A Study on Optimizing the Maximal Product in Cubic Fuzzy Graphs for Multifaceted Applications

Abstract

Graphs in the field of science and technology make considerable use of theoretical concepts. When dealing with numerous links and circumstances in which there are varying degrees of ambiguity or robustness in the connections between aspects, rather than purely binary interactions, cubic fuzzy graphs ($\mathcal{C}\mathcal{F}\mathcal{G}s)$ are more adaptable and compatible than fuzzy graphs. To better represent the complexity of interactions or linkages in the real world, an emerging $\mathcal{C}\mathcal{F}\mathcal{G}$ can be very helpful in achieving better problem-solving abilities that specialize in domains like network analysis, the social sciences, information retrieval, and decision support systems. This idea can be used for a variety of uncertainty-related issues and assist decision-makers in selecting the best course of action through the use of a $\mathcal{C}\mathcal{F}\mathcal{G}$. Enhancing the maximized network of three cubic fuzzy graphs’ decision-making efficiency was the ultimate objective of this study. We introduced the maximal product of three cubic fuzzy graphs to investigate how interval-valued fuzzy membership, fuzzy membership, and the miscellany of relations are all simultaneously supported through the aspect of degree and total degree of a vertex. Furthermore, the domination on the maximal product of three $\mathcal{C}\mathcal{F}\mathcal{G}s$ was illustrated to analyze the minimum domination number of the weighted $\mathcal{C}\mathcal{F}\mathcal{G}$, and the proposed approach is illustrated with applications.

1. Introduction

Graph theory in mathematics is practically applied to modeling various problems and occurrences from real-world scenarios. Graphs provide a straightforward method for presenting and visualizing data, such as the interconnections among various entities. Models based on fuzzy environments must be used when there is a lack of confidence in the link between items. A pivotal concept in the domain emerged from the innovative perspective that Zadeh [1] introduced in 1965 with his exposition of fuzzy sets. This notion swiftly garnered widespread momentum across diverse scientific disciplines, encompassing computer science, artificial intelligence, information science, system science, control engineering, management science, operations research, robotics, and theoretical mathematics. A decade after the initial introduction of fuzzy sets, Zadeh presented the concept of an interval-valued fuzzy set, a specialized subset of fuzzy sets characterized by membership values that are intervals ranging from 0 to 1, instead of fuzzy integers. In 1973, Kauffman [2] introduced the notion of fuzzy graphs, which gave rise to different types of graphs. Rosenfeld [3] established fuzzy graph theory in 1975, advancing the field of graph theory. He elucidated several concepts within fuzzy graphs, including trees, cut vertices, cycles, bridges, and terminal vertices. Researchers have explored various types of fuzzy graphs extensively. Talebi [4] conducted a study on Kayley fuzzy graphs. Borzooei et al. [5,6] undertook numerous studies on indeterminate graphs. As a further development of fuzzy sets, Atanassov [7] introduced the notion of intuitionistic fuzzy sets. In 2011, Akram and Dudek [8] proposed the concept of interval-valued fuzzy graphs. Subsequently, Talebi et al. [9,10] presented some innovative concepts concerning interval-valued intuitionistic fuzzy graphs. Kosari et al. [11,12,13] investigated recent developments in the areas of vague graphs and indistinct graph topologies. Rao et al. [14,15] defined the concepts of equitable dominant sets and dominating sets within ambiguous graphs. Shi et al. [16] explored several dominating properties in product-ambiguous graphs. The initial introduction of intuitionistic fuzzy graphs was by Shao et al. [17], and it found applications in supply systems. In 2006, Sampathkumar [18] expanded the concept of graph structures to include signed graphs and graphs with colored or annotated edges. Given that uncertainty and ambiguity often manifest through two or more distinct interactions in numerous real-world scenarios, fuzzy graph structures hold greater significance than traditional graph structures. Dinesh et al. [19] introduced the notion of fuzzy graph structures, discussing several related properties. Akram [20] presented new insights into m-polar fuzzy graphs. Akram and Akmal investigated the concepts of intuitionistic and bipolar fuzzy graph structures in their work [21]. Akram et al. expanded upon this by defining novel operations for fuzzy graph structures [22,23,24]. Kou et al. delved into the structure of fuzzy graphs [25]. In 2020, Dinesh introduced the concept of fuzzy incidence graph structures as a continuation of his research [26]. Utilizing q-rung ortho-pair fuzzy graph structures, Akram and Sitara developed a decision-making method [27]. Furthermore, Sitara and Zafar explored the application of q-rung ortho-pair fuzzy graph structures in airline services [28]. Previously, the only options for fuzzy graphs were interval-valued fuzzy membership or degrees of fuzzy membership. To provide a more versatile tool for simulating uncertainty and ambiguity, Jun et al. [29] introduced the concept of a cubic fuzzy set ($\mathcal{C}\mathcal{F}\mathcal{S}$), which combines a fuzzy set and an interval-valued fuzzy set. Utilizing a $\mathcal{C}\mathcal{F}\mathcal{S}$ can help address various problems related to uncertainty and assist decision-makers in selecting the best option. Furthermore, Jun et al. [30] extended this concept by combining the neutrosophic complex with a $\mathcal{C}\mathcal{F}\mathcal{S}$, proposing the idea of a neutrosophic $\mathcal{C}\mathcal{F}\mathcal{S}$. They also explored $\mathcal{C}\mathcal{F}\mathcal{S}$-based algebraic properties such as cubic interval-valued intuitionistic fuzzy graphs [31], cubic structures [32], cubic sets in semigroups [33], cubic soft sets [34], and cubic intuitionistic structures [35]. Muhiuddin et al. [36] introduced the concept of stable $\mathcal{C}\mathcal{F}\mathcal{S}$s. The study of cubic fuzzy graphs ($\mathcal{C}\mathcal{F}\mathcal{G}s$) and the concept of regularity were explored by Krishna Kumar et al. [37]. Rashid et al. [38] were the first to present the notion of a $\mathcal{C}\mathcal{F}\mathcal{G}$, proposing several novel graph types and their applications. Furthermore, Muhiuddin et al. [39] offered a revised definition of a $\mathcal{C}\mathcal{F}\mathcal{G}$, along with concepts concerning strong edges, paths, path strength, bridges, and cut vertices. Rashmanlou et al. [40,41] discussed some principles of $\mathcal{C}\mathcal{F}\mathcal{G}$s. Rao et al. [42] investigated the maximal product in cubic fuzzy graph structures. Meenakshi et al. [43] studied the intuitionistic fuzzy network using efficient domination and continued the study on equitable domination in neutrosophic graphs using strong arcs [44]. Muhiuddin et al. [45] defined new concepts of domination in cubic graphs. Meenakshi et al. [46] introduced the graph operation of a maximal product on three single-valued neutrosophic graphs and attained the most efficient network for collaboration by finding the minimal spanning tree of the weighted social network. And later, they studied the various operations of single-valued neutrosophic graphs to find an optimal network [47]. When it becomes unfeasible to assign either two fuzzy memberships or two interval-valued memberships to indeterminate variables, the cubic fuzzy graph ($\mathcal{C}\mathcal{F}\mathcal{G}$) emerges as a preferable modeling alternative. This is because it allocates two membership values to each vertex simultaneously. The complexity of the issue increases when numerous connections exist between vertices. In this context, one novel fuzzy graph that could prove extremely beneficial for problem-solving is the $\mathcal{C}\mathcal{F}\mathcal{G}$. Researchers conducted a study on two cubic fuzzy graph operations on Cartesian, maximal, residue, composition, and direct products. In this study, the concept of a $\mathcal{C}\mathcal{F}\mathcal{G}$ was introduced using three $\mathcal{C}\mathcal{F}\mathcal{G}s$ and briefly described. Subsequently, we examined the attributes of the maximal product derived from three $\mathcal{C}\mathcal{F}\mathcal{G}$s. The research encompasses various aspects, including the determination of a vertex’s degree and total degree within the maximal product of three $\mathcal{C}\mathcal{F}\mathcal{G}$s, along with its computation methods, which were used to identify the decision-making efficiency of the network using the domination parameter. Finally, an application of a $\mathcal{C}\mathcal{F}\mathcal{G}$ is illustrated.

2. Preliminaries

In this section, a few foundational concepts are reviewed before the main discussion begins.

A graph, $\mathbb{G}=\left(\mathbb{V},\mathbb{E}\right)$, consists of a non-empty set of $\mathbb{V}$ with ${\mathbb{E}}_1,{\mathbb{E}}_2,{\mathbb{E}}_3,\dots , {\mathbb{E}}_{\mathcal{k}}$ relations with $\mathbb{V}$, which are all mutually disjoint, and each ${\mathbb{E}}_{\mathcal{i}}$ is symmetric and irreflexive for $\mathcal{i}$ = 1,2,…, $\mathcal{k}$.

Definition 1 ([3]). 

A pair, $\mathbb{G}$ = ($\varphi$, $\mu$), represents a fuzzy graph on a non-empty set of $\mathbb{V}$, where $\varphi$ is the fuzzy subset of $\mathbb{V}$, and $\mu$ is the fuzzy relation with ϕ, such that

$$\mu (\mathcal{x}, \mathcal{y})\le \varphi \left(\mathcal{x}\right)\wedge \varphi \left(\mathcal{y}\right), \forall (\mathcal{x}, \mathcal{y})\in \mathbb{V}.$$

Definition 2 ([42]). 

