk−12 Nk−1) = O(Nk−1Nk) time. Summing over k gives the first term. • Step 2: For each c ∈ C, assigning (TV , IV , FV ) to all Nn vertices costs O(Nn), and assigning (TE , IE , FE) to all Nn edges and their Nn vertices costs O(N2 n). Hence O(|C|(Nn +N2 n)). • Step 3: A final pass over all (c, v) and (c, e, v) again costs O(|C|(Nn +N2 n)), which is absorbed into the previous bound. Since Nk = 2Nk−1 , the overall complexity grows exponentially in n (and doubly exponen- tially in |V0|). 5. Conclusion In this paper, we have introduced the Neutrosophic Soft Super-HyperGraph, a novel framework that synergistically combines neutrosophic logic, soft set theory, and Super- HyperGraph structures to capture both hierarchical interdependencies and uncertainty in complex systems. By unifying hierarchical connectivity, parameterized substructure selec- tion, and neutrosophic uncertainty, our model enables flexible multi-level representations, targeted subgraph analysis, and informed decision-making under uncertain and large-scale network conditions. For future work, we intend to extend this framework to additional graph general- izations, including bidirected graphs [66–68], multigraphs [69, 70], pseudographs [71], and multidirected graphs [72, 73]. Further extensions of the Neutrosophic Soft Super- HyperGraph could also leverage HyperUncertain Sets[22, 74, 75], HyperRough Sets[76, 77], HyperWeighted Sets[78], and HyperSoft Sets[79, 80]. Moreover, we will perform empiri- cal evaluations on real-world datasets using appropriate computational tools to assess the practical applicability, scalability, and robustness of the proposed approach. Acknowledgments We wish to thank everyone whose guidance, ideas, and assistance contributed to this research. Our appreciation goes to the readers for their interest and to the scholars whose publications provided the groundwork for this study. We are also indebted to the individuals and institutions that supplied the resources and infrastructure necessary for completing and disseminating this paper. Lastly, we extend our gratitude to all who offered their support in various capacities. Data Availability This paper is purely theoretical and does not involve any empirical data. We welcome future empirical studies that build upon and test the concepts presented here. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 30 of 35 Ethical Approval As this work is entirely conceptual and involves no human or animal subjects, ethical approval was not required. Conflicts of Interest The authors declare no conflicts of interest in connection with this study or its publi- cation. Funding No external or organizational funding supported this work. Author Contributions This paper was solely authored by the corresponding author. Research Integrity The authors affirm that, to the best of their knowledge, this manuscript represents their original research. It has not been previously published in any journal, nor is it currently being considered for publication elsewhere. Disclaimer on Computational Tools No computer-based tools—such as symbolic computation systems, automated theorem provers, or proof assistants (e.g., Mathematica, SageMath, Coq)—were employed in the development, analysis, or verification of the results contained in this paper. All derivations and proofs were conducted manually through analytical methods by the authors. Disclaimer on Scope and Accuracy The theoretical models and concepts proposed in this manuscript have not yet under- gone empirical testing or practical deployment. Future work may investigate their utility in applied or experimental contexts. While the authors have taken care to maintain ac- curacy and provide appropriate citations, inadvertent errors or omissions may remain. Readers are encouraged to consult original references for confirmation and further study. The authors assert that all mathematical results and justifications included in this work have been carefully reviewed and are believed to be correct. Should any inaccuracies or ambiguities be discovered, the authors welcome constructive feedback and will provide clarification upon request. