Fuzzy Graph Hyperoperations and Path-Based Algebraic Structures

Abstract

This paper introduces a framework of hypercompositional algebra on fuzzy graphs by defining and analyzing fuzzy path-based hyperoperations. Building on the notion of strongest strong paths (paths that are both strength-optimal and composed exclusively of strong edges, where each edge achieves maximum connection strength between its endpoints), we define two operations: a vertex-based fuzzy path hyperoperation and an edge-based variant. These operations generalize classical graph hyperoperations to the fuzzy setting while maintaining compatibility with the underlying topology. We prove that the vertex fuzzy path hyperoperation is associative, forming a fuzzy hypersemigroup, and establish additional properties such as reflexivity and monotonicity with respect to $\alpha$-cuts. Structural features such as fuzzy strong cut vertices and edges are examined, and a fuzzy distance function is introduced to quantify directional connectivity strength. We define an equivalence relation based on mutual full-strength reachability and construct a quotient fuzzy graph that reflects maximal closed substructures under the vertex fuzzy path hyperoperation. Applications are discussed in domains such as trust networks, biological systems, and uncertainty-aware communications. This work aims to lay the algebraic foundations for further exploration of fuzzy hyperstructures that support modeling, analysis, and decision-making in systems governed by partial and asymmetric relationships.

1. Introduction

Graphs are powerful tools for modeling relationships and dependencies across diverse fields, including network analysis [1,2], transport optimization [3,4], natural language processing [5,6], pattern recognition [7,8,9,10], knowledge representation [11,12,13], liver disease analysis [14,15], neuroscience [16,17,18], and the construction of bioinformatics networks [19,20,21,22]. Algebraic hyperstructures were introduced by Marty in 1934 as a generalization of classical algebraic structures [23]. They were extensively explored by Corsini [24,25,26], and later by other authors [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42]. The connection between algebraic hyperstructures and binary relations has been rigorously investigated [43,44,45,46,47,48,49,50,51,52,53,54], establishing a framework for representing and analyzing graph properties through hyperstructures [55,56,57,58,59,60,61,62,63,64]. In this direction, the introduction and investigation of algebraic hyperoperations deriving from directed graphs offer an integration of hypercompositional algebra with graph theory [65,66]. These graph hyperoperations exhibit characteristics dictated by the structure of their underlying graphs, which, in turn, define corresponding graph classes. This synergy provides novel insights and methodologies for analyzing complex graph properties [67,68,69], further enriching the study of graph theory through the lens of algebraic hyperstructures.

In this paper, we extend these ideas by examining fuzzy graph hyperoperations. The concept of fuzzy graphs, first introduced by Rosenfeld in 1975 [70], extends classical graph theory by incorporating fuzzy set theory to model uncertainty and imprecision in complex systems. A fuzzy graph is typically defined as a pair consisting of a fuzzy set of vertices and a fuzzy set of edges, with the constraint that the membership value of an edge cannot exceed the membership values of its connecting vertices [71,72]. By associating fuzzy membership values with nodes and edges, fuzzy graphs provide a robust framework for analyzing systems where interactions and elements exhibit inherent vagueness or ambiguity. This allows fuzzy graphs to represent partially connected vertices and varying strengths of relationships, making them particularly valuable in applications where traditional graph models fail to capture underlying complexities [73,74,75,76].

Fuzzy graphs have been applied across diverse domains, including social network analysis, bioinformatics, transportation systems, and more. In social network analysis, they model relationships with varying strength or certainty, such as trust or influence [77,78,79,80,81]. In medical diagnosis, fuzzy graphs help identify uncertain symptom-disease correlations [82]. In transportation systems, they facilitate traffic optimization by modeling routes with varying congestion levels or reliability [83]. Additionally, fuzzy graphs have been used in pattern recognition to identify complex structures [84,85] and in disease spread modeling to address uncertainties in transmission dynamics [86].

In this paper, we introduce hypercompositional structures on fuzzy graphs by extending classical path-based hyperoperations to a fuzzy framework. Building on the notion of strongest strong paths—those that are both strength-optimal and composed exclusively of strong edges—we define two types of fuzzy hyperoperations: a vertex-based and an edge-based fuzzy path hyperoperation. These operations generalize classical crisp hyperoperations and are compatible with fuzzy graph topology. The proposed framework allows us to reason algebraically about fuzzy connectivity and uncertainty-aware navigation in networks, providing a rich theoretical foundation for applications in social systems, biological regulation, communication networks, and medical diagnostics [74,75,76,77,78,79,81,82,86].

While classical fuzzy graph models typically focus on scalar-valued connectivity—such as determining the degree of reachability between two nodes using relational composition or $\alpha$-cut-based adjacency—they often do not preserve intermediate structural information, nor do they support compositional reasoning over sets of paths or substructures. In contrast, the vertex-based and edge-based fuzzy path hyperoperations introduced in this paper generate fuzzy subsets of vertices or edges lying on strongest strong paths, enabling the algebraic manipulation of connectivity regions rather than scalar summaries. This set-valued perspective permits the formalization of algebraic operations that are closed, associative, and structurally meaningful, thereby integrating fuzzy graph theory with hypercompositional algebra. Additionally, our framework supports the construction of quotient graphs that preserve directional strength semantics and admit interpretation as condensed fuzzy systems, offering advantages in modular analysis and abstraction. These features distinguish the proposed model from traditional fuzzy connectivity approaches and motivate its development as a foundational tool for uncertainty-aware reasoning in complex networks.

In the introduced framework, we develop a comprehensive set of structural and algebraic properties for the associated fuzzy hyperoperations. In particular, for the vertex fuzzy path hyperoperation, we prove associativity, thereby establishing a fuzzy hypersemigroup over the vertex set. We demonstrate reflexivity and $\alpha$-cut monotonicity, enabling threshold-sensitive reasoning. We characterize fuzzy strong cut vertices and edges, introduce a fuzzy distance function quantifying path strength, and analyze how these features behave under equivalence relations derived from mutual full-strength connectivity. By constructing quotient fuzzy graphs and establishing the closure properties of their equivalence classes, we derive coarse-grained models that preserve key connectivity semantics.

The paper is organized as follows. Section 2 reviews algebraic hyperstructures and classical graph hyperoperations. Section 3 extends these ideas to fuzzy graphs, defining fuzzy paths, strongest paths, and strongest strong paths. Section 4 introduces vertex- and edge-based fuzzy path hyperoperations and proves their key algebraic properties. Section 5 defines the mutual full-strength connectivity relation $R_V$ and studies the quotient fuzzy graph. Section 6 introduces a fuzzy distance function and examines its behavior on both the original and quotient graphs. Section 7 presents application scenarios, including trust networks and biological systems. Section 8 concludes the paper with a summary and future research directions.

2. Algebraic Background and Classical Graph Hyperoperations

We start with some necessary notions from hypercompositional algebra. Given a nonempty set V, a hyperoperation (hyperproduct) on V is a mapping

$$★:V\times V\to \mathcal{P}\left(V\right)$$

where $\mathcal{P}\left(V\right)$ denotes the powerset of V. As usual we set

$$(u,w)\mapsto u★w.$$

We define the hyperproduct between two subsets $U,W\in \mathcal{P}\left(V\right)$ by

$$U★W=\bigcup _{u\in U,w\in W}u★w.$$

A hypergroupoid is a pair $(V,★)$, where ★ is a hyperoperation on the set V. Following [37], a hypergroupoid $(V,★)$ is called

• Nonpartial: If the hyperproduct $u★v$ is nonempty for all $u,v\in V$.
• Degenerative: If $u★v=\varnothing$ for all $u,v\in V$.
• Weakly degenerative: If $u★v=\varnothing$ for all $u,v\in V$, with $u\ne v$.
• Total: If $u★v=V$ for all $u,v\in V$.
• Commutative: If, for all $u,v\in V$, it holds that $u★v=v★u$.
• Weakly commutative: If, for all $u,v\in V$, it holds that $u★v\cap v★u\ne \varnothing$.

A directed crisp graph (or simply graph) $G=(V,E)$ consists of a set of vertices (nodes) V and a set of edges $E\subseteq V\times V$. A subgraph of such a graph G is a graph $H=(V_H,E_H)$ with $V_H\subseteq V$ and $E_H\subseteq E$, such that every edge in $E_H$ has both endpoints in $V_H$.

Given a graph $G=(V,E)$, a path P inside G from a vertex $v_0$ to a vertex $v_m$ is a sequence of edges:

$$(v_0,v_1),(v_1,v_2),\dots ,(v_{m-1},v_m),$$

such that for every i, it holds that $(v_i,v_{i+1})\in E$. In this case, we say that the path has length m, and that the nodes $v_0,v_1,\dots ,v_m$ lie on the path P and similarly for the edges in the sequence. Note that nodes and edges may appear more than once in a path. Given a path P, we write $v\in P$ and $(v_i,v_j)\in P$ to express that vertex v and edge $(v_i,v_j)$ lie on P, respectively. We denote a path P from $v_0$ to $v_m$ by

$$P:v_0⇝v_m.$$

If all nodes appear at most once in a path, the path is called simple. A cycle is a path that starts and ends at the same node. The empty sequence $ϵ$ is a cycle from u to itself, containing only u.

The set of all nodes that lie in a path from $v_0$ to $v_m$ is denoted by

$$path(v_0,v_m)=\{x\mid x\mathrm{lies}\mathrm{on}\mathrm{a}\mathrm{path}\mathrm{from}{v}_0\mathrm{to}{v}_m\}.$$

A graph G is called strongly connected if, for any two nodes $v_1$ and $v_2$, there exists at least one path from $v_1$ to $v_2$. A strongly connected subgraph of a graph G is called a Strongly Connected Component of G.

Given a graph $G=(V,E)$, the path hyperoperation (see [65,66,67,68,69]) is a mapping

$${★}_G:V\times V\to \mathcal{P}\left(V\right)$$

defined by

$$u{★}_{G}w=\{v\in V\mid v\mathrm{lies}\mathrm{on}\mathrm{a}\mathrm{path}\mathrm{between}u\mathrm{and}w\}.$$

The hyperstructure $(V,{★}_G)$ is called the path hypergroupoid corresponding to G.

3. Fuzzy Graphs and Strongest Strong Paths

In this section we extend the definition of directed graphs to introduce fuzzy directed graphs, which incorporate the concept of partial membership for nodes and edges.

Fuzzy sets generalize traditional (crisp) sets, which allow only binary membership values of 0 or 1—an element either entirely belongs to the set or does not belong at all. By contrast, fuzzy sets capture a continuum of membership, enabling the representation of uncertain or imprecise data.