An interval-valued fuzzy set ($ℐ\mathcal{V}\mathcal{F}\mathcal{S}$), $\mathfrak{J}$, is defined as $\mathfrak{J}=\left\{\left[\alpha \left(\mathcal{k}\right),\beta \left(\mathcal{k}\right)\right]|\mathcal{k}\in \mathbb{V}\right\}$, where $\alpha$ and $\beta$ are a fuzzy set of $\mathbb{V}$ so that $\alpha \left(\mathcal{k}\right)$ ≤ $\beta \left(\mathcal{k}\right)$, for all $\mathcal{k}\in \mathbb{V}$.

For two IVFSs, $\mathfrak{L}$and $\mathfrak{M}$, we define the following:

$$\mathfrak{L}{{\displaystyle \cup }}^{ }\mathfrak{M} \{[{\alpha }_{\mathfrak{L}}\left(\mathcal{k}\right)\vee {\alpha }_{\mathfrak{M}}\left(\mathcal{k}\right), {\beta }_{\mathfrak{L}}\left(\mathcal{k}\right)\vee {\beta }_{\mathfrak{M}}\left(\mathcal{k}\right)] | \mathcal{k}\in \mathbb{V}\},$$

$$\mathfrak{L}{{\displaystyle \cap }}^{ }\mathfrak{M} \{[{\alpha }_{\mathfrak{L}}\left(\mathcal{k}\right)\wedge {\alpha }_{\mathfrak{M}}\left(\mathcal{k}\right), {\beta }_{\mathfrak{L}}\left(\mathcal{k}\right)\wedge {\beta }_{\mathfrak{M}}\left(\mathcal{k}\right)] | \mathcal{k}\in \mathbb{V}\}.$$

Definition 3 ([17]). 

A cubic fuzzy set ($\mathcal{C}\mathcal{F}\mathcal{S})$ of ${\mathcal{C}}_{\mathfrak{s}}$ on a non-empty set of $\mathbb{V}$ is defined as ${\mathcal{C}}_{\mathfrak{s}}=\{\langle \left[\alpha \left(\mathcal{k}\right),\beta \left(\mathcal{k}\right)\right], \gamma \left(\mathcal{k}\right)\rangle |\mathcal{k}\in \mathbb{V}\},$ where $\left[\alpha \left(\mathcal{k}\right),\beta \left(\mathcal{k}\right)\right]$ is the interval-valued fuzzy membership degree, and $\gamma \left(\mathcal{k}\right)$ is the fuzzy membership degree of $\mathcal{k}$. If $\gamma \left(\mathcal{k}\right)$ ∈ $\left[\alpha \left(\mathcal{k}\right),\beta \left(\mathcal{k}\right)\right]$, then it is called an internal cubic fuzzy set, and if $\gamma \left(\mathcal{k}\right)$ ∈ $\left[\alpha \left(\mathcal{k}\right),\beta \left(\mathcal{k}\right)\right]$, then it is called as external cubic fuzzy set.

Definition 4 ([42]). 

Let ${\mathcal{G}}_{\mathbb{c}}=\left(\mathfrak{L},\mathfrak{W}\right)$ be a pair on a non-empty set of $\mathbb{V}$ that is said to be a cubic fuzzy graph ($\mathcal{C}\mathcal{F}\mathcal{G}$) where $\mathfrak{L}$ is a cubic fuzzy set on $\mathbb{V}$, and $\mathfrak{W}$ is a cubic fuzzy set on $\mathbb{V}\times \mathbb{V}$, so that, for all, $\mathcal{p}\mathcal{q}$ ∈ ${\mathfrak{M}}^{\ast }$,

$${\alpha }_{\mathfrak{M}}\left(\mathcal{p}\mathcal{q}\right)\le {\alpha }_{\mathfrak{L}}\left(\mathcal{p}\right)\wedge {\alpha }_{\mathfrak{L}}\left(\mathcal{q}\right),$$

$${\beta }_{\mathfrak{L}}\left(\mathcal{p}\mathcal{q}\right)\le {\beta }_{\mathfrak{L}}\left(\mathcal{p}\right)\wedge {\beta }_{\mathfrak{L}}\left(\mathcal{q}\right),$$

$${\gamma }_{\mathfrak{M}}\left(\mathcal{p}\mathcal{q}\right)\le {\gamma }_{\mathfrak{L}}\left(\mathcal{p}\right)\wedge {\gamma }_{\mathfrak{L}}\left(\mathcal{q}\right).$$

Definition 5 ([45]). 

The $\mathcal{F}\mathcal{G}$ ${\mathcal{G}}_{\mathbb{c}}$ is said to be complete if ∀ $\mathcal{p},\mathcal{q}$ ∈ $\mathbb{V}$

$${\alpha }_{\mathfrak{M}}\left(\mathcal{p}\mathcal{q}\right)= {\alpha }_{\mathfrak{L}}\left(\mathcal{p}\right)\wedge {\alpha }_{\mathfrak{L}}\left(\mathcal{q}\right),$$

$${\beta }_{\mathfrak{M}}\left(\mathcal{p}\mathcal{q}\right)= {\beta }_{\mathfrak{L}}\left(\mathcal{p}\right)\wedge {\beta }_{\mathfrak{L}}\left(\mathcal{q}\right),$$

$${\gamma }_{\mathfrak{M}}\left(\mathcal{p}\mathcal{q}\right)={\gamma }_{\mathfrak{L}}\left(\mathcal{p}\right)\wedge {\gamma }_{\mathfrak{L}}\left(\mathcal{q}\right).$$

Definition 6 ([42]). 

The $\mathcal{F}\mathcal{G}$ ${\mathcal{G}}_{\mathbb{c}}=\left(\mathfrak{L},{\mathfrak{M}}_1, {\mathfrak{M}}_2,\dots , {\mathfrak{M}}_{\mathcal{r}}\right)$ is said to be ${\mathfrak{M}}_{\mathcal{j}}-$strong if ∀ $\mathcal{p}\mathcal{q}$ ∈ ${\mathbb{E}}_{\mathcal{j}}$

$${\alpha }_{{\mathfrak{M}}_{\mathcal{j}}}\left(\mathcal{p}\mathcal{q}\right)={\alpha }_{\mathfrak{L}}\left(\mathcal{p}\right)\wedge {\alpha }_{\mathfrak{L}}\left(\mathcal{q}\right),$$

$${\beta }_{{\mathfrak{M}}_{\mathcal{j}}}\left(\mathcal{p}\mathcal{q}\right)= {\beta }_{\mathfrak{L}}\left(\mathcal{p}\right)\wedge {\beta }_{\mathfrak{L}}\left(\mathcal{q}\right),$$

$${\gamma }_{{\mathfrak{M}}_{\mathcal{j}}}\left(\mathcal{p}\mathcal{q}\right)={\gamma }_{\mathfrak{L}}\left(\mathcal{p}\right)\wedge {\gamma }_{\mathfrak{L}}\left(\mathcal{q}\right).$$

If ${\mathcal{G}}_{\mathbb{c}}$, is ${\mathfrak{M}}_{\mathcal{j}}-$strong for all $\mathcal{j}=1,2,\dots ,\mathcal{r}$, then ${\mathcal{G}}_{\mathbb{c}}$is called a strong$\mathcal{C}\mathcal{F}\mathcal{G}$.

Definition 7 ([42]). 

Let $\mathbb{V}$ be a non-empty set and $\mathbb{G}=(\mathbb{V}$, ${\mathbb{E}}_1, {\mathbb{E}}_2,\dots , {\mathbb{E}}_{\mathcal{r}}$) be a graph. Then, ${\mathfrak{G}}_{\mathfrak{c}}=\left({\mathfrak{L}}^{\ast },{{\mathfrak{M}}^{\ast }}_1, {{\mathfrak{M}}^{\ast }}_2,\dots , {{\mathfrak{M}}^{\ast }}_{\mathcal{r}}\right)$ is a cubic fuzzy graph ($\mathcal{C}\mathcal{F}\mathcal{G}$) on $\mathbb{G}$ if ${\mathfrak{L}}^{\ast }$ is a cubic fuzzy set on $\mathbb{V}$, and ${\mathfrak{M}}^{\ast }$ is a cubic fuzzy set on ${\mathbb{E}}_1, {\mathbb{E}}_2,\dots , {\mathbb{E}}_{\mathcal{r}}$. Then, the maximal product of two $\mathcal{C}\mathcal{F}\mathcal{G}s is defined as follows:$

$${\alpha }_{{\mathfrak{M}}_{\mathcal{j}}}\left(\mathcal{p}\mathcal{q}\right)\le {\alpha }_{\mathfrak{L}}\left(\mathcal{p}\right)\wedge {\alpha }_{\mathfrak{L}}\left(\mathcal{q}\right),$$

$${\beta }_{{\mathfrak{M}}_{\mathcal{j}}}\left(\mathcal{p}\mathcal{q}\right)\le {\beta }_{\mathfrak{L}}\left(\mathcal{p}\right)\wedge {\beta }_{\mathfrak{L}}\left(\mathcal{q}\right),$$

$${\gamma }_{{\mathfrak{M}}_{\mathcal{j}}}\left(\mathcal{p}\mathcal{q}\right)\le {\gamma }_{\mathfrak{L}}\left(\mathcal{p}\right)\wedge {\gamma }_{\mathfrak{L}}\left(\mathcal{q}\right). \mathit{For\; all} \mathcal{p}\mathcal{q} \in {\mathfrak{M}}_{\mathcal{j}}\mathit{and} \mathcal{j}=1,2,\dots ,\mathcal{r}.$$

If $\mathcal{p}\mathcal{q}$ ∈ $Supp\left({\mathfrak{M}}_{\mathcal{j}}\right)$, then $\mathcal{p}\mathcal{q}$ is known as the ${\mathfrak{M}}_{\mathcal{j}}$-edge of $\mathcal{F}\mathcal{G} {\mathcal{G}}_{\mathbb{c}}$. Moreover, ${\mathfrak{M}}_1, {\mathfrak{M}}_2,\dots , {\mathfrak{M}}_{\mathcal{r}}$ are mutually disjoint such that each ${\alpha }_{\mathcal{j}},{\beta }_{\mathcal{j}} and {\gamma }_{\mathcal{j}}$ is symmetric and reflexive for $1\le \mathcal{j}\le \mathcal{r}$.