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 31 of 35 The conclusions presented are valid only within the specific theoretical framework and assumptions described in the text. Generalizing these results to other mathematical contexts may require further investigation. All opinions and interpretations expressed herein are solely those of the authors and do not necessarily reflect the views of their respective institutions. Consent to Publish All authors have given their consent for submission of this manuscript to the journal. References [1] Reinhard Diestel. Graph theory. Springer (print edition); Reinhard Diestel (eBooks), 2024. [2] Jonathan L Gross, Jay Yellen, and Mark Anderson. Graph theory and its applications. Chapman and Hall/CRC, 2018. [3] Claude Berge. Hypergraphs: combinatorics of finite sets, volume 45. Elsevier, 1984. [4] Florentin Smarandache. Introduction to the n-SuperHyperGraph-the most general form of graph today. Infinite Study, 2022. [5] Mohammad Hamidi, Florentin Smarandache, and Mohadeseh Taghinezhad. Decision Making Based on Valued Fuzzy Superhypergraphs. Infinite Study, 2023. [6] Lotfi A Zadeh. Fuzzy sets. Information and control, 8(3):338–353, 1965. [7] Dmitriy Molodtsov. Soft set theory-first results. Computers & mathematics with applications, 37(4-5):19–31, 1999. [8] Krassimir T Atanassov and Krassimir T Atanassov. Intuitionistic fuzzy sets. Springer, 1999. [9] Krassimir T Atanassov. On intuitionistic fuzzy sets theory, volume 283. Springer, 2012. [10] Krassimir T Atanassov. Circular intuitionistic fuzzy sets. Journal of Intelligent & Fuzzy Systems, 39(5):5981–5986, 2020. [11] Reinhard Diestel. Graph theory 3rd ed. Graduate texts in mathematics, 173(33):12, 2005. [12] Yifan Feng, Haoxuan You, Zizhao Zhang, Rongrong Ji, and Yue Gao. Hypergraph neural networks. In Proceedings of the AAAI conference on artificial intelligence, volume 33, pages 3558–3565, 2019. [13] Derun Cai, Moxian Song, Chenxi Sun, Baofeng Zhang, Shenda Hong, and Hongyan Li. Hypergraph structure learning for hypergraph neural networks. In IJCAI, pages 1923–1929, 2022. [14] Yifan Feng, Jiashu Han, Shihui Ying, and Yue Gao. Hypergraph isomorphism com- putation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2024. [15] Florentin Smarandache. Extension of HyperGraph to n-SuperHyperGraph and to Plithogenic n-SuperHyperGraph, and Extension of HyperAlgebra to n-ary (Classical- /Neutro-/Anti-) HyperAlgebra. Infinite Study, 2020. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 32 of 35 [16] Takaaki Fujita and Florentin Smarandache. Directed n-superhypergraphs incorpo- rating bipolar fuzzy information: A multi-tier framework for modeling bipolar uncer- tainty in complex networks. Neutrosophic Sets and Systems, 88:164–183, 2025. [17] Takaaki Fujita and Florentin Smarandache. Fundamental computational problems and algorithms for superhypergraphs. HyperSoft Set Methods in Engineering, 3:32– 61, 2025. [18] N. B. Nalawade, M. S. Bapat, S. G. Jakkewad, G. A. Dhanorkar, and D. J. Bhosale. Structural properties of zero-divisor hypergraph and superhypergraph over Zn: Girth and helly property. Panamerican Mathematical Journal, 35(4S):485, 2025. [19] Pradip Kumar Maji, Ranjit Biswas, and A Ranjan Roy. Soft set theory. Computers & mathematics with applications, 45(4-5):555–562, 2003. [20] W-L Gau and Daniel J Buehrer. Vague sets. IEEE transactions on systems, man, and cybernetics, 23(2):610–614, 1993. [21] Humberto Bustince and P Burillo. Vague sets are intuitionistic fuzzy sets. Fuzzy sets and systems, 79(3):403–405, 1996. [22] Florentin Smarandache. Hyperuncertain, superuncertain, and superhyperuncertain sets/logics/probabilities/statistics. Infinite Study, 2017. [23] Florentin Smarandache. A unifying field in logics: Neutrosophic logic. In Philosophy, pages 1–141. American Research Press, 1999. [24] Madeleine Al Tahan, Saba Al-Kaseasbeh, and Bijan Davvaz. Neutrosophic quadruple hv-modules and their fundamental module. Neutrosophic Sets and Systems, 72:304– 325, 2024. [25] Florentin Smarandache. Plithogenic set, an extension of crisp, fuzzy, intuitionistic fuzzy, and neutrosophic sets-revisited. Infinite study, 2018. [26] Muhammad Azeem, Humera Rashid, Muhammad Kamran Jamil, Selma Gütmen, and Erfan Babaee Tirkolaee. Plithogenic fuzzy graph: A study of fundamental properties and potential applications. Journal of Dynamics and Games, pages 0–0, 2024. [27] Azriel Rosenfeld. Fuzzy graphs. In Fuzzy sets and their applications to cognitive and decision processes, pages 77–95. Elsevier, 1975. [28] Hossein Rashmanlou, Sovan Samanta, Madhumangal Pal, and Rajab Ali Borzooei. Intuitionistic fuzzy graphs with categorical properties. Fuzzy information and Engi- neering, 7(3):317–334, 2015. [29] Said Broumi, Mohamed Talea, Assia Bakali, and Florentin Smarandache. Single valued neutrosophic graphs. Journal of New theory, (10):86–101, 2016. [30] Said Broumi, Mohamed Talea, Assia Bakali, Florentin Smarandache, and PK Kishore Kumar. Shortest path problem on single valued neutrosophic graphs. In 2017 interna- tional symposium on networks, computers and communications (ISNCC), pages 1–6. IEEE, 2017. [31] Takaaki Fujita. A short note on the basic graph construction algorithm for plithogenic graphs. Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond, page 274, 2025. [32] R. Jahir Hussain and M. S. Afya Farhana. Fuzzy chromatic number of fuzzy soft cycle and complete fuzzy soft graphs. AIP Conference Proceedings, 2023. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 33 of 35 [33] Jyoti D Thenge, B Surendranath Reddy, and Rupali S Jain. Contribution to soft graph and soft tree. New Mathematics and Natural Computation, 15(01):129–143, 2019. [34] P Thangavelu, Saeid Jafari, et al. On quasi soft sets in soft topology. Tamsui Oxford Journal of Information and Mathematical Sciences, 2018. [35] Abbas Amini, Narjes Firouzkouhi, Ahmad Gholami, Anju R Gupta, Chun Cheng, and Bijan Davvaz. Soft hypergraph for modeling global interactions via social media networks. Expert Systems with Applications, 203:117466, 2022. [36] BOBIN GEORGE, K THUMBAKARA RAJESH, and JOSE JINTA. A study on soft hypergraphs and their and & or operations. Journal of the Calcutta Mathematical Society, 19(1):29–44, 2023. [37] Muhammad Akram and Hafiza Saba Nawaz. Implementation of single-valued neu- trosophic soft hypergraphs on human nervous system. Artificial Intelligence Review, 56(2):1387–1425, 2023. [38] KK Myithili and RD Beulah. Recognizing emotions using elementary hyperedges in intuitionistic fuzzy soft hypergraphs. 2021. [39] Takaaki Fujita. Review of some superhypergraph classes: Directed, bidirected, soft, and rough. Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond (Second Vol- ume), 2024. [40] Muhammad Akram, Sundas Shahzadi, and AB Saeid. Single-valued neutrosophic hypergraphs. TWMS Journal of Applied and Engineering Mathematics, 8(1):122– 135, 2018. [41] Alain Bretto. Hypergraph theory. An introduction. Mathematical Engineering. Cham: Springer, 1, 2013. [42] Yue Gao, Zizhao Zhang, Haojie Lin, Xibin Zhao, Shaoyi Du, and Changqing Zou. Hypergraph learning: Methods and practices. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(5):2548–2566, 2020. [43] Muhammad Akram and Muzzamal Sitara. Decision-making with q-rung orthopair fuzzy graph structures. Granular Computing, 7(3):505–526, 2022. [44] Muhammad Akram and Gulfam Shahzadi. Hypergraphs in m-polar fuzzy environ- ment. Mathematics, 6(2):28, 2018. [45] Takaaki Fujita. Directed acyclic superhypergraphs (dash): A general framework for hierarchical dependency modeling. Neutrosophic Knowledge, 6:72–86, 2025. [46] Takaaki Fujita. Review of probabilistic hypergraph and probabilistic superhyper- graph. Spectrum of Operational Research, 3(1):319–338, 2025. [47] Takaaki Fujita and Florentin Smarandache. A concise study of some superhypergraph classes. Neutrosophic Sets and Systems, 77:548–593, 2024. [48] Florentin Smarandache. SuperHyperFunction, SuperHyperStructure, Neutrosophic SuperHyperFunction and Neutrosophic SuperHyperStructure: Current understanding and future directions. Infinite Study, 2023. [49] Ajoy Kanti Das, Rajat Das, Suman Das, Bijoy Krishna Debnath, Carlos Granados, Bimal Shil, and Rakhal Das. A comprehensive study of neutrosophic superhyper bci- T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 34 of 35 semigroups and their algebraic significance. Transactions on Fuzzy Sets and Systems, 8(2):80, 2025. [50] Mohammad Hamidi, Florentin Smarandache, and Elham Davneshvar. Spectrum of superhypergraphs via flows. Journal of Mathematics, 2022(1):9158912, 2022. [51] Florentin Smarandache. Introduction to the n-SuperHyperGraph-the most general form of graph today. Infinite Study, 2022. [52] HaibinWang, Florentin Smarandache, Yanqing Zhang, and Rajshekhar Sunderraman. Single valued neutrosophic sets. Infinite study, 2010. [53] Juanjuan Ding, Wenhui Bai, and Chao Zhang. A new multi-attribute decision mak- ing method with single-valued neutrosophic graphs. International Journal of Neutro- sophic Science, 2021. [54] M Hamidi and A Borumand Saeid. Accessible single-valued neutrosophic graphs. Journal of Applied Mathematics and Computing, 57:121–146, 2018. [55] Muhammad Aslam Malik, Ali Hassan, Said Broumi, Assia Bakali, Mohamed Talea, and Florentin