Formally, a fuzzy set $A_{\mu }$ is defined as a nonempty crisp set A paired with a function $\mu :A\to [0,1]$, known as membership function. For every element $s\in A$, the value $\mu \left(s\right)$ represents the membership grade of s in the fuzzy set. This grade indicates the degree to which the element belongs to the set, where $\mu \left(s\right)=0$ implies no membership, $\mu \left(s\right)=1$ implies full membership, and intermediate values reflect partial membership. From this definition it is clear that crisp sets are a special case of fuzzy sets.

If $A_{\mu }$ is a fuzzy set and $B\subseteq A$, then $B_{\mu }$ denotes the fuzzy set obtained by taking the restriction of $\mu$ to B. Given a finite universal set $A=\{x_1,\dots ,x_n\}$, the fuzzy set $A_{\mu }$ can be denoted as

$$\{\mu \left(x_1\right)/x_1,\dots ,\mu \left(x_n\right)/x_n\}.$$

The set of all fuzzy sets that can be defined on a universe set A is denoted by $\mathrm{\Phi }\left(A\right)$. The support of a fuzzy set $A_{\mu }$ is defined as the crisp subset of A containing all elements with a non-zero membership grade. Formally, the support of $A_{\mu }$, denoted as $\mathrm{supp}\left(A_{\mu }\right)$, is given by

$$\mathrm{supp}\left(A_{\mu }\right)=\{s\in A\mid \mu \left(s\right)>0\}.$$

In other words, the support of a fuzzy set includes all elements of the universe A that have a positive degree of membership.

The $\alpha$-cut (or $\alpha$-level set) of a fuzzy set $A_{\mu }$ is a crisp subset of A containing all elements whose membership grades are greater than or equal to a given threshold $\alpha$, where $\alpha \in [0,1]$. The $\alpha$-cut is a fundamental concept in fuzzy set theory, providing a way to analyze and approximate fuzzy sets using crisp subsets. It is particularly useful in applications where decision-making or reasoning is based on threshold levels. Formally, the $\alpha$-cut of $A_{\mu }$, denoted as $A_{\mu }^{\alpha }$, is defined as

$$A_{\mu }^{\alpha }=\{s\in A\mid \mu \left(s\right)\ge \alpha \}.$$

and the strict $\alpha$-cut of $A_{\mu }$, denoted as $A_{\mu }^{\underline{\alpha }}$, is

$$A_{\mu }^{\underline{\alpha }}=\{s\in A\mid \mu \left(s\right)>\alpha \}.$$

For $\alpha =0$, the strict $\alpha$-cut corresponds to the support of the fuzzy set, i.e.,

$$A_{\mu }^{\underline{0}}=\mathrm{supp}\left(A_{\mu }\right).$$

For any $a,b\in [0,1]$, we denote by $a\wedge b$ the minimum of a and b and by $a\vee b$ their maximum.

A fuzzy directed graph (or simply fuzzy graph) [71] is a generalization of a directed crisp graph, where the nodes and the edges are characterized by membership functions that represent the degree of membership. Formally, a fuzzy graph is defined as a pair $F=(V_{\mu },\lambda )$ where

-

V is the set of vertices (nodes) of the graph.

-

$\mu :V\to [0,1]$ is the vertex membership function, which assigns to each vertex $v\in V$ a membership grade $\mu \left(v\right)$, indicating the degree to which v belongs to the graph.

-

$\lambda :V\times V\to [0,1]$ is the edge membership function, which assigns to each pair of vertices $(u,v)\in V\times V$ a membership grade $\lambda (u,v)$, indicating the degree of connectivity between u and v.

In addition, the following condition is required:

$$\lambda (u,v)\le min\left(\mu \right(u),\mu (v\left)\right)\forall u,v\in V,$$

ensuring that the degree of membership of an edge cannot exceed the membership grades of its endpoints. Given a fuzzy graph $F=(V_{\mu },\lambda )$, the set $V\times V$ together with the edge membership function $\lambda$, is called the edge set of F and is denoted by $E_{\lambda }$.

Given a fuzzy graph $F=(V_{\mu },\lambda )$, the underlying crisp graph of F is the graph $F^c=(V^c,E^c)$, where $V^c=V$ is the underlying crisp vertex set and

$$E^c=\{(x,y)\in V\times V\mid \lambda (x,y)>0\},$$

is the underlying crisp edge set. A fuzzy graph $F=(V_{\mu },\lambda )$ is called complete if $\lambda (x,y)=\mu \left(x\right)\wedge \mu \left(y\right)$ for every $x,y\in V$.

Now we introduce the concept of a fuzzy path within a fuzzy graph. Given a fuzzy graph $F=(V_{\mu },\lambda )$ and two nodes $v_0,v_k\in V$, a fuzzy path P of length k, from the node $v_0$ to the node $v_k$, is a sequence of edges

$$(v_0,v_1)(v_1,v_2)\cdots (v_{k-1},v_k),$$

such that $\lambda (u_i,u_{i+1})>0$ for every $i=0,\dots ,k-1$. We say that the vertices $v_0,\dots ,v_k$ and the edges $(v_0,v_1),(v_1,v_2),\dots ,(v_{k-1},v_k)$ belong to the fuzzy path P or, equivalently, that F contains them. The set of all fuzzy paths from v to u, inside a fuzzy graph F, is denoted by $F_p(v,w)$. For every vertex $u\in V$, the empty sequence $\epsilon$ is a path from u to itself, containing only the vertex u and no edges.

Given a fuzzy graph $F=(V_{\mu },\lambda )$ and a set $S\subseteq V$, a path in $F-S$ is path in F that does not contain any vertices from S, and similarly, given a set $E\subseteq V\times V$, a path in $F-E$ does not contain any edges from E.

The strength of a path P inside a fuzzy graph $F=(V_{\mu },\lambda )$ is denoted by $s\left(P\right)$, and it is the minimum membership grade among all of the edges of P, i.e.,

$$s\left(P\right)={\displaystyle \underset{i=0}{\overset{k-1}{\bigwedge }}}\lambda (v_i,v_{i+1}).$$

The strength of connectedness between two nodes u and v of a fuzzy graph F is defined as the maximum of the strengths of all paths $P\in F_p(u,v)$ and is denoted by $Co{n}_F(u,v)$. If no path exists between u and v, we set $Co{n}_F(u,v)=0$.

A path $P\in F_p(u,v)$ of a fuzzy graph F is called a strongest path of F if $s\left(P\right)=Co{n}_F(u,v)$. Equivalently, P is a strongest fuzzy path of F if and only if

$$s\left(P\right)\ge s\left(P^{\prime }\right),\mathrm{for}\mathrm{all}{P}^{\prime }\in F_p(u,v).$$

A fuzzy graph $F=(V_{\mu },\lambda )$ is strongly connected if for every $u,v\in V^c$, it holds that $Co{n}_F(u,v)>0$. An edge $(u,v)$ is called strong if $\lambda (u,v)=Co{n}_F(u,v)$. A path consisting entirely of strong edges is called a strong path. In general, a path may be the strongest without being strong and vice versa. Fuzzy paths that admit both properties are called strongest strong paths (see [72]). Formally we have

$$s{s}_F(u,v)=\{P\in F_p(u,v)\mid s\left(P\right)={Con}_F(u,v)\mathrm{and}\mathrm{all}\mathrm{edges}\mathrm{in}P\mathrm{are}\mathrm{strong}.\}$$

Intuitively, the strongest strong paths represent the most reliable routes in a fuzzy network, combining the optimal overall strength with the local edge optimality. The set of all strongest strong paths from u to v inside a fuzzy graph F is denoted by $s{s}_F(u,v)$.

Fuzzy strong cuts were introduced for undirected fuzzy graphs in [87] by employing the strongest strong paths. We define them in a similar way for directed fuzzy graphs as follows. Consider a fuzzy graph $F=(V_{\mu },\lambda )$ with an underlying crisp graph $F^c=(V^c,E^c)$. A subset $M\subseteq V^c$ is called a fuzzy strong vertex cut set if for some vertices $u,v\in V^c-M$, there exists $P\in s{s}_F(u,v)$, but there exists no strongest strong path from u to v in $F-M$. In other words, this means that there is a strongest strong path P from u to v inside F if and only if we allow P to go through the vertices of M. If $M=\left\{u\right\}$, then u is called a fuzzy strong cut vertex of F. A subset $\mathcal{E}\subseteq E^c$ is called a fuzzy strong edge cut set if for some vertices $u,v\in V^c$ there exists $P\in s{s}_F(u,v)$ but there exists no strongest strong path from u to v in $G-\mathcal{E}$. If $\mathcal{E}=\left\{\right(x,y\left)\right\}$, then $(x,y)$ is called a fuzzy strong cut edge of F.

Example 1.

Consider the directed fuzzy graph $F_1=(V_{\mu },\lambda )$ of Figure 1, with vertices $V=\{a,b,c,d,e\}$, vertex membership function

$$\mu \left(a\right)=1.0,\mu \left(b\right)=0.8,\mu \left(c\right)=0.9,\mu \left(d\right)=0.7,\mu \left(e\right)=0.9$$

and edge membership function:

$$\lambda (a,b)=0.6,\lambda (a,c)=0.4,\lambda (c,b)=0.6,\lambda (b,d)=0.7$$

$$\lambda (c,d)=0.5,\lambda (b,e)=0.6,\lambda (d,e)=0.6.$$

The fuzzy paths from a to e are

• $P_1:a\to b\to e$ with strength $s\left(P_1\right)=0.6$
• $P_2:a\to b\to d\to e$ with strength $s\left(P_2\right)=0.6$
• $P_3:a\to c\to d\to e$ with strength $s\left(P_3\right)=0.4$
• $P_4:a\to c\to b\to e$ with strength $s\left(P_4\right)=0.4$
• $P_5:a\to c\to b\to d\to e$ with strength $s\left(P_5\right)=0.4$

The strength of connectedness between a and e is $Co{n}_F(a,e)=0.6$; hence, the strongest paths from a to e are $P_1$ and $P_2$. From these two only $P_2$ is strong. The path $P_1$ is not strong since it includes the edge $(b,e)$, which is not strong because it holds $Co{n}_{F_1}(b,e)=0.7\ge \lambda (b,e)=0.6$. Thus, the only strongest strong path from a to e is $P_2$. From this we can see that b and d are fuzzy strong cut vertices and that the edges $(a,b),(b,d),(d,e)$ are fuzzy strong cut edges of $F_1$.

Figure 1. The fuzzy graph $F_1$ of Example 1, with vertex memberships and edge weights shown.

Figure 1. The fuzzy graph $F_1$ of Example 1, with vertex memberships and edge weights shown.