Definition 8 ([42]). 

The degree of a vertex in the maximal product ${\mathfrak{G}}_{\mathfrak{c}}=\left({\mathfrak{L}}^{\ast },{{\mathfrak{M}}^{\ast }}_1, {{\mathfrak{M}}^{\ast }}_2,\dots , {{\mathfrak{M}}^{\ast }}_{\mathcal{r}}\right)$ of two $\mathcal{F}\mathcal{G}s,$ ${\mathcal{G}}_{\mathbb{c}}=\left(\mathfrak{L},{\mathfrak{M}}_1, {\mathfrak{M}}_2,\dots , {\mathfrak{M}}_{\mathcal{r}}\right)$ and ${\mathcal{G}}_{\mathbb{c}}^{\prime }=\left({\mathfrak{L}}^{\prime },{\mathfrak{M}}_1^{\prime }, {\mathfrak{M}}_2^{\prime },\dots , {\mathfrak{M}}_{\mathcal{r}}^{\prime }\right)$, is defined as follows:

$${\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right)=\langle \left[{\mathfrak{D}}_{\alpha }\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right),{\mathfrak{D}}_{\beta }\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right)\right],{\mathfrak{D}}_{\gamma }\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right)\rangle where,$$

$${\mathfrak{D}}_{\alpha }\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right)={\displaystyle \sum }_{{\mathcal{j}}_{\mathcal{a}}{\mathcal{j}}_{\mathcal{p}}\in {\mathbb{E}}_{\mathfrak{i}},{\mathcal{k}}_{\mathcal{b}}={\mathcal{k}}_{\mathcal{q}}}{\alpha }_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{j}}_{\mathcal{a}}{\mathcal{j}}_{\mathcal{p}}\right)\vee {\alpha }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_{\mathcal{q}}\right)+{\displaystyle \sum }_{{\mathcal{k}}_{\mathcal{b}}{\mathcal{k}}_{\mathcal{q}}\in {\mathbb{E}}_{\mathfrak{i}}^{\prime },{\mathcal{j}}_{\mathcal{a}}={\mathcal{j}}_{\mathcal{p}}}{\alpha }_{{{\mathfrak{M}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{k}}_{\mathcal{b}}{\mathcal{k}}_{\mathcal{q}}\right)\vee {\alpha }_{\mathfrak{L}}\left({\mathcal{j}}_{\mathcal{p}}\right)$$

$${\mathfrak{D}}_{\beta }\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right)={\displaystyle \sum }_{{\mathcal{j}}_{\mathcal{a}}{\mathcal{j}}_{\mathcal{p}}\in {\mathbb{E}}_{\mathfrak{i}},{\mathcal{k}}_{\mathcal{b}}={\mathcal{k}}_{\mathcal{q}}}{\beta }_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{j}}_{\mathcal{a}}{\mathcal{j}}_{\mathcal{p}}\right)\vee {\beta }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_{\mathcal{q}}\right)+{\displaystyle \sum }_{{\mathcal{k}}_{\mathcal{b}}{\mathcal{k}}_{\mathcal{q}}\in {\mathbb{E}}_{\mathfrak{i}}^{\prime },{\mathcal{j}}_{\mathcal{a}}={\mathcal{j}}_{\mathcal{p}}}{\beta }_{{{\mathfrak{M}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{k}}_{\mathcal{b}}{\mathcal{k}}_{\mathcal{q}}\right)\vee {\beta }_{\mathfrak{L}}\left({\mathcal{j}}_{\mathcal{p}}\right)$$

$${\mathfrak{D}}_{\gamma }\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right)={\displaystyle \sum }_{\begin{array}{c}{\mathcal{j}}_{\mathcal{a}}{\mathcal{j}}_{\mathcal{p}}\\ \in {\mathbb{E}}_{\mathfrak{i}},{\mathcal{k}}_{\mathcal{b}}={\mathcal{k}}_{\mathcal{q}}\end{array}}{\gamma }_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{j}}_{\mathcal{a}}{\mathcal{j}}_{\mathcal{p}}\right)\vee {\gamma }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_{\mathcal{q}}\right)+{\displaystyle \sum }_{{\mathcal{k}}_{\mathcal{b}}{\mathcal{k}}_{\mathcal{q}}\in {\mathbb{E}}_{\mathfrak{i}}^{\prime },{\mathcal{j}}_{\mathcal{a}}={\mathcal{j}}_{\mathcal{p}}}{\gamma }_{{{\mathfrak{M}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{k}}_{\mathcal{b}}{\mathcal{k}}_{\mathcal{q}}\right)\vee {\gamma }_{\mathfrak{L}}\left({\mathcal{j}}_{\mathcal{p}}\right)$$

Definition 9 ([42]). 

The total degree of a vertex in the maximal product, ${\mathfrak{G}}_{\mathfrak{c}}=\left({\mathfrak{L}}^{\ast },{{\mathfrak{M}}^{\ast }}_1, {{\mathfrak{M}}^{\ast }}_2,\dots , {{\mathfrak{M}}^{\ast }}_{\mathcal{r}}\right)$, of two $\mathcal{F}\mathcal{G}s$, ${\mathcal{G}}_{\mathbb{c}}=\left(\mathfrak{L},{\mathfrak{M}}_1, {\mathfrak{M}}_2,\dots , {\mathfrak{M}}_{\mathcal{r}}\right)$ and ${\mathcal{G}}_{\mathbb{c}}^{\prime }=\left({\mathfrak{L}}^{\prime },{\mathfrak{M}}_1^{\prime }, {\mathfrak{M}}_{\prime }^2,\dots , {\mathfrak{M}}_{\mathcal{r}}^{\prime }\right)$, is defined as follows:

$$\mathbb{T}{\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right)=\langle \left[\mathbb{T}{\mathfrak{D}}_{\alpha }\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right),\mathbb{T}{\mathfrak{D}}_{\beta }\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right)\right],\mathbb{T}{\mathfrak{D}}_{\gamma }\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right)\rangle \mathit{where,}$$

$$\mathbb{T}{\mathfrak{D}}_{\alpha }\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right)={\displaystyle \sum }_{{\mathcal{j}}_{\mathcal{a}}{\mathcal{j}}_{\mathcal{p}}\in {\mathbb{E}}_{\mathfrak{i}},{\mathcal{k}}_{\mathcal{b}}={\mathcal{k}}_{\mathcal{q}}}{\alpha }_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{j}}_{\mathcal{a}}{\mathcal{j}}_{\mathcal{p}}\right)\vee {\alpha }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_{\mathcal{q}}\right)+{\displaystyle \sum }_{{\mathcal{k}}_{\mathcal{b}}{\mathcal{k}}_{\mathcal{q}}\in {\mathbb{E}}_{\mathfrak{i}}^{\prime },{\mathcal{j}}_{\mathcal{a}}={\mathcal{j}}_{\mathcal{p}}}{\alpha }_{{{\mathfrak{M}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{k}}_{\mathcal{b}}{\mathcal{k}}_{\mathcal{q}}\right)\vee {\alpha }_{\mathfrak{L}}\left({\mathcal{j}}_{\mathcal{p}}\right)$$

$$\mathbb{T}{\mathfrak{D}}_{\beta }\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right)={\displaystyle \sum }_{{\mathcal{j}}_{\mathcal{a}}{\mathcal{j}}_{\mathcal{p}}\in {\mathbb{E}}_{\mathfrak{i}},{\mathcal{k}}_{\mathcal{b}}={\mathcal{k}}_{\mathcal{q}}}{\beta }_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{j}}_{\mathcal{a}}{\mathcal{j}}_{\mathcal{p}}\right)\vee {\beta }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_{\mathcal{q}}\right)+{\displaystyle \sum }_{{\mathcal{k}}_{\mathcal{b}}{\mathcal{k}}_{\mathcal{q}}\in {\mathbb{E}}_{\mathfrak{i}}^{\prime },{\mathcal{j}}_{\mathcal{a}}={\mathcal{j}}_{\mathcal{p}}}{\beta }_{{\mathfrak{M}}_{\mathfrak{i}}^{\prime }}\left({\mathcal{k}}_{\mathcal{b}}{\mathcal{k}}_{\mathcal{q}}\right)\vee {\beta }_{\mathfrak{L}}\left({\mathcal{j}}_{\mathcal{p}}\right)$$

$$\mathbb{T}{\mathfrak{D}}_{\gamma }\left({\mathcal{j}}_{\mathcal{p}},{\mathcal{k}}_{\mathcal{q}}\right)={\displaystyle \sum }_{{\mathcal{j}}_{\mathcal{a}}{\mathcal{j}}_{\mathcal{p}}\in {\mathbb{E}}_{\mathfrak{i}},{\mathcal{k}}_{\mathcal{b}}={\mathcal{k}}_{\mathcal{q}}}{\gamma }_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{j}}_{\mathcal{a}}{\mathcal{j}}_{\mathcal{p}}\right)\vee {\gamma }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_{\mathcal{q}}\right)+{\displaystyle \sum }_{{\mathcal{k}}_{\mathcal{b}}{\mathcal{k}}_{\mathcal{q}}\in {\mathbb{E}}_{\mathfrak{i}}^{\prime },{\mathcal{j}}_{\mathcal{a}}={\mathcal{j}}_{\mathcal{p}}}{\gamma }_{{{\mathfrak{M}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{k}}_{\mathcal{b}}{\mathcal{k}}_{\mathcal{q}}\right)\vee {\gamma }_{\mathfrak{L}}\left({\mathcal{j}}_{\mathcal{p}}\right)$$

Definition 10 ([45]). 