Smarandache. Isomorphism of bipolar single valued neutrosophic hyper- graphs. Collected Papers. Volume IX: On Neutrosophic Theory and Its Applications in Algebra, page 72, 2022. [56] Muhammad Akram and Anam Luqman. Intuitionistic single-valued neutrosophic hypergraphs. Opsearch, 54:799–815, 2017. [57] Anam Luqman, Muhammad Akram, and Florentin Smarandache. Complex neutro- sophic hypergraphs: New social network models. Algorithms, 12:234, 2019. [58] Muhammad Akram, Sundas Shahzadi, and Arsham Borumand Saeid. Single-valued neutrosophic hypergraphs. viXra, pages 1–14, 2018. [59] Sumbal Ali, Asad Ali, Ahmad Bin Azim, Ahmad Aloqaily, and Nabil Mlaiki. Utilizing aggregation operators based on q-rung orthopair neutrosophic soft sets and their applications in multi-attributes decision making problems. Heliyon, 2024. [60] Shawkat Alkhazaleh. n-valued refined neutrosophic soft set theory. Journal of Intel- ligent & Fuzzy Systems, 32(6):4311–4318, 2017. [61] Mumtaz Ali, Le Hoang Son, Irfan Deli, and Nguyen Dang Tien. Bipolar neutrosophic soft sets and applications in decision making. J. Intell. Fuzzy Syst., 33:4077–4087, 2017. [62] Vakkas Ulucay. Q-neutrosophic soft graphs in operations management and commu- nication network. Soft Computing, 25:8441 – 8459, 2021. [63] Muhammad Akram and Sundas Shahzadi. Neutrosophic soft graphs with application. J. Intell. Fuzzy Syst., 32:841–858, 2017. [64] Satham Hussain, Jahir Hussain, Isnaini Rosyida, and Said Broumi. Quadripartitioned neutrosophic soft graphs. In Handbook of Research on Advances and Applications of Fuzzy Sets and Logic, pages 771–795. IGI Global, 2022. [65] Muhammad Akram and Hafiza Saba Nawaz. Algorithms for the computation of regular single-valued neutrosophic soft hypergraphs applied to supranational asian bodies. Journal of Applied Mathematics and Computing, 68(6):4479–4506, 2022. [66] Laura Gellert and Raman Sanyal. On degree sequences of undirected, directed, and bidirected graphs. European Journal of Combinatorics, 64:113–124, 2017. T. Fujita, F.Smarandache / Eur. J. Pure Appl. Math, 18 (3) (2025), 6621 35 of 35 [67] Takaaki Fujita. Review of plithogenic directed, mixed, bidirected, and pangene off- graph. Advancing Uncertain Combinatorics through Graphization, Hyperization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond, page 120, 2024. [68] Rui Xu and Cun-Quan Zhang. On flows in bidirected graphs. Discrete mathematics, 299(1-3):335–343, 2005. [69] Hiroshi Nagamochi, Takashi Shiraki, and Toshihide Ibaraki. Augmenting a submod- ular and posi-modular set function by a multigraph. Journal of Combinatorial Opti- mization, 5:175–212, 2001. [70] Mordechai Lewin. On intersection multigraphs of hypergraphs. Journal of Combina- torial Theory, Series B, 34(2):229–232, 1983. [71] Hongye Yang, Yuzhang Gu, Jianchao Zhu, Keli Hu, and Xiaolin Zhang. Pgcn-tca: Pseudo graph convolutional network with temporal and channel-wise attention for skeleton-based action recognition. IEEE Access, 8:10040–10047, 2020. [72] Sebastian Pardo-Guerra, Vivek Kurien George, and Gabriel A Silva. On the graph isomorphism completeness of directed and multidirected graphs. Mathematics, 13(2):228, 2025. [73] Sebastian Pardo-Guerra, Vivek Kurien George, Vikash Morar, Joshua Roldan, and Gabriel Alex Silva. Extending undirected graph techniques to directed graphs via category theory. Mathematics, 12(9):1357, 2024. [74] Takaaki Fujita. The hyperfuzzy vikor and hyperfuzzy dematel methods for multi- criteria decision-making. Spectrum of Decision Making and Applications, 3(1):292– 315, 2026. [75] Zohreh Nazari and Batool Mosapour. The entropy of hyperfuzzy sets. Journal of Dynamical Systems and Geometric Theories, 16:173 – 185, 2018. [76] Takaaki Fujita. Hyperrough cubic set and superhyperrough cubic set. Prospects for Applied Mathematics and Data Analysis, 4(1):28–35, 2024. [77] Takaaki Fujita. Advancing Uncertain Combinatorics through Graphization, Hyper- ization, and Uncertainization: Fuzzy, Neutrosophic, Soft, Rough, and Beyond. Biblio Publishing, 2025. [78] Takaaki Fujita. Hyperweighted graph, superhyperweighted graph, and multiweighted graph. Pure Mathematics for Theoretical Computer Science, 5(1):21–33, 2025. [79] Florentin Smarandache. Extension of soft set to hypersoft set, and then to plithogenic hypersoft set. Neutrosophic sets and systems, 22(1):168–170, 2018. [80] Sagvan Y Musa and Baravan A Asaad. Topological structures via bipolar hypersoft sets. Journal of Mathematics, 2022(1):2896053, 2022.