4. Fuzzy Hyperoperations and Structural Properties

A fuzzy hypergroupoid is a pair $(V,★)$, where V is a nonempty set and ★ is a binary fuzzy hyperoperation

$$★:V\times V\to \mathrm{\Phi }\left(V\right).$$

Given a fuzzy graph $F=(V_{\mu },\lambda )$, we define a binary and a unary fuzzy hyperoperation as follows. The vertex fuzzy path hyperoperation

$${\odot }_F:V\times V\to \mathrm{\Phi }\left(V\right)$$

is defined by setting

$$v{\odot }_{F}u={\left\{w\in V|\exists P\in s{s}_F(v,u)\mathrm{such}\mathrm{that}w\in P\right\}}_{\mu }$$

Hence, $v{\odot }_{F}u$ is a fuzzy subset of V with membership function $\mu \left(w\right)$ if $w\in V$ lies on some $P\in s{s}_F(v,u)$, and 0 otherwise. Note that $v{\odot }_{F}u$ yields a fuzzy subset where membership grades reflect the original vertex reliabilities of nodes participating in optimal signal transmission paths.

Intuitively, the vertex fuzzy path hyperoperation ${\odot }_F$ captures all nodes that lie on strongest strong paths from vertices v to u. It produces a fuzzy subset of vertices weighted by their original credibility scores $\mu \left(w\right)$, thereby modeling the spread or support of optimal communication under full-strength constraints. This operation generalizes binary connectivity to a graded, path-based relation central to our fuzzy algebraic framework.

The edge fuzzy path hyperoperation is a unary hyperoperation on the set $V\times V$

$${\oslash }_F:V\times V\to \mathrm{\Phi }(V\times V)$$

defined by

$$v{\oslash }_{F}u={\left\{(x,y)\in V\times V|\exists P\in s{s}_F(v,u)\mathrm{such}\mathrm{that}(x,y)\in P\right\}}_{\lambda }$$

Hence, $v{\oslash }_{F}u$ is a fuzzy subset of $V\times V$ with membership function $\lambda (x,y)$ if $(x,y)\in V\times V$ lies on some $P\in s{s}_F(v,u)$, and 0 otherwise.

Remark 1.

Although the domain of the edge fuzzy path hyperoperation ${\oslash }_F$ is formally $V\times V$, which would suggest a binary operation in the classical sense, we interpret ${\oslash }_F$ as a unary hyperoperation over the edge universe $V\times V$. This perspective aligns with the structural analogy to the vertex-based fuzzy path hyperoperation ${\odot }_F$, which acts on the vertex set V. In this framework, the set $V\times V$ serves as the domain and codomain of the hyperoperation, thereby justifying its classification as unary within the context of edge-based fuzzy algebraic structures.

The proposed fuzzy hyperoperations generalize classical graph hyperoperations while preserving their structure when applied to crisp graphs. The following proposition formally establishes this by showing that the vertex fuzzy path hyperoperation coincides with the classical path hyperoperation when all memberships are binary.

Proposition 1.

The path hyperoperation and the vertex fuzzy path hyperoperation coincide for crisp graphs.

Proof. 

Let $G=(V,E)$ be a directed crisp graph. The path hyperoperation ${★}_G$ is defined as

$$u{★}_{G}v=\{w\in V\mid w\mathrm{lies}\mathrm{on}\mathrm{a}\mathrm{path}\mathrm{between}u\mathrm{and}v\}.$$

Now consider the corresponding fuzzy graph $F=(V_{\mu },\lambda )$ where

$$\mu \left(w\right)=1\mathrm{for}\mathrm{all}w\in V,\mathrm{and}\lambda (x,y)=\left\{\begin{array}{cc}1\hfill & \mathrm{if}(x,y)\in E,\hfill \\ 0\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

That is, F is a crisp graph viewed as a fuzzy graph with binary membership functions. In this case, every existing path P from u to v consists entirely of edges with membership 1. Therefore, each such path has strength $s\left(P\right)=1$, the maximum possible and since $s\left(P\right)={Con}_F(u,v)$, all such paths are strongest.

To determine whether these paths are also strong, recall that an edge $(x,y)$ is called strong if

$$\lambda (x,y)={Con}_F(x,y).$$

For all $(x,y)\in E$, we have $\lambda (x,y)=1={Con}_F(x,y)$. Thus, all edges in the graph are strong, and every path composed of such edges is also strong. Therefore, every path in the crisp graph is both strongest and strong, which means

$${ss}_F(u,v)=\{P\mid P\mathrm{is}\mathrm{a}\mathrm{path}\mathrm{from}u\mathrm{to}v\mathrm{in}G\}.$$

The vertex fuzzy path hyperoperation is defined by

$$u{\odot }_{F}v={\{w\in V\mid w\mathrm{lies}\mathrm{on}\mathrm{some}P\in {ss}_F(u,v)\}}_{\mu }.$$

Since $\mu \left(w\right)=1$ for all $w\in V$, this reduces to the crisp set

$$u{\odot }_{F}v=\{w\in V\mid w\mathrm{lies}\mathrm{on}\mathrm{a}\mathrm{path}\mathrm{from}u\mathrm{to}v\}.$$

Thus, $u{\odot }_{F}v=u{★}_{G}v$. □

Given a fuzzy graph $F=(V_{\mu },\lambda )$ and a set $M\subseteq V$, the strongest fuzzy subgraph of F generated by M is the fuzzy graph $S_F\left(M\right)=(V_{{\mu }_M},{\lambda }_M)$, where

$${\mu }_M\left(w\right)=\left\{\begin{array}{c}\mu \left(w\right),\mathrm{if}w\in supp\left(v{\odot }_{F}u\right)\mathrm{for}\mathrm{some}v,u\in M\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

and

$${\lambda }_M(x,y)=\left\{\begin{array}{c}\lambda (x,y),\mathrm{if}(x,y)\in supp\left(v{\oslash }_{F}u\right)\mathrm{for}\mathrm{some}v,u\in M\hfill \\ 0,\mathrm{otherwise}.\hfill \end{array}\right.$$

A graph $F=(V_{\mu },\lambda )$ is called strongest if $F=S_F\left(V\right)$. We now give an example showing that even if a graph is strongest, not every path is a strongest strong path.

Example 2.

Let $F_2=(V_{\mu },\lambda )$ be the fuzzy graph of Figure 2, with a vertex set $V=\{v_0,v_1,v_2,v_3\}$ and uniform vertex membership $\mu \left(v_i\right)=1\mathrm{for}\mathrm{all}i$. Let the non-zero edge membership values be

$$\lambda (v_0,v_1)=0.6,\lambda (v_1,v_3)=0.6,$$

$$\lambda (v_0,v_2)=0.9,\lambda (v_2,v_3)=0.9.$$

The graph $F_2$ is strongest since each edge $(x,y)$ in the graph satisfies

$$\lambda (x,y)={Con}_{F_2}(x,y).$$

The paths

$$P=(v_0,v_1,v_3),\mathit{and}Q=(v_0,v_2,v_3),$$

have strengths

$$s\left(P\right)=0.6\mathit{and}s\left(Q\right)=0.9.$$

Therefore, even though P is a path in a strongest graph, it is not a strongest strong path between $v_0$ and $v_3$.

Figure 2. The fuzzy graph $F_2$ of Example 2, illustrating that not every path in a strongest graph is a strongest strong path.

Figure 2. The fuzzy graph $F_2$ of Example 2, illustrating that not every path in a strongest graph is a strongest strong path.

The following proposition provides a precise characterization of fuzzy strong cut edges by identifying them as those edges that appear in every strongest strong path between a given pair of vertices.

Proposition 2.

Let $F=(V_{\mu },\lambda )$ be a strongest fuzzy graph. An edge $(x,y)\in supp\left(\lambda \right)$ is a fuzzy strong cut edge if and only if it appears in every strongest strong path between some pair of vertices $u,v\in V$.

Proof. 

(⇒) Suppose that $(x,y)$ is a fuzzy strong cut edge. By definition, there exist vertices $u,v\in V$ such that

(i)

$s{s}_F(u,v)\ne \varnothing$,

(ii)

$s{s}_{F^{\prime }}(u,v)=\varnothing$, where $F^{\prime }=F-\left\{(x,y)\right\}$.

This means that every strongest strong path from u to v in F must contain the edge $(x,y)$. Otherwise, if there existed a path $P\in s{s}_F(u,v)$ that did not include $(x,y)$, then P would still be present in $F^{\prime }$, contradicting the assumption that $s{s}_{F^{\prime }}(u,v)=\varnothing$.

(⇐) Conversely, suppose that $(x,y)$ appears in every strongest strong path from u to v. Then, removing $(x,y)$ from F eliminates all such paths. That is, in the graph $F^{\prime }=F-\left\{(x,y)\right\}$, we have $s{s}_{F^{\prime }}(u,v)=\varnothing$. Hence, by definition, $(x,y)$ is a fuzzy strong cut edge. □

We now turn to a fundamental structural behavior of strongest strong paths in fuzzy graphs. In many practical contexts, especially in decision-making and network resilience, it is important to understand how the set of available paths changes as we impose stricter constraints on the strength of connections. This motivates the following proposition, which formalizes the monotonicity of the $\alpha$-cuts of the vertex fuzzy path hyperoperation.

Proposition 3.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph, and let $u,v\in V$. If $0\le \alpha \le \beta \le 1$, then

$${\left(u{\odot }_{F}v\right)}^{\beta }\subseteq {\left(u{\odot }_{F}v\right)}^{\alpha },$$

where ${\left(u{\odot }_{F}v\right)}^{\gamma }$ denotes the γ-cut of the fuzzy set $u{\odot }_{F}v$, for any $\gamma \in [0,1]$.

Proof. 

By definition, the vertex fuzzy path hyperoperation $u{\odot }_{F}v$ is a fuzzy subset of V, with membership function

$${\mu }_{u{\odot }_{F}v}\left(w\right)=\left\{\begin{array}{cc}\mu \left(w\right),\hfill & \mathrm{if}w\mathrm{lies}\mathrm{on}\mathrm{a}\mathrm{path}P\in s{s}_F(u,v),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

The $\alpha$-cut of this fuzzy set is defined by

$${\left(u{\odot }_{F}v\right)}^{\alpha }=\{w\in V\mid {\mu }_{u{\odot }_{F}v}\left(w\right)\ge \alpha \}.$$

Now suppose $0\le \alpha \le \beta \le 1$, and let $w\in {\left(u{\odot }_{F}v\right)}^{\beta }$. Then

$${\mu }_{u{\odot }_{F}v}\left(w\right)\ge \beta \ge \alpha ,$$

which implies $w\in {\left(u{\odot }_{F}v\right)}^{\alpha }$. Therefore, ${\left(u{\odot }_{F}v\right)}^{\beta }\subseteq {\left(u{\odot }_{F}v\right)}^{\alpha }.$ □

An identity property of the vertex fuzzy path hyperoperation can be established on the diagonal.