The cardinality of a vertex, $\mathcal{x}\in \mathbb{V}$, in a $\mathcal{C}\mathcal{F}\mathcal{G}, {\mathfrak{G}}_{\mathfrak{c}}=\left({\mathfrak{L}}^{\ast },{\mathfrak{M}}^{\ast }\right)$ is defined using

$$ℒ={\displaystyle \sum }_{\mathcal{x}\in \mathbb{V}}\left(\frac{1-{\mu }_{{\mathfrak{L}}^{\ast }}^L\left(\mathcal{x}\right)+{\mu }_{{\mathfrak{L}}^{\ast }}^U\left(\mathcal{x}\right)+{\sigma }_{{\mathfrak{L}}^{\ast }}\left(\mathcal{x}\right)}{3}\right)$$

The cardinality of an edge, $\mathcal{x}\mathcal{y}\in \mathbb{E}$, in a $\mathcal{C}\mathcal{F}\mathcal{G}$, ${\mathfrak{G}}_{\mathfrak{c}}=\left({\mathfrak{L}}^{\ast },{\mathfrak{M}}^{\ast }\right)$, is defined using

$$\mathcal{M}={\displaystyle \sum }_{\mathcal{x}\mathcal{y}\in \mathbb{E}}\left(\frac{1-{\mu }_{{\mathfrak{M}}^{\ast }}^L\left(\mathcal{x}\mathcal{y}\right)+{\mu }_{{\mathfrak{M}}^{\ast }}^U\left(\mathcal{x}\mathcal{y}\right)+{\sigma }_{{\mathfrak{M}}^{\ast }}\left(\mathcal{x}\mathcal{y}\right)}{3}\right)$$

Definition 11 ([45]). 

Let $\mathcal{C}\mathcal{F}\mathcal{G}, {\mathfrak{G}}_{\mathfrak{c}}=\left({\mathfrak{L}}^{\ast },{\mathfrak{M}}^{\ast }\right),$ be a $\mathcal{C}\mathcal{F}\mathcal{G}$ over $\mathbb{V}$ and $\mathcal{x},\mathcal{y}\in \mathbb{V}$, and then $\mathcal{x}$ dominates if there exists a strong cubic edge between $\mathcal{x}$ and $\mathcal{y}$.

Definition 12 ([45]). 

Let $\mathcal{C}\mathcal{F}\mathcal{G}, {\mathfrak{G}}_{\mathfrak{c}}=\left({\mathfrak{L}}^{\ast },{\mathfrak{M}}^{\ast }\right),$ be a $\mathcal{C}\mathcal{F}\mathcal{G}$ over $\mathbb{V}$, and $\mathbb{S}\subseteq \mathbb{V}$ is called the dominating set in ${\mathfrak{G}}_{\mathfrak{c}}$ if, for every $\mathcal{y}\notin \mathbb{S}$, there exists $\mathcal{x}\in \mathbb{S}$ such that $\mathcal{x}$ dominates $\mathcal{y}$. The minimum cardinality of all dominating sets in ${\mathfrak{G}}_{\mathfrak{c}}$ is called the domination number of ${\mathfrak{G}}_{\mathfrak{c}}$ and denoted as ${\gamma }_{ \mathcal{C}\mathcal{F}\mathcal{G}}\left({\mathfrak{G}}_{\mathfrak{c}}\right)$.

3. Maximal Product of Three Cubic Fuzzy Graphs

Let $\mathcal{G}$ = ($\varphi$, $\mu$) be a fuzzy graph with the underlying crisp graph $\mathbb{G}=(\mathbb{V}$, ${\mathbb{E}}_1, {\mathbb{E}}_2,\dots , {\mathbb{E}}_{\mathcal{r}}$) if we frame the maximal product of two fuzzy graphs, ${\mathcal{G}}_1$ and ${\mathcal{G}}_2$, which maximizes only the membership value of the maximal product, whereas in an intuitionistic fuzzy graph, it defines both truth and falsity membership values of the maximal product. In a neutrosophic graph, it maximizes the truth, indeterminacy, and falsity membership values of the maximal product. But in a cubic fuzzy graph, the membership value is defined in both interval-valued and fuzzy values. When three distinct types of cubic fuzzy graph attributes are maximized, their optimal level is characterized as the interval-valued membership value or the fuzzy membership value, which is more efficient than other types of fuzzy graphs. Maximizing the product operation in cubic fuzzy graphs offers us an abundance of opportunities to solve multidimensional attributes. This strategy not only broadens the range of applications but also improves the efficiency and efficacy of solutions across several areas.

Let ${\mathcal{G}}_{\mathbb{c}}=\left(\mathfrak{L},{\mathfrak{M}}_1, {\mathfrak{M}}_2,\dots , {\mathfrak{M}}_{\mathcal{r}}\right)$, ${\mathcal{G}}_{\mathbb{c}}^{\prime }=\left({\mathfrak{L}}^{\prime },{\mathfrak{M}}_1^{\prime }, {\mathfrak{M}}_2^{\prime },\dots , {\mathfrak{M}}_{\mathcal{r}}^{\prime }\right)$, and ${\mathcal{G}}_{\mathbb{c}}^{″}=\left({\mathfrak{L}}^{″},{\mathfrak{M}}_1^{″}, {\mathfrak{M}}_2^{″},\dots , {\mathfrak{M}}_{\mathcal{r}}^{″}\right)$ be three $\mathcal{C}\mathcal{F}\mathcal{G}$s with underlying crisp graphs $\mathbb{G}=(\mathbb{V}$, ${\mathbb{E}}_1, {\mathbb{E}}_2,\dots , {\mathbb{E}}_{\mathcal{r}}$), ${\mathbb{G}}^{\prime }=({\mathbb{V}}^{\prime }$, ${\mathbb{E}}_1^{\prime }, {\mathbb{E}}_2^{\prime },\dots , {\mathbb{E}}_{\mathcal{r}}^{\prime }$), and ${\mathbb{G}}^{″}=({\mathbb{V}}^{″}$, ${\mathbb{E}}_1^{″}, {\mathbb{E}}_2^{″},\dots , {\mathbb{E}}_{\mathcal{r}}^{″}$), respectively. Then, ${\mathcal{G}}_{\mathbb{c}}{\ast }_{\mathcal{M}\mathcal{P}}{\mathcal{G}}_{\mathbb{c}}^{\prime }{\ast }_{\mathcal{M}\mathcal{P}}{\mathcal{G}}_{\mathbb{c}}^{″}=\left(\mathfrak{L}{\ast }_{\mathcal{M}\mathcal{P}}{\mathfrak{L}}^{\prime }{\ast }_{\mathcal{M}\mathcal{P}}{\mathfrak{L}}^{″}, \mathfrak{M}{\ast }_{\mathcal{M}\mathcal{P}}{\mathfrak{M}}^{\prime }{\ast }_{\mathcal{M}\mathcal{P}}{\mathfrak{M}}^{″}\right)=\left({\mathfrak{L}}^{\ast },{{\mathfrak{M}}^{\ast }}_1, {{\mathfrak{M}}^{\ast }}_2,\dots , {{\mathfrak{M}}^{\ast }}_{\mathcal{r}}\right)$ is the maximal product of ${\mathcal{G}}_{\mathbb{c}}, {\mathcal{G}}_{\mathbb{c}}^{\prime }$ and ${\mathcal{G}}_{\mathbb{c}}^{″}$. Where $\mathcal{V}=\mathbb{V}\times {\mathbb{V}}^{\prime }\times {\mathbb{V}}^{″}$ and ${\epsilon }_{\mathcal{r}}=\left\{\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right)\right|{\mathcal{j}}_1={\mathcal{j}}_2,{\mathcal{k}}_1={\mathcal{k}}_2,{\mathcal{s}}_1{\mathcal{s}}_2\in {\mathbb{E}}_{\mathcal{r}}^{″} \mathrm{or} {\mathcal{j}}_1={\mathcal{j}}_2,{\mathcal{s}}_1={\mathcal{s}}_2,{\mathcal{k}}_1{\mathcal{k}}_2\in {\mathbb{E}}_{\mathcal{r}}^{\prime } \mathrm{or} {\mathcal{k}}_1={\mathcal{k}}_2,{\mathcal{s}}_1={\mathcal{s}}_2,{\mathcal{j}}_1{\mathcal{j}}_2\in {\mathbb{E}}_{\mathcal{r}}$}, and it is defined as