Proposition 4.

Let $F=(V_{\mu },\lambda )$ be a fuzzy directed graph. If F is acyclic, then for every $v\in V$, we have

$$v{\odot }_{F}v={\left\{v\right\}}_{\mu },$$

i.e., the fuzzy set resulting from the vertex fuzzy path hyperoperation on the diagonal contains only the node v with its original membership grade.

Proof. 

Since F is acyclic, any path from v to v must be the trivial path $ϵ$, which is both strong and strongest by definition and, hence, belongs to $s{s}_F(v,v)$. Since the only node appearing in $ϵ$ is v, the fuzzy set $v{\odot }_{F}v$ is ${\left\{v\right\}}_{\mu }$, as required. □

The following result highlights that the support of the fuzzy path hyperoperation is always contained within the classical path hyperoperation. In effect, the fuzzy vertex hyperoperation respects the topology of the underlying graph and supports hybrid models that combine both crisp and fuzzy structures.

Proposition 5.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph, and let $G=(V,E)$ be its underlying crisp graph. Then, for any $u,v\in V$, we have

$$supp\left(u{\odot }_{F}v\right)\subseteq u{★}_{G}v,$$

where $u{★}_{G}v$ denotes the classical path hyperoperation in G.

Proof. 

By definition, $u{\odot }_{F}v$ is a fuzzy subset of V such that

$${\mu }_{u{\odot }_{F}v}\left(w\right)=\left\{\begin{array}{cc}\mu \left(w\right),\hfill & \mathrm{if}w\in P\mathrm{for}\mathrm{some}P\in s{s}_F(u,v),\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Thus, the support of $u{\odot }_{F}v$ is the set of all nodes lying on some strongest strong path from u to v in F:

$$supp\left(u{\odot }_{F}v\right)=\{w\in V\mid w\in P\mathrm{for}\mathrm{some}P\in s{s}_F(u,v)\}.$$

Since all paths in $s{s}_F(u,v)$ are also paths in the underlying crisp graph G, we have

$$w\in supp\left(u{\odot }_{F}v\right)\Rightarrow w\in u{★}_{G}v.$$

Hence,

$$supp\left(u{\odot }_{F}v\right)\subseteq u{★}_{G}v.$$

□

To assess the utility of a vertex in fuzzy connectivity without entirely removing it from the graph, we adopt a notion of structural removal. The following definition formalizes this operation.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph and let $v\in V$. We define the structural removal of v, denoted by $F^{\setminus v}=(V_{\mu },{\lambda }^{\setminus v})$, as the fuzzy graph obtained by removing all edges incident to v while keeping v in the vertex set. Formally,

$${\lambda }^{\setminus v}(x,y)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}x=v\mathrm{or}y=v,\hfill \\ \lambda (x,y),\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

The presence of fuzzy strong cut vertices influences the composition of strongest strong paths in a fuzzy graph. Specifically, if a vertex lies on every such path between two nodes, then it serves as a critical connector whose structural removal partitions the path space. Hence, these nodes effectively segment the network into components that are joined solely through them.

Theorem 1.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph and let $v\in V$ be a fuzzy strong cut vertex. Then, there exist vertices $u,w\in V$ such that

(i) 

$s{s}_F(u,w)\ne \varnothing$,

(ii) 

$s{s}_{F^{\setminus v}}(u,w)=\varnothing$,

(iii) 

v lies on every path in $s{s}_F(u,w)$,

(iv) 

The set $s{s}_F(u,w)$ is partitioned into components that are connected only through v.

Proof. 

By definition of a fuzzy strong cut vertex, there exist $u,w\in V$ such that every path in $s{s}_F(u,w)$ contains v, and $s{s}_{F^{\setminus v}}(u,w)=\varnothing$. Since each path in $s{s}_F(u,w)$ must pass through v, removing all edges incident to v (as in $F^{\setminus v}$) breaks the connectivity of each such path at the point that links these two components. □

We now establish a relationship between vertex and edge membership values along strongest strong paths. Specifically, the membership of any node appearing in such a path is bounded below by the minimal edge strength of that path.

Proposition 6.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph and let

$$P=((v_0,v_1),(v_1,v_2),\dots ,(v_{k-1},v_k))\in s{s}_F(u,v)$$

be a strongest strong path. Then, for any vertex $v_i$ that appears in P, we have

$$\mu \left(v_i\right)\ge min\{\lambda (v_j,v_{j+1})\mid 0\le j<k\}.$$

Proof. 

Let $P=((v_0,v_1),(v_1,v_2),\dots ,(v_{k-1},v_k))\in s{s}_F(u,v)$, and let

$$\alpha =min\{\lambda (v_j,v_{j+1})\mid 0\le j<k\}=s\left(P\right).$$

Since P is a strongest strong path, each edge $(v_j,v_{j+1})$ is strong, so

$$\lambda (v_j,v_{j+1})={Con}_F(v_j,v_{j+1})\le min\{\mu \left(v_j\right),\mu \left(v_{j+1}\right)\}.$$

Thus, for every i, we have $\mu \left(v_i\right)\ge \alpha$. □

Remark 2.

This result ensures that fuzzy path hyperoperations preserve minimum reliability thresholds—if a signal can traverse a path of strength α, then all intermediate nodes have reliability of at least α.

Corollary 1.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph, and let

$$P=((v_0,v_1),(v_1,v_2),\dots ,(v_{k-1},v_k))\in s{s}_F(u,v)$$

be a strongest strong path of strength α. Then, for every vertex $v_i$ that appears in P, we have

$$\mu \left(v_i\right)\ge \alpha ,$$

i.e., $v_i\in V_{\mu }^{\alpha }$.

We now prove that the vertex fuzzy path hyperoperation is associative.

Proposition 7.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph. Then, the vertex fuzzy path hyperoperation ${\odot }_F$ is associative; i.e., for all $u,v,w\in V$,

$$\left(u{\odot }_{F}v\right){\odot }_{F}w=u{\odot }_F\left(v{\odot }_{F}w\right).$$

Proof. 

Recall that for any $x,y\in V$, the fuzzy set $x{\odot }_{F}y$ is defined by

$$x{\odot }_{F}y={\left\{z\in V\mid z\in P\mathrm{for}\mathrm{some}P\in s{s}_F(x,y)\right\}}_{\mu }.$$

Let us compute both sides of the identity and show that they yield the same fuzzy set.

Left-hand side: $\left(u{\odot }_{F}v\right){\odot }_{F}w$. Let

$$A=u{\odot }_{F}v=\bigcup _{P\in s{s}_F(u,v)}{\{x\in P\}}_{\mu }.$$

Then,

$$\left(u{\odot }_{F}v\right){\odot }_{F}w=\bigcup _{x\in supp\left(A\right)}x{\odot }_{F}w=\bigcup _{x\in supp\left(u{\odot }_{F}v\right)}\left(\bigcup _{Q\in s{s}_F(x,w)}{\{z\in Q\}}_{\mu }\right).$$

The above formula describes how to compute the left-hand side of the associativity identity using the definition of the vertex fuzzy path hyperoperation.

Here, $A=u{\odot }_{F}v$ is the fuzzy set of all vertices lying on some strongest strong path from u to v, with membership values inherited from $\mu$. The support $supp\left(u{\odot }_{F}v\right)$ consists of all vertices that occur with positive membership in A.

For each such vertex $x\in supp\left(u{\odot }_{F}v\right)$, we compute $x{\odot }_{F}w$, which includes all vertices lying on some strongest strong path from x to w. Taking the union over all such x yields the fuzzy set of vertices that lie on some strongest strong path from u to v and then from x to w.

Since the family of strongest strong paths is closed under concatenation when intermediate vertices are shared, this construction recovers all vertices lying on some strongest strong path from u to w through v, preserving the membership structure. This justifies the associativity of the operation ${\odot }_F$.

Right-hand side: $u{\odot }_F\left(v{\odot }_{F}w\right)$. Let

$$B=v{\odot }_{F}w=\bigcup _{Q\in s{s}_F(v,w)}{\{y\in Q\}}_{\mu }.$$

Then,

$$u{\odot }_F\left(v{\odot }_{F}w\right)=\bigcup _{y\in supp\left(B\right)}u{\odot }_{F}y=\bigcup _{y\in supp\left(v{\odot }_{F}w\right)}\left(\bigcup _{P\in s{s}_F(u,y)}{\{z\in P\}}_{\mu }\right).$$

In this case, for each y lying on a strongest strong path from v to w, we consider all strongest strong paths from u to y, and their union yields the same support and membership degrees as the LHS.

In both cases, the construction effectively accumulates all nodes lying on a strongest strong path from u to w. The union over all intermediate decompositions covers all such paths, and due to maximality in the strength selection of $s{s}_F$, we do not introduce weaker paths. Hence,

$$\left(u{\odot }_{F}v\right){\odot }_{F}w=u{\odot }_F\left(v{\odot }_{F}w\right).$$

□

Intuitively, this result reflects that fuzzy path composition is independent of grouping; hence, intermediate aggregation points do not affect the final reachability analysis.

The associativity of the vertex fuzzy path hyperoperation establishes a foundational algebraic structure over the fuzzy vertex set. Specifically, it ensures that $(V_{\mu },{\odot }_F)$ forms a fuzzy hypersemigroup, a hypercompositional analogue of semigroups in which the operation yields fuzzy subsets rather than single elements. This result provides a robust algebraic framework for modeling compositional reasoning in systems with uncertainty, where reachability and influence propagate through multiple layers of interaction. It also enables the application of tools from hyperstructure theory to analyze the dynamics and resilience of fuzzy networks in a mathematically rigorous manner.

Corollary 2.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph. Then, the structure $(V_{\mu },{\odot }_F)$ forms a fuzzy hypersemigroup.

This algebraic structure opens the door to applying tools from hyperalgebra and fuzzy semigroup theory in the analysis of fuzzy graphs. It also connects the proposed hyperoperation to a broader mathematical context, enabling generalizations to fuzzy hypermonoids or hypergroupoids when identity or inverse properties are explored.

The following proposition highlights a fundamental structural property of the vertex fuzzy path hyperoperation. Path decomposition reflects the internal coherence of the hyperoperation and enables localized analysis of fuzzy connectivity through intermediate nodes.

Proposition 8

(Path Decomposition). Consider a fuzzy graph $F=(V_{\mu },\lambda )$ and vertices $u,v,w\in V$. If $w\in u{\odot }_{F}v$, then

$$u{\odot }_{F}v\subseteq \left(u{\odot }_{F}w\right)\cup \left(w{\odot }_{F}v\right).$$

Proof. 

Suppose $x\in u{\odot }_{F}v$. Then, by definition, x lies on a path $P\in s{s}_F(u,v)$.

Since $w\in P$ by assumption, the path P can be decomposed into two subpaths

$$P_1:u⇝w,P_2:w⇝v,$$

each of which is a subpath of a strongest strong path. Because all edges in P are strong and the overall path is the strongest, both $P_1$ and $P_2$ are themselves strongest strong paths.

$$P_1\in s{s}_F(u,w),P_2\in s{s}_F(w,v).$$

Therefore, any vertex $x\in P$ must either belong to $P_1$ or $P_2$, implying

$$x\in u{\odot }_{F}w\mathrm{or}x\in w{\odot }_{F}v.$$

Thus,

$$x\in \left(u{\odot }_{F}w\right)\cup \left(w{\odot }_{F}v\right),$$

and since x was arbitrary in $u{\odot }_{F}v$, we conclude that

$$u{\odot }_{F}v\subseteq \left(u{\odot }_{F}w\right)\cup \left(w{\odot }_{F}v\right).$$

□

The inclusion in the above proposition becomes an equality when the intermediate vertex is traversed by every strongest strong path between the two endpoints.

Proposition 9.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph, and let $u,v,w\in V$. If every path in $s{s}_F(u,v)$ contains w, then

$$u{\odot }_{F}v=\left(u{\odot }_{F}w\right)\cup \left(w{\odot }_{F}v\right).$$

Proof. 

From Proposition 8, we already know that

$$u{\odot }_{F}v\subseteq \left(u{\odot }_{F}w\right)\cup \left(w{\odot }_{F}v\right).$$

To prove the reverse inclusion, let $x\in \left(u{\odot }_{F}w\right)\cup \left(w{\odot }_{F}v\right)$. Then, either

(i)

x lies on a path $P_1\in s{s}_F(u,w)$, or

(ii)

x lies on a path $P_2\in s{s}_F(w,v)$.

By the hypothesis, every path in $s{s}_F(u,v)$ contains w. Let P be any such path; then, P can be decomposed as

$$P=P_1^{\prime }\circ P_2^{\prime },$$

where $P_1^{\prime }$ is a strongest strong path from u to w, and $P_2^{\prime }$ is a strongest strong path from w to v. Since strongest strong paths from u to w and from w to v exist and include x in at least one of them, then x must lie on some $P\in s{s}_F(u,v)$. Thus,

$$x\in u{\odot }_{F}v.$$

Therefore,

$$\left(u{\odot }_{F}w\right)\cup \left(w{\odot }_{F}v\right)\subseteq u{\odot }_{F}v.$$

Combined with the earlier inclusion, this proves the equality

$$u{\odot }_{F}v=\left(u{\odot }_{F}w\right)\cup \left(w{\odot }_{F}v\right).$$

□

This result shows that under a strong centrality condition—where all strongest strong paths from u to v must pass through w—the fuzzy vertex path operation satisfies an exact decomposability property.

Proposition 10

(Edge Support Decomposition). Let $F=(V_{\mu },\lambda )$ be a fuzzy graph, and let $u,v,w\in V$. If for every $P\in {\mathrm{ss}}_F(u,v)$, it holds that $w\in P$, then,

$$supp(u\oslash v)\subseteq supp(u\oslash w)\cup supp(w\oslash v),$$

with equality if every $P\in {\mathrm{ss}}_F(u,v)$ can be decomposed as $P=P_1\circ P_2$ with $P_1\in {\mathrm{ss}}_F(u,w)$ and $P_2\in {\mathrm{ss}}_F(w,v)$.

Proof. 

Let $(x,y)\in supp(u\oslash v)$. Then, by definition, there exists a strongest strong path $P\in {\mathrm{ss}}_F(u,v)$ such that $(x,y)$ is an edge of P. By hypothesis, w lies on every such path P, so P contains w and can be written as the concatenation $P=P_1\circ P_2$, where $P_1$ is a subpath from u to w, and $P_2$ is a subpath from w to v.

Since P is a strongest strong path and all of its subpaths are also strongest strong paths, we have $P_1\in {\mathrm{ss}}_F(u,w)$ and $P_2\in {\mathrm{ss}}_F(w,v)$. Hence, the edge $(x,y)$ must appear in either $P_1$ or $P_2$, which implies that

$$(x,y)\in supp(u\oslash w)\cup supp(w\oslash v),$$

Therefore,

$$supp(u\oslash v)\subseteq supp(u\oslash w)\cup supp(w\oslash v).$$

If, in addition, every $P\in {\mathrm{ss}}_F(u,v)$ decomposes into a concatenation of paths in ${\mathrm{ss}}_F(u,w)$ and ${\mathrm{ss}}_F(w,v)$, then the reverse inclusion holds, yielding equality. □

The following result formalizes the monotonicity of $\alpha$-cuts in the context of the edge fuzzy path hyperoperation.

Proposition 11

(Monotonicity of $\alpha$-Cuts of the Edge Fuzzy Path Hyperoperation). Let $F=(V_{\mu },\lambda )$ be a fuzzy graph, and let $u,v\in V$. For any $0\le \alpha \le \beta \le 1$, denote by ${(u\oslash v)}^{\gamma }$ the γ-cut of the fuzzy set $u\oslash v$, defined as

$${(u\oslash v)}^{\gamma }=\{(x,y)\in V\times V\mid \lambda (x,y)\ge \gamma \mathit{and}(x,y)\in P\in {\mathrm{ss}}_F(u,v)\}.$$

Then,

$${(u\oslash v)}^{\beta }\subseteq {(u\oslash v)}^{\alpha }.$$

Proof. 

Let $(x,y)\in {(u\oslash v)}^{\beta }$. By definition, there exists a strongest strong path $P\in {\mathrm{ss}}_F(u,v)$ such that $(x,y)\in P$ and $\lambda (x,y)\ge \beta$. Since $\alpha \le \beta$, it follows that $\lambda (x,y)\ge \alpha$ as well. Thus, $(x,y)$ also belongs to the $\alpha$-cut:

$$(x,y)\in {(u\oslash v)}^{\alpha }.$$

Therefore, we conclude that

$${(u\oslash v)}^{\beta }\subseteq {(u\oslash v)}^{\alpha }.$$

□

5. Equivalence Classes, Quotient Graphs, and Connectivity Structures

We now introduce a relation on the vertex set of a fuzzy graph that captures mutual full-strength connectivity. Let $F=(V_{\mu },\lambda )$ be a fuzzy graph, we define the binary relation $R_V\subseteq V\times V$ by setting $(u,v)\in R_V$ if and only if there exists a path P from u to v and a path $P^{\prime }$ from v to u such that $s\left(P\right)=1$ and $s\left(P^{\prime }\right)=1$. Equivalently,

$$R_V=\left\{(u,v)\in V\times V|\exists P:u⇝v,\exists P^{\prime }:v⇝u,\mathrm{with}s\left(P\right)=s\left(P^{\prime }\right)=1\right\}.$$

The following proposition shows that $R_V$ is an equivalence relation on the set V.

Proposition 12.

The relation $R_V$ is an equivalence relation on the vertex set V of the fuzzy graph $F=(V_{\mu },\lambda )$.

Proof. 

We verify the three defining properties of an equivalence relation:

Reflexivity: For every $u\in V$, consider the trivial path $\epsilon$ from u to itself (i.e., a path of length zero). Since this path contains no edges, its strength is defined to be 1. Hence, $(u,u)\in R_V$.

Symmetry: Suppose $(u,v)\in R_V$. Then, by definition, there exists a path $P:u⇝v$ and a path $P^{\prime }:v⇝u$ such that both $s\left(P\right)=1$ and $s\left(P^{\prime }\right)=1$. Therefore, by symmetry of the existence of such paths, $(v,u)\in R_V$.

Transitivity: Suppose $(u,v)\in R_V$ and $(v,w)\in R_V$. Then, there exist paths

• $P_1:u⇝v$ and $P_2:v⇝u$, both with strength 1,
• $Q_1:v⇝w$ and $Q_2:w⇝v$, both with strength 1.

Let $R_1=P_1\circ Q_1$ be the concatenation of paths from u to v and then from v to w. Similarly, define $R_2=Q_2\circ P_2$, a path from w to v and then from v to u.

Since each edge in $P_1$ and $Q_1$ has a membership value of 1 (as their strengths are 1), all edges in $R_1$ have a membership value of 1, and thus $s\left(R_1\right)=1$. The same applies to $R_2$.

Therefore, there exists a path from u to w and from w to u, both of strength 1, and so $(u,w)\in R_V$.

Hence, $R_V$ is reflexive, symmetric, and transitive, and it is, therefore, an equivalence relation. □

The equivalence relation $R_V$ partitions the vertex set V into disjoint subsets, each consisting of vertices that are mutually connected via paths of maximal edge strength. These equivalence classes can be viewed as the strongly connected components of the crisp digraph obtained by taking the $\alpha$-cut of the fuzzy graph $F=(V_{\mu },\lambda )$ at level $\alpha =1$.

Proposition 13.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph, and let $G_1=(V,E_1)$ be the directed graph obtained by the α-cut at level $\alpha =1$, where

$$E_1=\{(u,v)\in V\times V\mid \lambda (u,v)=1\}.$$

Then, the equivalence classes of the relation $R_V$ coincide with the strongly connected components of the crisp digraph $G_1$.

Proof. 

By definition, two vertices $u,v\in V$ satisfy $(u,v)\in R_V$ if and only if there exist paths $P:u⇝v$ and $P^{\prime }:v⇝u$ in F such that $s\left(P\right)=s\left(P^{\prime }\right)=1$. This implies that every edge on these paths must have a membership value of 1, i.e., each edge belongs to $E_1$. Therefore, u and v are connected by directed paths in both directions in the graph $G_1$, which means they lie in the same strongly connected component of $G_1$.

Conversely, if u and v lie in the same strongly connected component of $G_1$, then there exist paths in $G_1$ from u to v and from v to u, consisting entirely of edges with a membership value of 1 in F, and, hence, of strength 1. Therefore, $(u,v)\in R_V$.

Thus, the equivalence classes of $R_V$ are exactly the strongly connected components of $G_1$. □

The following proposition shows that the equivalence classes determined by the relation $R_V$ are ${\odot }_F$-closed components.

Proposition 14.

Given a fuzzy graph $F=(V_{\mu },\lambda )$, every equivalence class $\left[v\right]$ under the relation $R_V$ is closed under ${\odot }_F$; that is, for all $u,w\in \left[v\right]$, we have

$$u{\odot }_{F}w\subseteq \left[v\right].$$

Proof. 

Let $u,w\in \left[v\right]$ and $z\in u{\odot }_{F}w$. By definition of the hyperoperation ${\odot }_F$, this means that z lies on some strongest strong path P from u to w, and the strength of P is ${\mathrm{Con}}_F(u,w)$.