$${\mathfrak{L}}^{\ast }\left(\mathcal{j},\mathcal{k},\mathcal{s}\right)=\left(\mathfrak{L}{\ast }_{\mathcal{M}\mathcal{P}}{\mathfrak{L}}^{\prime }{\ast }_{\mathcal{M}\mathcal{P}}{\mathfrak{L}}^{″}\right)\left(\mathcal{j},\mathcal{k},\mathcal{s}\right)=\{\begin{array}{c}{\mathsf{\alpha }}_{\mathfrak{L}}\left(\mathcal{j}\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left(\mathcal{k}\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left(\mathcal{s}\right)\\ {\mathsf{\beta }}_{\mathfrak{L}}\left(\mathcal{j}\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{\prime }}\left(\mathcal{k}\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{″}}\left(\mathcal{s}\right)\\ {\mathsf{\gamma }}_{\mathfrak{L}}\left(\mathcal{j}\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{\prime }}\left(\mathcal{k}\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{″}}\left(\mathcal{s}\right)\\ \end{array}$$

$$\begin{gathered} {{\mathfrak{M}}^{\ast }}_{\mathfrak{i}}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right)= \\ \{\begin{array}{c}\{\begin{array}{c}{\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\mathsf{\alpha }}_{{{\mathfrak{M}}_{\mathfrak{i}}}^{″}}\left({\mathcal{s}}_1{\mathcal{s}}_2\right), {\mathcal{j}}_1={\mathcal{j}}_2, {\mathcal{k}}_1={\mathcal{k}}_2, {\mathcal{s}}_1{\mathcal{s}}_2\in {\mathbb{E}}_{\mathfrak{i}}^{″}\\ {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{M}}_{\mathfrak{i}}}\left({\mathcal{j}}_1{\mathcal{j}}_2\right), {\mathcal{k}}_1={\mathcal{k}}_2, {\mathcal{s}}_1={\mathcal{s}}_2, {\mathcal{j}}_1{\mathcal{j}}_2\in {\mathbb{E}}_{\mathfrak{i}}\\ {\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)\vee {\mathsf{\alpha }}_{{{\mathfrak{M}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{k}}_1{\mathcal{k}}_2\right), {\mathcal{j}}_1={\mathcal{j}}_2, {\mathcal{s}}_1={\mathcal{s}}_2, {\mathcal{k}}_1{\mathcal{k}}_2\in {\mathbb{E}}_{\mathfrak{i}}^{\prime }\end{array}\\ \{\begin{array}{c}{\mathsf{\beta }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\mathsf{\beta }}_{{{\mathfrak{M}}_{\mathfrak{i}}}^{″}}\left({\mathcal{s}}_1{\mathcal{s}}_2\right), {\mathcal{j}}_1={\mathcal{j}}_2, {\mathcal{k}}_1={\mathcal{k}}_2, {\mathcal{s}}_1{\mathcal{s}}_2\in {\mathbb{E}}_{\mathfrak{i}}^{″}\\ {\mathsf{\beta }}_{\mathfrak{L}}\left({\mathcal{k}}_1\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{s}}_1\right)\vee {\mathsf{\beta }}_{{\mathfrak{M}}_{\mathfrak{i}}}\left({\mathcal{j}}_1{\mathcal{j}}_2\right), {\mathcal{k}}_1={\mathcal{k}}_2, {\mathcal{s}}_1={\mathcal{s}}_2, {\mathcal{j}}_1{\mathcal{j}}_2\in {\mathbb{E}}_{\mathfrak{i}}\\ {\mathsf{\beta }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{s}}_1\right)\vee {\mathsf{\beta }}_{{{\mathfrak{M}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{k}}_1{\mathcal{k}}_2\right), {\mathcal{j}}_1={\mathcal{j}}_2, {\mathcal{s}}_1={\mathcal{s}}_2, {\mathcal{k}}_1{\mathcal{k}}_2\in {\mathbb{E}}_{\mathfrak{i}}^{\prime }\end{array}\\ \{\begin{array}{c}{\mathsf{\gamma }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\mathsf{\vee }{\gamma }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\mathsf{\gamma }}_{{{\mathfrak{M}}_{\mathfrak{i}}}^{″}}\left({\mathcal{s}}_1{\mathcal{s}}_2\right), {\mathcal{j}}_1={\mathcal{j}}_2, {\mathcal{k}}_1={\mathcal{k}}_2, {\mathcal{s}}_1{\mathcal{s}}_2\in {\mathbb{E}}_{\mathfrak{i}}^{″}\\ {\mathsf{\gamma }}_{\mathfrak{L}}\left({\mathcal{k}}_1\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{s}}_1\right)\vee {\mathsf{\gamma }}_{{\mathfrak{M}}_{\mathfrak{i}}}\left({\mathcal{j}}_1{\mathcal{j}}_2\right), {\mathcal{k}}_1={\mathcal{k}}_2, {\mathcal{s}}_1={\mathcal{s}}_2, {\mathcal{j}}_1{\mathcal{j}}_2\in {\mathbb{E}}_{\mathfrak{i}}\\ {\mathsf{\gamma }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{s}}_1\right)\vee {\mathsf{\gamma }}_{{{\mathfrak{M}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{k}}_1{\mathcal{k}}_2\right), {\mathcal{j}}_1={\mathcal{j}}_2, {\mathcal{s}}_1={\mathcal{s}}_2, {\mathcal{k}}_1{\mathcal{k}}_2\in {\mathbb{E}}_{\mathfrak{i}}^{\prime }\end{array}\end{array} \end{gathered}$$

$\mathrm{for} \mathrm{all} \left(\mathcal{j},\mathcal{k},\mathcal{s}\right)\in \mathcal{V}$ and $\mathfrak{i}=1,2,\dots ,\mathcal{r}$.

Theorem 1. 

The maximal product of three strong $\mathcal{C}\mathcal{F}\mathcal{G}$s is also a strong $\mathcal{C}\mathcal{F}\mathcal{G}$.

Proof. 

Let ${\mathcal{G}}_{\mathbb{c}}=\left(\mathfrak{L},{\mathfrak{M}}_1, {\mathfrak{M}}_2,\dots , {\mathfrak{M}}_{\mathcal{r}}\right)$, ${\mathcal{G}}_{\mathbb{c}}^{\prime }=\left({\mathfrak{L}}^{\prime },{{\mathfrak{M}}^{\prime }}_1, {{\mathfrak{M}}^{\prime }}_2,\dots , {{\mathfrak{M}}^{\prime }}_{\mathcal{r}}\right),$ and ${\mathcal{G}}_{\mathbb{c}}^{″}=\left({\mathfrak{L}}^{″},{\mathfrak{M}}_1^{″}, {\mathfrak{M}}_2^{″},\dots , {\mathfrak{M}}_{\mathcal{r}}^{″}\right)$ be three strong $\mathcal{C}\mathcal{F}\mathcal{G}$s. Then, ${\mathsf{\alpha }}_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{j}}_1{\mathcal{j}}_2\right)={\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\wedge {\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_2\right)$ for any ${\mathcal{j}}_1{\mathcal{j}}_2\in {\mathbb{E}}_{\mathfrak{i}}$, ${\mathsf{\alpha }}_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{k}}_1{\mathcal{k}}_2\right)={\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\wedge {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_2\right)$ for any ${\mathcal{k}}_1{\mathcal{k}}_2\in {\mathbb{E}}_{\mathfrak{i}}^{\prime }$, and ${\mathsf{\alpha }}_{{{\mathfrak{W}}^{″}}_{\mathfrak{i}}}\left({\mathcal{s}}_1{\mathcal{s}}_2\right)={\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)\wedge {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_2\right)$ for any ${\mathcal{s}}_1{\mathcal{s}}_2\in {\mathbb{E}}_{\mathfrak{i}}^{″}$, $\mathfrak{i}=1,2,\dots ,\mathcal{r}.$ Then, according to the definition of the maximal product,

Case (i): ${\mathcal{j}}_1={\mathcal{j}}_2,{\mathcal{k}}_1={\mathcal{k}}_2,{\mathcal{s}}_1{\mathcal{s}}_2\in {\mathbb{E}}_{\mathfrak{i}}^{″}$, then

$$\begin{gathered} {\mathsf{\alpha }}_{{{\mathfrak{M}}^{\ast }}_{\mathfrak{i}}}\left(\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right)\right) \\ ={\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\mathsf{\alpha }}_{{{\mathfrak{M}}_{\mathfrak{i}}}^{″}}={\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee \left[{\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)\wedge {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_2\right)\right] \\ =[{\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)]\wedge [{\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_2\right)] \\ ={\mathsf{\alpha }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\wedge { \mathsf{\alpha }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right) \end{gathered}$$

Case (ii): ${\mathcal{j}}_1={\mathcal{j}}_2,{\mathcal{s}}_1={\mathcal{s}}_2,{\mathcal{k}}_1{\mathcal{k}}_2\in {\mathbb{E}}_{\mathfrak{i}}^{\prime }$, then

$$\begin{gathered} { \mathsf{\alpha }}_{{{\mathfrak{M}}^{\ast }}_{\mathfrak{i}}}\left(\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right)\right)={\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)\vee {\mathsf{\alpha }}_{{{\mathfrak{M}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{k}}_1{\mathcal{k}}_2\right) \\ ={\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)\vee \left[{\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\wedge {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_2\right)\right] \\ =[{\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)]\wedge [{\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_2\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)] \\ ={\mathsf{\alpha }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\wedge { \mathsf{\alpha }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right) \end{gathered}$$