Since $u,w\in \left[v\right]$, we have ${\mathrm{Con}}_F(u,w)={\mathrm{Con}}_F(w,u)=1$ by the definition of $R_V$. Hence, the path P is a strongest strong path from u to w with strength 1. Therefore, every edge of P has a membership value of exactly 1.

Let $P_1$ be the subpath of P from u to z, and $P_2$ the subpath from z to w. Since all edges in P have a membership of 1, the same holds for $P_1$ and $P_2$, and both have a strength of 1; hence,

$${\mathrm{Con}}_F(u,z)={\mathrm{Con}}_F(z,w)=1.$$

Since ${\mathrm{Con}}_F(w,u)=1$, there exists a strongest strong path Q from w to u of strength 1. Consider the concatenation of the paths $P_2$ and Q. This forms a path R from z to u, and all its edges have a membership of 1, i.e.,

$${\mathrm{Con}}_F(z,u)=1.$$

Since ${\mathrm{Con}}_F(u,z)={\mathrm{Con}}_F(z,u)=1$, we have $z\in \left[u\right]$. But $u\in \left[v\right]$ and, hence, $z\in \left[v\right]$. It follows that

$$u{\odot }_{F}w\subseteq \left[v\right].$$

□

Corollary 3.

The vertex set of the quotient graph $F/R_V$ consists precisely of the maximal ${\odot }_F$-closed subsets of V. That is, each equivalence class under $R_V$ is a maximal subset of V that is closed under the hyperoperation ${\odot }_F$, and distinct classes are disjoint with respect to ${\odot }_F$.

Proof. 

By the previous proposition, each equivalence class $\left[v\right]$ under $R_V$ is closed under ${\odot }_F$. To show maximality, suppose there exists $u\notin \left[v\right]$ such that $u\in x{\odot }_{F}y$ for some $x,y\in \left[v\right]$. Then, by the definition of ${\odot }_F$, u lies on a strongest strong path of strength 1 between x and y. This implies ${\mathrm{Con}}_F(x,u)={\mathrm{Con}}_F(u,x)=1$, so $u\in \left[v\right]$, contradicting the assumption. Hence, no element outside $\left[v\right]$ can be generated by ${\odot }_F$ from elements inside it, and the class is maximal.

Distinct classes are disjoint by the definition of $R_V$, and since each is ${\odot }_F$-closed, the equivalence classes partition V into maximal ${\odot }_F$-closed subsets, which form the vertex set of $F/R_V$. □

Remark 3.

The above corollary suggests that the quotient graph $F/R_V$ admits an interpretation as an algebraic reduction of the fuzzy graph F into its maximal ${\odot }_F$-closed components, and one may view the quotient as the base set for an induced hyperstructure.

The relation $R_V$ also admits a natural interpretation in terms of fuzzy connectivity measures. First, if $u{R}_{V}v$, then there exist bidirectional paths of strength 1 between u and v, which implies that the connection strengths satisfy

$${Con}_F(u,v)={Con}_F(v,u)=1.$$

Moreover, in this case, any strongest strong path connecting u and v necessarily consists entirely of edges with membership value 1. As a result, every vertex lying on such a path inherits a membership of at least 1 under the vertex fuzzy path hyperoperation, and in fact retains its original membership value exactly. If no edge in the graph has a membership value of 1, then $R_V$ reduces to the identity relation. Conversely, if all edges have a membership value of 1, then $R_V$ coincides with classical strong connectivity. Therefore, $R_V$ captures the top tier of certainty in fuzzy connectivity and can be used to isolate fully reliable components within a fuzzy network.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph, and let $R_V$ be the equivalence relation on V defined by mutual full-strength connectivity. The quotient fuzzy graph of F with respect to $R_V$, denoted $F/R_V$, is the fuzzy graph $(W_{\nu },\mathrm{\Lambda })$ defined as follows:

-

The vertex set W is the set of equivalence classes of V under $R_V$; that is,

$$W=V/R_V=\{\left[v\right]\mid v\in V\}.$$

-

The fuzzy vertex membership function $\nu :W\to [0,1]$ is given by

$$\nu \left(\right[v\left]\right)=max\left\{\mu \right(u)\mid u\in [v\left]\right\}.$$

-

The fuzzy edge membership function $\mathrm{\Lambda }:W\times W\to [0,1]$ is defined by

$$\mathrm{\Lambda }(\left[u\right],\left[v\right])=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}\left[u\right]=\left[v\right],\hfill \\ max\left\{\lambda \right(x,y)\mid x\in [u],y\in [v\left]\right\},\hfill & \mathrm{if}\left[u\right]\ne \left[v\right].\hfill \end{array}\right.$$

The equivalence relation $R_V$ identifies groups of vertices that are mutually reachable through strongest strong paths, i.e., fully reliable and symmetric fuzzy connections. The corresponding quotient fuzzy graph $F/R_V$ abstracts each equivalence class as a single node and preserves the strongest interclass connections. This reduction simplifies the structure of the fuzzy graph while retaining essential path-based connectivity information, making it easier to analyze large or modular fuzzy systems.

Proposition 15.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph; then, the quotient fuzzy graph $F/R_V=(W_{\nu },\mathrm{\Lambda })$ is well-defined.

Proof. 

The vertex membership function $\nu \left(\right[v\left]\right)=max\left\{\mu \right(u)\mid u\in [v\left]\right\}$ is clearly well-defined since each equivalence class $\left[v\right]$ is nonempty and $\mu :V\to [0,1]$.

The edge membership function is also well-defined, since it considers all pairs $(x,y)$ with $x\in \left[u\right]$, $y\in \left[v\right]$ and selects the maximum of their membership grades:

$$\mathrm{\Lambda }\left(\right[u],[v\left]\right)=max\left\{\lambda \right(x,y)\mid x\in [u],y\in [v\left]\right\}.$$

We now verify that the fuzzy graph condition is preserved; that is,

$$\mathrm{\Lambda }\left(\right[u],[v\left]\right)\le min\left(\nu \right(\left[u\right]),\nu (\left[v\right]\left)\right).$$

Let $x\in \left[u\right]$ and $y\in \left[v\right]$ be such that $\mathrm{\Lambda }\left(\right[u],[v\left]\right)=\lambda (x,y)$. Since $F=(V_{\mu },\lambda )$ is a fuzzy graph, it satisfies

$$\lambda (x,y)\le min\left(\mu \right(x),\mu (y\left)\right).$$

But by the definition of $\nu$, we have $\mu \left(x\right)\le \nu \left(\right[u\left]\right)$ and $\mu \left(y\right)\le \nu \left(\right[v\left]\right)$; hence,

$$\lambda (x,y)\le min\left(\mu \right(x),\mu (y\left)\right)\le min\left(\nu \right(\left[u\right]),\nu (\left[v\right]\left)\right).$$

Therefore,

$$\mathrm{\Lambda }\left(\right[u],[v\left]\right)=\lambda (x,y)\le min\left(\nu \right(\left[u\right]),\nu (\left[v\right]\left)\right).$$

This shows that both $\nu$ and $\mathrm{\Lambda }$ are well-defined and satisfy the fuzzy graph condition. Hence, the quotient $F/R_V=(W_{\nu },\mathrm{\Lambda })$ is a well-defined fuzzy graph. □

The quotient fuzzy graph $F/R_V=(W_{\nu },\mathrm{\Lambda })$ can be interpreted as the condensation of the fuzzy graph F at strength level 1. Each vertex of $F/R_V$ represents a maximal subset of vertices of F that are mutually connected by paths of strength 1, that is, an equivalence class under $R_V$. The internal structure within each $R_V$-class is collapsed into a single node, and the edges between classes reflect the strongest possible fuzzy connections between different full-strength components. In this way, $F/R_V$ provides a simplified representation of the overall connectivity of F among its fully reliable substructures.

We illustrate the quotient fuzzy graph construction process using a flowchart in Figure 3. The diagram outlines the key computational steps, starting from a fuzzy graph $F=(V_{\mu },\lambda )$, through the identification of mutual full-strength connectivity and equivalence classes to the construction of the resulting coarse-grained graph $F/R_V$. Each step preserves directional trust semantics and fuzzy strength while simplifying the graph’s structure.

Corollary 4.

The quotient fuzzy graph $F/R_V=(W_{\nu },\mathrm{\Lambda })$ is the condensation of the fuzzy graph F at strength level 1. Each vertex of $F/R_V$ corresponds to a maximal subset of vertices of F that are mutually connected by paths of strength 1, and the edges of $F/R_V$ represent the strongest fuzzy connections between different such subsets.

Remark 4.

If all edges of a fuzzy graph $F=(V_{\mu },\lambda )$ have a membership value of 1, that is, if $\lambda (u,v)=1$ for all $u,v\in V$, then the relation $R_V$ becomes the universal relation on V. In this case, the quotient fuzzy graph $F/R_V$ consists of a single vertex representing the entire vertex set V and no edges. Thus, $F/R_V$ reduces to a fuzzy graph with one vertex whose membership is the maximum of all vertex memberships in F.

Example 3.

Consider the fuzzy graph $F_3=(V_{\mu },\lambda )$ shown in Figure 4. The vertices $\{v_0,v_1,v_2,v_3\}$ form a fully strongly connected component via full-strength paths and define an equivalence class under $R_V$, denoted $C_1$. The maximum vertex membership in this class is

$$\mu \left(C_1\right)=max\{\mu \left(v_i\right)\mid i=0,1,2,3\}=1.$$

Similarly, $\{v_4,v_5\}$ and $\{v_7,v_8\}$ form strongly connected components, defining the classes $C_2$ and $C_4$ with $\mu \left(C_2\right)=\mu \left(C_4\right)=1.$ The vertex $v_6$, not reachable via any full-strength path from or to another vertex, forms its own singleton class $C_3=\left\{v_6\right\}$, with $\mu \left(C_3\right)=0.8$.

The quotient fuzzy graph $F_3/R_V$, shown in Figure 5, has a vertex set $\{C_1,C_2,C_3,C_4\}$. Each vertex is labeled with its corresponding membership value. An edge is drawn from $C_i$ to $C_j$ whenever there exists an edge from some vertex in $C_i$ to some vertex in $C_j$ in the original graph $F_3$, and the membership of the edge in the quotient is defined as the maximum among those edges.

In this case, there is an edge from $C_1$ to $C_2$ of strength 0.8, coming from $v_3\to v_4$. An edge from $C_2$ to $C_3$ arises from $v_5\to v_6$, with strength 0.6. Finally, an edge from $C_3$ to $C_4$ comes from $v_6\to v_7$, with strength 0.9.

We now introduce a fuzzy distance function based on the strength of the strongest strong path between two vertices in a fuzzy graph. This measure reflects the reliability or certainty of connection between nodes, i.e., the stronger the path, the shorter the distance.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph. The fuzzy distance function$d_F:V\times V\to [0,1]$ is defined by

$$d_F(u,v)=1-{Con}_F(u,v),$$

where ${Con}_F(u,v)$ denotes the strength of the strongest strong path from u to v.

Proposition 16.

The function $d_F$ is a fuzzy semi-metric on V.

Proof. 

We verify the properties of a semi-metric.

(i)

Non-negativity and boundedness. For all $u,v\in V$, since $0\le {Con}_F(u,v)\le 1$, we have

$$0\le d_F(u,v)=1-{Con}_F(u,v)\le 1.$$

(ii)

Identity of indiscernibles. Since the strongest strong path from a vertex to itself is the trivial path with strength 1, we have

$$d_F(u,u)=1-{Con}_F(u,u)=1-1=0.$$

However, $d_F(u,v)=0$ does not imply $u=v$; it only implies that there exists a strongest strong path from u to v with maximum strength. Hence, the identity of indiscernibles holds only in one direction.

(iii)

Symmetry does not hold in general. In directed fuzzy graphs, it is possible that

$${Con}_F(u,v)\ne {Con}_F(v,u),$$

and, therefore,

$$d_F(u,v)\ne d_F(v,u).$$

Thus, $d_F$ is not symmetric unless F is symmetric.

□

This distance may be useful in applications where reliability or confidence of interaction is key, such as in fuzzy social networks, communication systems, and uncertain biological networks. Its inverse relation to connection strength makes it suitable for optimization, ranking, and clustering tasks in fuzzy environments.

The distance function $d_F$ on a fuzzy graph does not, in general, satisfy the identity of indiscernibles in the strong form. However, the quotient graph obtained by the equivalence relation $R_V$ admits this property. This is formalized in the following proposition.

Proposition 17.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph, and let $F/R_V=(W_{\nu },\mathrm{\Lambda })$. The fuzzy distance function $d_{F/R_V}$ satisfies the identity of indiscernibles in the strong form:

$$d_{F/R_V}(\left[u\right],\left[v\right])=0\iff \left[u\right]=\left[v\right].$$

Proof. 

Assume $\left[u\right]=\left[v\right]$. Then, $d_{F/R_V}(\left[u\right],\left[v\right])=1-{\mathrm{Con}}_{F/R_V}(\left[u\right],\left[u\right])=1-1=0$, since the strongest strong path from a class to itself is trivially of strength 1.

Conversely, suppose $\left[u\right]\ne \left[v\right]$. By the construction of the quotient graph, any edge from a vertex in $\left[u\right]$ to a vertex in $\left[v\right]$ must have a strength that is strictly less than 1; otherwise, u and v would belong to the same $R_V$-class, contradicting the assumption. Therefore, every path from $\left[u\right]$ to $\left[v\right]$ in the quotient has a strength strictly less than 1, and, hence,

$${\mathrm{Con}}_{F/R_V}(\left[u\right],\left[v\right])<1\Rightarrow d_{F/R_V}(\left[u\right],\left[v\right])>0.$$

Thus, $d_{F/R_V}(\left[u\right],\left[v\right])=0$ implies $\left[u\right]=\left[v\right]$, completing the proof. □

6. Fuzzy Distance and Metric Interpretations

This interpretation extends naturally to the quotient graph $F/R_V$, where each equivalence class represents a maximal hyperoperation-closed component. The quotient distance function $d_{F/R_V}$ thus captures the separation between these structurally aggregated units, measuring how distant entire hyperoperation closures are from one another in terms of fuzzy connectivity.

The fuzzy distance function $d_F$ can be interpreted in terms of the vertex fuzzy path hyperoperation ${\odot }_F$. The value $d_F(u,v)=1-{\mathrm{Con}}_F(u,v)$ quantifies the extent to which the vertex v is reachable from u via high-strength paths. When $d_F(u,v)=0$, we have ${\mathrm{Con}}_F(u,v)=1$, so $v\in u{\odot }_{F}v$. If in addition, $d_F(v,u)=0$, then $u\in v{\odot }_{F}u$, and the two vertices lie on mutual full-strength strongest strong paths. In that case, they belong to the same equivalence class under $R_V$, which is closed under ${\odot }_F$.

In contrast, when $d_F(u,v)>0$, a strongest strong path from u to v may still exist, but its strength is strictly less than 1. In this case, v may still belong to $u{\odot }_{F}v$, but mutual inclusion and equivalence are not guaranteed. Thus, the distance function reflects how close a pair of vertices is to being symmetrically and fully connected within a ${\odot }_F$-closed structure, and it provides a graded measure of deviation from this ideal.

Proposition 18.

Let $F=(V_{\mu },\lambda )$ be a fuzzy graph; then, for any vertices $u,v\in V$, the following are equivalent:

(i) 

$d_F(u,v)=0$ and $d_F(v,u)=0$,

(ii) 

${\mathrm{Con}}_F(u,v)={\mathrm{Con}}_F(v,u)=1$,

(iii) 

$v\in u{\odot }_{F}v$ and $u\in v{\odot }_{F}u$,

(iv) 

u and v belong to the same equivalence class under $R_V$.

Example 4.

Consider the fuzzy graph $F_4=(V_{\mu },\lambda )$ shown in Figure 6, where each vertex is labeled by its fuzzy membership grade $\mu \left(v\right)$, and each edge is labeled by its membership value $\lambda (u,v)$. Its equivalence classes with respect to $R_V$ are

$$\begin{array}{cc}\hfill C_1& =\{v_1,v_2\},\hfill \\ \hfill C_2& =\{v_3,v_4,v_5\},\hfill \\ \hfill C_3& =\left\{v_6\right\},C_4=\left\{v_7\right\},C_5=\left\{v_8\right\}.\hfill \end{array}$$

The behavior of the vertex fuzzy path hyperoperation ${\odot }_F$ confirms the structural coherence of the equivalence classes under $R_V$. Proposition 14 guarantees that each class $C_i$ is closed under ${\odot }_F$. This aligns with the observation that within each class, all paths contributing to the hyperoperation closure are of full strength and remain entirely internal.

Furthermore, Corollary 3 implies that each equivalence class can be viewed as a ${\odot }_F$-closed component—meaning that $C_i=u{\odot }_{F}w\cup w{\odot }_{F}u$ for all $u,w\in C_i$.

The quotient graph $F_4/R_V$ is shown in Figure 7. For instance, the cycle $v_3\to v_4\to v_5\to v_3$, composed entirely of edges with strength 1, becomes condensed into the single node $C_2$, and the resulting quotient graph shows the directional relationship among the classes with appropriate edge strengths preserved.

The above example illustrates the interpretation of the fuzzy distance $d_F$; as established in Proposition 16, $d_F$ defines a fuzzy semi-metric on the vertex set V, capturing the loss of connectivity strength from one node to another. In the original graph, several pairs of distinct vertices $u\ne v$ satisfy $d_F(u,v)=0$, indicating that there exists a strongest strong path of full strength from u to v, though not necessarily reciprocally. Thus, while $d_F$ quantifies directed closeness, it lacks full symmetry and fails the strong form of identity of indiscernibles.

However, as Proposition 17 confirms, the induced distance $d_{F/R_V}$ on the quotient graph satisfies the identity of indiscernibles in its strong form. In the context of Example 4, this means that the distance between two quotient nodes is zero if and only if they correspond to the same equivalence class under $R_V$. The quotient graph, therefore, provides a coarse-grained metric abstraction in which each node represents a maximal zero-distance component of the original graph.

In practice, this metric can be useful in applications such as information flow modeling, fault-tolerant communication networks, or biological regulatory analysis. The original distance function $d_F$ supports fine-grained ranking and uncertainty-aware navigation, while the quotient distance $d_{F/R_V}$ supports cluster identification and top-level structural reasoning.

7. Applications of Fuzzy Connectivity Structures

The fuzzy hyperoperational framework introduced in this paper provides an algebraic foundation for analyzing systems with partial, asymmetric, or uncertain connectivity. By combining strongest strong path analysis with hypercompositional operations, the proposed model captures local and global properties of interaction networks.

7.1. Trust-Aware Social Systems

We illustrate a simulated trust network modeled as a fuzzy graph with directional fuzzy path analysis. Consider a distributed decision-making system consisting of eight agents $V=\{v_1,v_2,\dots ,v_8\}$, where each agent $v_i$ is assigned a fuzzy credibility score $\mu \left(v_i\right)\in [0,1]$ and directional fuzzy trust scores $\lambda (v_i,v_j)\in [0,1]$ toward others based on prior collaborative experience.

The credibility scores (vertex membership values) are as follows:

$$\mu \left(v_1\right)=0.9,\mu \left(v_2\right)=0.85,\mu \left(v_3\right)=0.8,\mu \left(v_4\right)=0.7,$$

$$\mu \left(v_5\right)=0.8,\mu \left(v_6\right)=0.95,\mu \left(v_7\right)=0.9,\mu \left(v_8\right)=0.9.$$

The directional trust scores (edge membership values) are as follows:

$$\begin{array}{cc}& \lambda (v_1,v_2)=0.8,\lambda (v_2,v_3)=0.8,\lambda (v_3,v_1)=0.8,\hfill \\ & \lambda (v_4,v_5)=\lambda (v_5,v_4)=0.7,\hfill \\ & \lambda (v_6,v_7)=0.9,\lambda (v_7,v_8)=0.85,\lambda (v_8,v_6)=0.9,\hfill \\ & \lambda (v_1,v_6)=0.6.\hfill \end{array}$$

The resulting fuzzy graph $F_5=(V_{\mu },\lambda )$ respects the structural constraint

$$\lambda (u,v)\le min\left(\mu \right(u),\mu (v\left)\right)\mathrm{for}\mathrm{all}u,v\in V,$$

and is visualized in Figure 8. Using the vertex fuzzy path hyperoperation ${\odot }_F$, we compute the strongest strong paths and identify the equivalence classes under the mutual full-strength connectivity relation $R_V$:

$$\begin{array}{cc}\hfill C_1& =\{v_1,v_2,v_3\},\hfill \\ \hfill C_2& =\{v_4,v_5\},\hfill \\ \hfill C_3& =\{v_6,v_7,v_8\}.\hfill \end{array}$$

Each class $C_i$ forms a closed component under ${\odot }_F$, meaning that every pair of nodes within $C_i$ is mutually reachable via full-strength paths. The corresponding quotient fuzzy graph $F_5/R_V$, shown in Figure 9, has a vertex set $\{C_1,C_2,C_3\}$. It includes a single inter-class edge from $C_1$ to $C_3$, derived from the link $v_1\to v_6$ of strength 0.6.