Case (iii): ${\mathcal{k}}_1={\mathcal{k}}_2,{\mathcal{s}}_1={\mathcal{s}}_2,{\mathcal{j}}_1{\mathcal{j}}_2\in {\mathbb{E}}_{\mathfrak{i}}$, then

$$\begin{gathered} {\mathsf{\alpha }}_{{{\mathfrak{M}}^{\ast }}_{\mathfrak{i}}}\left(\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right)\right)={\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{k}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{s}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{M}}_{\mathfrak{i}}}\left({\mathcal{j}}_1{\mathcal{j}}_2\right) \\ ={\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)\vee \left[{\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\wedge {\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_2\right)\right] \\ =[{\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)]\wedge [{\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_2\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)] \\ ={\mathsf{\alpha }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\wedge { \mathsf{\alpha }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right) \end{gathered}$$

Therefore, ${\mathsf{\alpha }}_{{{\mathfrak{M}}^{\ast }}_{\mathfrak{i}}}\left(\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right)\right)$ = ${\mathsf{\alpha }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\wedge {\mathsf{\alpha }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right)$ for all edges of the maximal product of ${\mathcal{G}}_{\mathbb{c}}$, ${\mathcal{G}}_{\mathbb{c}}^{\prime }$ and ${\mathcal{G}}_{\mathbb{c}}^{″}$. Similarly,

$${\mathsf{\beta }}_{{{\mathfrak{M}}^{\ast }}_{\mathfrak{i}}}\left(\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right)\right)={\mathsf{\beta }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\wedge { \mathsf{\beta }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right),$$

$${\mathsf{\gamma }}_{{{\mathfrak{M}}^{\ast }}_{\mathfrak{i}}}\left(\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right)\right)={\mathsf{\gamma }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)\wedge { \mathsf{\gamma }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2\right)$$

Hence, ${\mathfrak{G}}_{\mathfrak{c}}={\mathcal{G}}_{\mathbb{c}}{\ast }_{\mathcal{M}\mathcal{P}}{{\mathcal{G}}^{\prime }}_{\mathbb{c}}{\ast }_{\mathcal{M}\mathcal{P}}{{\mathcal{G}}^{″}}_{\mathbb{c}}=\left({\mathfrak{L}}^{\ast },{{\mathfrak{M}}^{\ast }}_1, {{\mathfrak{M}}^{\ast }}_2,\dots , {{\mathfrak{M}}^{\ast }}_{\mathcal{r}}\right)$ is a strong $\mathcal{C}\mathcal{F}\mathcal{G}.$ □

Theorem 2. 

The maximal product of three connected $\mathcal{C}\mathcal{F}\mathcal{G}$s is also a connected $\mathcal{C}\mathcal{F}\mathcal{G}$.

Proof. 

Let ${\mathcal{G}}_{\mathbb{c}}=\left(\mathfrak{L},{\mathfrak{M}}_1, {\mathfrak{M}}_2,\dots , {\mathfrak{M}}_{\mathcal{r}}\right)$, ${\mathcal{G}}_{\mathbb{c}}^{\prime }=\left({\mathfrak{L}}^{\prime },{\mathfrak{M}}_1^{\prime }, {\mathfrak{M}}_2^{\prime },\dots , {\mathfrak{M}}_{\mathcal{r}}^{\prime }\right)$, and ${\mathcal{G}}_{\mathbb{c}}^{″}=\left({\mathfrak{L}}^{″},{\mathfrak{M}}_1^{″}, {\mathfrak{M}}_2^{″},\dots , {\mathfrak{M}}_{\mathcal{r}}^{″}\right)$ be three finite and connected $\mathcal{C}\mathcal{F}\mathcal{G}$s with underlying crisp graphs $\mathbb{G}=(\mathbb{V}$, ${\mathbb{E}}_1, {\mathbb{E}}_2,\dots , {\mathbb{E}}_{\mathcal{r}}$), ${\mathbb{G}}^{\prime }=({\mathbb{V}}^{\prime }$, ${\mathbb{E}}_1^{\prime }, {\mathbb{E}}_2^{\prime },\dots , {\mathbb{E}}_{\mathcal{r}}^{\prime }$), and ${\mathbb{G}}^{″}=({\mathbb{V}}^{″}$, ${\mathbb{E}}_1^{″}, {\mathbb{E}}_2^{″},\dots , {\mathbb{E}}_{\mathcal{r}}^{″}$), respectively.

Since, ${\mathcal{G}}_{\mathbb{c}},{\mathcal{G}}_{\mathbb{c}}^{\prime }$, and ${\mathcal{G}}_{\mathbb{c}}^{″}$ are three connected $\mathcal{C}\mathcal{F}\mathcal{G}$s,

$${\mathsf{\alpha }}_{{\mathfrak{M}}_{\mathfrak{i}}}^{\infty }\left({\mathcal{o}}_{\mathfrak{a}},{\mathcal{o}}_{\mathfrak{b}}\right)>0, {\mathsf{\beta }}_{{\mathfrak{M}}_{\mathfrak{i}}}^{\infty }\left({\mathcal{o}}_{\mathfrak{a}},{\mathcal{o}}_{\mathfrak{b}}\right)>0, {\mathsf{\gamma }}_{{\mathfrak{M}}_{\mathfrak{i}}}^{\infty }\left({\mathcal{o}}_{\mathfrak{a}},{\mathcal{o}}_{\mathfrak{b}}\right)>0, for all {\mathcal{o}}_{\mathfrak{a}},{\mathcal{o}}_{\mathfrak{b}}\in \mathbb{V}$$

$${\mathsf{\alpha }}_{{\mathfrak{M}}_{\mathfrak{i}}}^{\infty }\left({\mathcal{p}}_{\mathfrak{m}},{\mathcal{p}}_{\mathfrak{n}}\right)>0, {\mathsf{\beta }}_{{\mathfrak{M}}_{\mathfrak{i}}}^{\infty }\left({\mathcal{p}}_{\mathfrak{m}},{\mathcal{p}}_{\mathfrak{n}}\right)>0, {\mathsf{\gamma }}_{{\mathfrak{M}}_{\mathfrak{i}}}^{\infty }\left({\mathcal{p}}_{\mathfrak{m}},{\mathcal{p}}_{\mathfrak{n}}\right)>0, for all {\mathcal{p}}_{\mathfrak{m}},{\mathcal{p}}_{\mathfrak{n}}\in {\mathbb{V}}^{\prime }$$

$${\mathsf{\alpha }}_{{\mathfrak{M}}_{\mathfrak{i}}}^{\infty }\left({\mathcal{q}}_{\mathfrak{e}},{\mathcal{q}}_{\mathfrak{f}}\right)>0, {\mathsf{\beta }}_{{\mathfrak{M}}_{\mathfrak{i}}}^{\infty }\left({\mathcal{q}}_{\mathfrak{e}},{\mathcal{q}}_{\mathfrak{f}}\right)>0, {\mathsf{\gamma }}_{{\mathfrak{M}}_{\mathfrak{i}}}^{\infty }\left({\mathcal{q}}_{\mathfrak{e}},{\mathcal{q}}_{\mathfrak{f}}\right)>0, for all {\mathcal{q}}_{\mathfrak{e}},{\mathcal{q}}_{\mathfrak{f}}\in {\mathbb{V}}^{″}$$

Let ${\mathfrak{G}}_{\mathfrak{c}}=\left({\mathfrak{L}}^{\ast },{{\mathfrak{M}}^{\ast }}_1, {{\mathfrak{M}}^{\ast }}_2,\dots , {{\mathfrak{M}}^{\ast }}_{\mathcal{o}}\right)$ be the maximal product of ${\mathcal{G}}_{\mathbb{c}},{\mathcal{G}}_{\mathbb{c}}^{\prime }$, and ${\mathcal{G}}_{\mathbb{c}}^{″}$.

For every vertex ${\mathcal{o}}_{\mathfrak{a}},{\mathcal{o}}_{\mathfrak{b}}\in \mathbb{V}, {\mathcal{p}}_{\mathfrak{m}},{\mathcal{p}}_{\mathfrak{n}}\in {\mathbb{V}}^{\prime }, {\mathcal{q}}_{\mathfrak{e}},{\mathcal{q}}_{\mathfrak{f}}\in {\mathbb{V}}^{″}$ and $\left({\mathcal{o}}_{\mathfrak{a}}{\mathcal{p}}_{\mathfrak{m}}{\mathcal{q}}_{\mathfrak{e}}),({\mathcal{o}}_{\mathfrak{b}}{\mathcal{p}}_{\mathfrak{n}}{\mathcal{q}}_{\mathfrak{f}}\right)\in \mathbb{V}\times {\mathbb{V}}^{\prime }\times {\mathbb{V}}^{″}$

$$\begin{gathered} {\mathsf{\alpha }}_{{{\mathfrak{M}}^{\ast }}_{\mathfrak{i}}}\left(\left({\mathcal{o}}_{\mathfrak{a}},{\mathcal{p}}_{\mathfrak{m}},{\mathcal{q}}_{\mathfrak{e}}\right)\left({\mathcal{o}}_{\mathfrak{b}},{\mathcal{p}}_{\mathfrak{n}},{\mathcal{q}}_{\mathfrak{f}}\right)\right) \\ ={\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathfrak{a}}\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathfrak{m}}\right)\vee {\mathsf{\alpha }}_{{{\mathfrak{M}}_{\mathfrak{i}}}^{″}}={\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathfrak{a}}\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathfrak{m}}\right)\vee \left[{\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathfrak{e}}\right)\wedge {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathfrak{f}}\right)\right] \\ ={\mathsf{\alpha }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{o}}_{\mathfrak{a}},{\mathcal{p}}_{\mathfrak{m}},{\mathcal{q}}_{\mathfrak{e}}\right)\wedge { \mathsf{\alpha }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{o}}_{\mathfrak{b}},{\mathcal{p}}_{\mathfrak{n}},{\mathcal{q}}_{\mathfrak{f}}\right) \end{gathered}$$

$$\mathrm{Similarly}, {\mathsf{\beta }}_{{{\mathfrak{M}}^{\ast }}_{\mathfrak{i}}}\left(\left({\mathcal{o}}_{\mathfrak{a}},{\mathcal{p}}_{\mathfrak{m}},{\mathcal{q}}_{\mathfrak{e}}\right)\left({\mathcal{o}}_{\mathfrak{b}},{\mathcal{p}}_{\mathfrak{n}},{\mathcal{q}}_{\mathfrak{f}}\right)\right)={\mathsf{\beta }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{o}}_{\mathfrak{a}},{\mathcal{p}}_{\mathfrak{m}},{\mathcal{q}}_{\mathfrak{e}}\right)\wedge { \mathsf{\beta }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{o}}_{\mathfrak{b}},{\mathcal{p}}_{\mathfrak{n}},{\mathcal{q}}_{\mathfrak{f}}\right),$$

$${\mathsf{\gamma }}_{{{\mathfrak{M}}^{\ast }}_{\mathfrak{i}}}\left(\left({\mathcal{o}}_{\mathfrak{a}},{\mathcal{p}}_{\mathfrak{m}},{\mathcal{q}}_{\mathfrak{e}}\right)\left({\mathcal{o}}_{\mathfrak{b}},{\mathcal{p}}_{\mathfrak{n}},{\mathcal{q}}_{\mathfrak{f}}\right)\right)={\mathsf{\gamma }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{o}}_{\mathfrak{a}},{\mathcal{p}}_{\mathfrak{m}},{\mathcal{q}}_{\mathfrak{e}}\right)\wedge { \mathsf{\gamma }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{o}}_{\mathfrak{b}},{\mathcal{p}}_{\mathfrak{n}},{\mathcal{q}}_{\mathfrak{f}}\right)$$

According to the definition of the maximal product, it is obvious that $\left({\mathcal{o}}_{\mathfrak{a}},{\mathcal{p}}_{\mathfrak{m}},{\mathcal{q}}_{\mathfrak{e}}\right)\left({\mathcal{o}}_{\mathfrak{b}},{\mathcal{p}}_{\mathfrak{n}},{\mathcal{q}}_{\mathfrak{f}}\right)$ is connected.

Therefore,

$${\mathsf{\alpha }}_{{\mathfrak{M}}_{\mathfrak{i}}}^{\infty }\left(\left({\mathcal{o}}_{\mathfrak{a}}{\mathcal{p}}_{\mathfrak{m}}{\mathcal{q}}_{\mathfrak{e}}),({\mathcal{o}}_{\mathfrak{b}}{\mathcal{p}}_{\mathfrak{n}}{\mathcal{q}}_{\mathfrak{f}}\right)\right)>0,{\mathsf{\beta }}_{{\mathfrak{M}}_{\mathfrak{i}}}^{\infty }\left(\left({\mathcal{o}}_{\mathfrak{a}}{\mathcal{p}}_{\mathfrak{m}}{\mathcal{q}}_{\mathfrak{e}}),({\mathcal{o}}_{\mathfrak{b}}{\mathcal{p}}_{\mathfrak{n}}{\mathcal{q}}_{\mathfrak{f}}\right)\right)>0,{\mathsf{\gamma }}_{{\mathfrak{M}}_{\mathfrak{i}}}^{\infty }\left(\left({\mathcal{o}}_{\mathfrak{a}}{\mathcal{p}}_{\mathfrak{m}}{\mathcal{q}}_{\mathfrak{e}}),({\mathcal{o}}_{\mathfrak{b}}{\mathcal{p}}_{\mathfrak{n}}{\mathcal{q}}_{\mathfrak{f}}\right)\right)>0$$

for all $\left({\mathcal{o}}_{\mathfrak{a}}{\mathcal{p}}_{\mathfrak{m}}{\mathcal{q}}_{\mathfrak{e}}),({\mathcal{o}}_{\mathfrak{b}}{\mathcal{p}}_{\mathfrak{n}}{\mathcal{q}}_{\mathfrak{f}}\right)\in$ ${\mathbb{E}}_{\mathcal{r}}$. Thus, ${\mathfrak{G}}_{\mathfrak{c}}$ is a connected $\mathcal{C}\mathcal{F}\mathcal{G}$. □

3.1. Definition

The degree of a vertex in the maximal product ${\mathfrak{G}}_{\mathfrak{c}}=\left({\mathfrak{L}}^{\ast },{{\mathfrak{M}}^{\ast }}_1, {{\mathfrak{M}}^{\ast }}_2,\dots , {{\mathfrak{M}}^{\ast }}_{\mathcal{r}}\right)$ of three $\mathcal{C}\mathcal{F}\mathcal{G}\mathrm{s}$, ${\mathcal{G}}_{\mathbb{c}}=\left(\mathfrak{L},{\mathfrak{M}}_1, {\mathfrak{M}}_2,\dots , {\mathfrak{M}}_{\mathcal{r}}\right)$, ${\mathcal{G}}_{\mathbb{c}}^{\prime }=\left({\mathfrak{L}}^{\prime },{\mathfrak{M}}_1^{\prime }, {\mathfrak{M}}_2^{\prime },\dots , {\mathfrak{M}}_{\mathcal{r}}^{\prime }\right)$, and ${\mathcal{G}}_{\mathbb{c}}^{″}=\left({\mathfrak{L}}^{″},{\mathfrak{M}}_1^{″}, {\mathfrak{M}}_2^{″},\dots , {\mathfrak{M}}_{\mathcal{r}}^{″}\right)$, represented in Figure 1, Figure 2 and Figure 3, is defined as

$${\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)=\langle \left[{\mathfrak{D}}_{\mathsf{\alpha }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right),{\mathfrak{D}}_{\mathsf{\beta }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)\right],{\mathfrak{D}}_{\mathsf{\gamma }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)\rangle$$

where

$$\begin{array}{cc}{\mathfrak{D}}_{\mathsf{\alpha }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)=& {\displaystyle {\displaystyle \sum }_{{\mathcal{o}}_{\mathcal{a}}{\mathcal{o}}_{\mathcal{x}}\in {\mathbb{E}}_{\mathfrak{i}},{\mathcal{p}}_{\mathcal{b}}={\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{c}}={\mathcal{q}}_{\mathcal{z}}}}{\mathsf{\alpha }}_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{o}}_{\mathcal{a}}{\mathcal{o}}_{\mathcal{x}}\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathcal{z}}\right)\hfill \\ & +{\displaystyle {\displaystyle \sum }_{{\mathcal{p}}_{\mathcal{b}}{\mathcal{p}}_{\mathcal{y}}\in {\mathbb{E}}_{\mathfrak{i}}^{\prime },{\mathcal{o}}_{\mathcal{a}}={\mathcal{o}}_{\mathcal{x}},{\mathcal{q}}_{\mathcal{c}}={\mathcal{q}}_{\mathcal{z}}}}{\mathsf{\alpha }}_{{{\mathfrak{W}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{p}}_{\mathcal{b}}{\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathcal{x}}\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathcal{z}}\right)\hfill \\ & +{\displaystyle {\displaystyle \sum }_{{\mathcal{q}}_{\mathcal{c}}{\mathcal{q}}_{\mathcal{z}}\in {\mathbb{E}}_{\mathfrak{i}}^{″},{\mathcal{p}}_{\mathcal{b}}={\mathcal{p}}_{\mathcal{y}},{\mathcal{o}}_{\mathcal{a}}={\mathcal{o}}_{\mathcal{x}}}}{\mathsf{\alpha }}_{{{\mathfrak{W}}_{\mathfrak{i}}}^{″}}\left({\mathcal{q}}_{\mathcal{c}}{\mathcal{q}}_{\mathcal{z}}\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathcal{x}}\right)\hfill \end{array}$$

$$\begin{array}{cc}{\mathfrak{D}}_{\mathsf{\beta }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)=& {\displaystyle {\displaystyle \sum }_{{\mathcal{o}}_{\mathcal{a}}{\mathcal{o}}_{\mathcal{x}}\in {\mathbb{E}}_{\mathfrak{i}},{\mathcal{p}}_{\mathcal{b}}={\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{c}}={\mathcal{q}}_{\mathcal{z}}}}{\mathsf{\beta }}_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{o}}_{\mathcal{a}}{\mathcal{o}}_{\mathcal{x}}\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathcal{z}}\right)\hfill \\ & +{\displaystyle {\displaystyle \sum }_{{\mathcal{p}}_{\mathcal{b}}{\mathcal{p}}_{\mathcal{y}}\in {\mathbb{E}}_{\mathfrak{i}}^{\prime },{\mathcal{o}}_{\mathcal{a}}={\mathcal{o}}_{\mathcal{x}},{\mathcal{q}}_{\mathcal{c}}={\mathcal{q}}_{\mathcal{z}}}}{\mathsf{\beta }}_{{{\mathfrak{W}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{p}}_{\mathcal{b}}{\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\beta }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathcal{x}}\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathcal{z}}\right)\hfill \\ & +{\displaystyle {\displaystyle \sum }_{{\mathcal{q}}_{\mathcal{c}}{\mathcal{q}}_{\mathcal{z}}\in {\mathbb{E}}_{\mathfrak{i}}^{″},{\mathcal{p}}_{\mathcal{b}}={\mathcal{p}}_{\mathcal{y}},{\mathcal{o}}_{\mathcal{a}}={\mathcal{o}}_{\mathcal{x}}}}{\mathsf{\beta }}_{{{\mathfrak{W}}_{\mathfrak{i}}}^{″}}\left({\mathcal{q}}_{\mathcal{c}}{\mathcal{q}}_{\mathcal{z}}\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\beta }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathcal{x}}\right)\hfill \end{array}$$