This abstraction highlights the formation of three coherent trust clusters, each internally robust in connectivity. The edge $C_1\to C_3$ signifies a unidirectional trust influence from the first group toward the third. In applied contexts such as team formation or consensus protocols, the quotient graph offers a compact yet semantically meaningful representation, preserving the directionality and strength of trust dynamics in a coarse-grained manner.

The fuzzy distance function $d_F(u_i,u_j)=1-{Con}_F(u_i,u_j)$ can now be applied to the trust network $F_5$ to quantify the absence or weakness of strong trust pathways between agents. In this context, the distance metric helps identify pairs of agents lacking reliable directional influence, cluster users based on proximity in trust propagation, and detect structural gaps or bottlenecks across classes. For example, the edge from $C_1$ to $C_3$ in the quotient graph highlights a weak inter-cluster trust dependency that may warrant strategic reinforcement. Thus, fuzzy distance not only complements hyperoperation-based reasoning but also enables applications in trust evaluation, decision-making optimization, and network resilience analysis.

7.2. Communication Networks

In communication networks or transportation systems, connections often vary in reliability or capacity due to physical limitations, congestion, or failures. Fuzzy graphs provide a natural way to model such uncertainties by allowing both nodes (e.g., stations or routers) and edges (e.g., links or routes) to have membership values representing their operational reliability.

Consider a communication backbone connecting eight relay nodes $\{r_1,r_2,\dots ,r_8\}$, where

$$\mu \left(r_1\right)=\mu \left(r_2\right)=1.0,\mu \left(r_3\right)=\mu \left(r_4\right)=\mu \left(r_5\right)=0.95,$$

$$\mu \left(r_6\right)=0.8,\mu \left(r_7\right)=0.7,\mu \left(r_8\right)=0.6.$$

The edge membership values represent the current quality or bandwidth availability of the connection between two nodes:

$$\lambda (r_1,r_2)=\lambda (r_2,r_1)=1.0,\lambda (r_2,r_3)=0.9,\lambda (r_3,r_4)=\lambda (r_4,r_5)=\lambda (r_5,r_3)=1,$$

$$\lambda (r_5,r_6)=0.8,\lambda (r_6,r_7)=0.7,\lambda (r_7,r_8)=\lambda (r_8,r_6)=0.6.$$

The vertex fuzzy path hyperoperation $r_i{\odot }_F{r}_j$ yields the fuzzy set of relay nodes lying on strongest strong communication paths between $r_i$ and $r_j$, respecting both the edge quality and node reliability. For instance, the strongest strong path from $r_1$ to $r_6$ traverses $r_2\to r_3\to r_4\to r_5\to r_6$, with strength $0.8$, and the involved nodes all have $\mu \ge 0.8$, as ensured by Proposition 6.

The equivalence relation $R_V$ reveals clusters of fully bidirectional, high-confidence communication, such as

$$C_1=\{r_1,r_2\},C_2=\{r_3,r_4,r_5\},C_3=\left\{r_6\right\},C_4=\left\{r_7\right\},C_5=\left\{r_8\right\}.$$

The quotient graph $F/R_V$ abstracts the relay infrastructure into a smaller graph with well-defined inter-cluster bandwidths. For instance, the path $r_2\to r_3$ of strength 0.9 becomes an edge from $C_1$ to $C_2$, and the link $r_5\to r_6$ induces an edge from $C_2$ to $C_3$ of strength 0.8.

The fuzzy distance function $d_F$ supports diagnosing degraded or failing paths, identifying redundant versus critical infrastructure and prioritizing high-strength relay chains for robust routing.

7.3. Biological Systems

In systems biology and epidemiology, the spread of signaling cascades, protein activation, or infectious agents through complex biological or social networks is often non-uniform and uncertain. Nodes (e.g., proteins, genes, or individuals) may differ in their activation readiness or susceptibility, while connections (e.g., reactions or interactions) vary in reliability or strength due to biological noise, mutation, or behavioral diversity. Fuzzy graphs offer a rigorous framework for capturing these graded interactions.

Consider a simplified biological network $F_6=(V_{\mu },\lambda )$ representing signal transduction between nine molecular or cellular entities $\{v_1,\dots ,v_9\}$. Node memberships $\mu \left(v_i\right)$ denote activation readiness or receptor density, and directed edge memberships $\lambda (u,v)$ represent the likelihood or reliability of downstream activation from u to v. The graph is shown in Figure 10 and satisfies all fuzzy graph consistency constraints.

The vertex fuzzy path hyperoperation $v_i{\odot }_{F_6}{v}_j$ identifies all biological units participating in the strongest strong signal paths between components $v_i$ and $v_j$, where both transmission fidelity and local reactivity are preserved. For example, the strongest strong paths, without repeating nodes, from $v_5$ to $v_8$ are

• $v_5\to v_2\to v_3\to v_6\to v_8$
• $v_5\to v_4\to v_1\to v_2\to v_3\to v_6\to v_8$

since they both achieve the maximum possible strength $Co{n}_{F_6}(v_5,v_8)=0.5$, and all edges in these paths are strong edges. Note also that all vertices lying in these paths have membership values $\mu \ge 0.5$, complying with Proposition 6. The vertex fuzzy path hyperoperation $v_5{\odot }_{F_6}{v}_8$ yields the fuzzy set

$$v_5{\odot }_{F_6}{v}_8=\{v_1/1.0,v_2/1.0,v_3/1.0,v_4/1.0,v_5/0.7,v_6/0.7,v_8/0.6\}$$

where each vertex that appears in any of the strongest strong paths retains its original membership value from the fuzzy graph $F_6$.

The equivalence relation $R_V$ captures bidirectional full-strength signaling loops. In this case, five equivalence classes emerge:

$$C_1=\{v_1,v_2,v_3,v_4\},C_2=\left\{v_5\right\},C_3=\left\{v_6\right\},C_4=\left\{v_8\right\},C_5=\{v_7,v_9\}.$$

These groups reflect integrated biological modules or compartments. The quotient fuzzy graph $F_6/R_V$ shown in Figure 11 reduces the system to 5 aggregate components and reveals dominant signal routes between them. The quotient model facilitates analysis of cross-module influence, redundancy, and critical weak points in the network.

In the quotient graph $F_6/R_V$, the only strongest strong path from the class of $v_5$ to the class of $v_8$, i.e., from $C_2$ to $C_4$ is

$$C_2\to C_1\to C_3\to C_4.$$

Therefore, the vertex fuzzy path hyperoperation $C_2{\odot }_{F_6/R_V}{C}_4$ yields the fuzzy set

$$C_2{\odot }_{F_6/R_V}{C}_4=\{C_1/1.0,C_2/0.7,C_3/0.7,C_4/0.6\}.$$

This result captures the signal propagation pathway from the $C_2$ module (containing $v_5$) through the integrated $C_1$ module ($\{v_1,v_2,v_3,v_4\}$) and intermediate $C_3$ module ($v_6$) to reach the target $C_4$ module ($v_8$). In biological terms, this means that signal transmission from module $C_2$ to $C_4$ must pass through the integrated signaling hub $C_1$ and regulatory checkpoint $C_3$, suggesting that these modules are critical for inter-compartment communication and represent potential therapeutic targets or regulatory control points in the biological system.

The introduced hyperoperational framework enables robust subsystem identification (via $R_V$ and $F_6/R_V$), failure impact analysis using fuzzy strong cut vertices and edges, and signal strength-based reachability (via fuzzy distances $d_{F_6}$). Such an approach supports targeted drug design, epidemiological intervention planning, and resilience testing in systems characterized by gradated structure and uncertain propagation.

8. Conclusions and Future Directions

In this paper, we introduce a novel framework of fuzzy graph hyperoperations based on the concept of strongest strong paths, which are both optimal in strength and entirely composed of strong edges. We define two key hyperoperations: the vertex-based fuzzy path hyperoperation ${\odot }_F$ and the edge-based fuzzy path hyperoperation ${\oslash }_F$, generalizing classical path-based operations in a fuzzy environment.

We prove that $(V_{\mu },{\odot }_F)$ forms a fuzzy hypersemigroup and establish fundamental properties such as reflexivity and $\alpha$-cut monotonicity. These results allow for the analysis of fuzzy connectivity at varying levels of confidence. We also characterize structural features such as fuzzy strong cut vertices and edges, which identify points of vulnerability and resilience in fuzzy networks.

A fuzzy distance function $d_F(u,v)=1-{Con}_F(u,v)$ is introduced to quantify directional closeness between vertices. While this function does not satisfy the strong identity of indiscernibles in general, its induced counterpart $d_F/R_V$ on the quotient graph respects this property. The construction of the quotient fuzzy graph $F/R_V$, based on mutual full-strength connectivity, enables a semantically faithful abstraction that preserves the topology of the original graph.

To demonstrate the real-world relevance of the proposed framework, we present detailed examples from trust-aware social networks, communication infrastructures, and biological regulatory systems. These applications illustrate the interpretability and flexibility of our model in domains where uncertainty and asymmetric influence are intrinsic.

While the primary aim of this work is to establish a rigorous algebraic framework for fuzzy graph hyperoperations, it is important to consider their algorithmic realizability in applied settings. The construction of strongest strong paths and the evaluation of the fuzzy hyperoperations can be implemented using adaptations of fuzzy relation composition algorithms, particularly those relying on max–min and max–product transitive closures. For example, the equivalence classes under $R_V$ can be computed via a repeated composition of the fuzzy adjacency matrix until a fixed point is reached, with a worst-case time complexity of $O\left(n^3\right)$, where n is the number of vertices. Once $R_V$ is derived, the construction of quotient graphs proceeds in polynomial time, with each inter-class edge determined via constrained maximum search.

Furthermore, computing the fuzzy distance function $d_F(u,v)$ for all vertex pairs can be achieved using a modified version of the Floyd–Warshall algorithm adapted for fuzzy conjunctions. These algorithmic pathways suggest that the proposed framework is not only mathematically elegant but also computationally tractable.

We envision extending this line of work by developing concrete implementations and integrating them into fuzzy trust modeling, social influence mapping, and graph-based neural architectures. Additional research directions merit further exploration, including developing efficient algorithms for large-scale fuzzy networks, extending to temporal fuzzy graphs where membership values evolve over time, formal development of fuzzy hypermonoid and fuzzy hypergroupoid structures, algorithmic implementations for the efficient computation of ${\odot }_F$ and ${\oslash }_F$, and exploring applications in machine learning where fuzzy hyperoperations could serve as novel graph neural network layers for uncertainty-aware deep learning architectures as well as application to recommendation systems, epidemic modeling, and robustness analysis in intelligent infrastructure.