$$\begin{array}{cc}{\mathfrak{D}}_{\mathsf{\gamma }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)=& {\displaystyle {\displaystyle \sum }_{{\mathcal{o}}_{\mathcal{a}}{\mathcal{o}}_{\mathcal{x}}\in {\mathbb{E}}_{\mathfrak{i}},{\mathcal{p}}_{\mathcal{b}}={\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{c}}={\mathcal{q}}_{\mathcal{z}}}}{\mathsf{\gamma }}_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{o}}_{\mathcal{a}}{\mathcal{o}}_{\mathcal{x}}\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathcal{z}}\right)\hfill \\ & +{\displaystyle {\displaystyle \sum }_{{\mathcal{p}}_{\mathcal{b}}{\mathcal{p}}_{\mathcal{y}}\in {\mathbb{E}}_{\mathfrak{i}}^{\prime },{\mathcal{o}}_{\mathcal{a}}={\mathcal{o}}_{\mathcal{x}},{\mathcal{q}}_{\mathcal{c}}={\mathcal{q}}_{\mathcal{z}}}}{\mathsf{\gamma }}_{{{\mathfrak{W}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{p}}_{\mathcal{b}}{\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\gamma }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathcal{x}}\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathcal{z}}\right)\hfill \\ & +{\displaystyle {\displaystyle \sum }_{{\mathcal{q}}_{\mathcal{c}}{\mathcal{q}}_{\mathcal{z}}\in {\mathbb{E}}_{\mathfrak{i}}^{″},{\mathcal{p}}_{\mathcal{b}}={\mathcal{p}}_{\mathcal{y}},{\mathcal{o}}_{\mathcal{a}}={\mathcal{o}}_{\mathcal{x}}}}{\mathsf{\gamma }}_{{{\mathfrak{W}}_{\mathfrak{i}}}^{″}}\left({\mathcal{q}}_{\mathcal{c}}{\mathcal{q}}_{\mathcal{z}}\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\gamma }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathcal{x}}\right)\hfill \end{array}$$

${\mathfrak{M}}_{\mathcal{j}}$, the degree of a vertex of maximal product, ${\mathfrak{G}}_{\mathfrak{c}}={\mathcal{G}}_{\mathbb{c}}{\ast }_{\mathcal{M}\mathcal{P}}{\mathcal{G}}_{\mathbb{c}}^{\prime }{\ast }_{\mathcal{M}\mathcal{P}}{\mathcal{G}}_{\mathbb{c}}^{″}$, is determined as ${\mathfrak{D}}_{{\mathfrak{M}}_{\mathcal{j}}}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)=\langle \left[{\mathfrak{D}}_{{\mathsf{\alpha }}_{\mathcal{j}}}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right),{\mathfrak{D}}_{{\mathsf{\beta }}_{\mathcal{j}}}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)\right],{\mathfrak{D}}_{{\mathsf{\gamma }}_{\mathcal{j}}}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)\rangle$.

3.2. Definition

The total degree of a vertex in the maximal product ${\mathfrak{G}}_{\mathfrak{c}}=\left({\mathfrak{L}}^{\ast },{{\mathfrak{M}}^{\ast }}_1, {{\mathfrak{M}}^{\ast }}_2,\dots , {{\mathfrak{M}}^{\ast }}_{\mathcal{r}}\right)$ of three $\mathcal{F}\mathcal{G}\mathrm{s}$, ${\mathcal{G}}_{\mathbb{c}}=\left(\mathfrak{L},{\mathfrak{M}}_1, {\mathfrak{M}}_2,\dots , {\mathfrak{M}}_{\mathcal{r}}\right)$, ${\mathcal{G}}_{\mathbb{c}}^{\prime }=\left({\mathfrak{L}}^{\prime },{\mathfrak{M}}_1^{\prime }, {\mathfrak{M}}_2^{\prime },\dots , {\mathfrak{M}}_{\mathcal{r}}^{\prime }\right)$, and ${\mathcal{G}}_{\mathbb{c}}^{″}=\left({\mathfrak{L}}^{″},{\mathfrak{M}}_1^{″}, {\mathfrak{M}}_2^{″},\dots , {\mathfrak{M}}_{\mathcal{r}}^{″}\right)$, is defined as

$$\mathcal{T}{\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)=\langle \left[\mathcal{T}{\mathfrak{D}}_{\mathsf{\alpha }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right),\mathcal{T}{\mathfrak{D}}_{\mathsf{\beta }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)\right],\mathcal{T}{\mathfrak{D}}_{\mathsf{\gamma }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)\rangle ,$$

where

$$\begin{array}{cc}\mathcal{T}{\mathfrak{D}}_{\mathsf{\alpha }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)=& {\displaystyle {\displaystyle \sum }_{{\mathcal{o}}_{\mathcal{a}}{\mathcal{o}}_{\mathcal{x}}\in {\mathbb{E}}_{\mathfrak{i}},{\mathcal{p}}_{\mathcal{b}}={\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{c}}={\mathcal{q}}_{\mathcal{z}}}}{\mathsf{\alpha }}_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{o}}_{\mathcal{a}}{\mathcal{o}}_{\mathcal{x}}\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathcal{z}}\right)\hfill \\ & +{\displaystyle {\displaystyle \sum }_{{\mathcal{p}}_{\mathcal{b}}{\mathcal{p}}_{\mathcal{y}}\in {\mathbb{E}}_{\mathfrak{i}}^{\prime },{\mathcal{o}}_{\mathcal{a}}={\mathcal{o}}_{\mathcal{x}},{\mathcal{q}}_{\mathcal{c}}={\mathcal{q}}_{\mathcal{z}}}}{\mathsf{\alpha }}_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{p}}_{\mathcal{b}}{\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathcal{x}}\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathcal{z}}\right)\hfill \\ & +{\displaystyle {\displaystyle \sum }_{{\mathcal{q}}_{\mathcal{c}}{\mathcal{q}}_{\mathcal{z}}\in {\mathbb{E}}_{\mathfrak{i}}^{″},{\mathcal{p}}_{\mathcal{b}}={\mathcal{p}}_{\mathcal{y}},{\mathcal{o}}_{\mathcal{a}}={\mathcal{o}}_{\mathcal{x}}}}{\mathsf{\alpha }}_{{{\mathfrak{W}}_{\mathfrak{i}}}^{″}}\left({\mathcal{q}}_{\mathcal{c}}{\mathcal{q}}_{\mathcal{z}}\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathcal{x}}\right)+{\mathsf{\alpha }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)\hfill \end{array}$$

$$\begin{array}{cc}\mathcal{T}{\mathfrak{D}}_{\mathsf{\beta }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)=& {\displaystyle {\displaystyle \sum }_{{\mathcal{o}}_{\mathcal{a}}{\mathcal{o}}_{\mathcal{x}}\in {\mathbb{E}}_{\mathfrak{i}},{\mathcal{p}}_{\mathcal{b}}={\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{c}}={\mathcal{q}}_{\mathcal{z}}}}{\mathsf{\beta }}_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{o}}_{\mathcal{a}}{\mathcal{o}}_{\mathcal{x}}\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathcal{z}}\right)\hfill \\ & +{\displaystyle {\displaystyle \sum }_{{\mathcal{p}}_{\mathcal{b}}{\mathcal{p}}_{\mathcal{y}}\in {\mathbb{E}}_{\mathfrak{i}}^{\prime },{\mathcal{o}}_{\mathcal{a}}={\mathcal{o}}_{\mathcal{x}},{\mathcal{q}}_{\mathcal{c}}={\mathcal{q}}_{\mathcal{z}}}}{\mathsf{\beta }}_{{{\mathfrak{W}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{p}}_{\mathcal{b}}{\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\beta }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathcal{x}}\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathcal{z}}\right)\hfill \\ & +{\displaystyle {\displaystyle \sum }_{{\mathcal{q}}_{\mathcal{c}}{\mathcal{q}}_{\mathcal{z}}\in {\mathbb{E}}_{\mathfrak{i}}^{″},{\mathcal{p}}_{\mathcal{b}}={\mathcal{p}}_{\mathcal{y}},{\mathcal{o}}_{\mathcal{a}}={\mathcal{o}}_{\mathcal{x}}}}{\mathsf{\beta }}_{{{\mathfrak{W}}_{\mathfrak{i}}}^{″}}\left({\mathcal{q}}_{\mathcal{c}}{\mathcal{q}}_{\mathcal{z}}\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\beta }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathcal{x}}\right)+{\mathsf{\beta }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)\hfill \end{array}$$

$$\begin{array}{cc}\mathcal{T}{\mathfrak{D}}_{\mathsf{\gamma }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)=& {\displaystyle {\displaystyle \sum }_{{\mathcal{o}}_{\mathcal{a}}{\mathcal{o}}_{\mathcal{x}}\in {\mathbb{E}}_{\mathfrak{i}},{\mathcal{p}}_{\mathcal{b}}={\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{c}}={\mathcal{q}}_{\mathcal{z}}}}{\mathsf{\gamma }}_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{o}}_{\mathcal{a}}{\mathcal{o}}_{\mathcal{x}}\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathcal{z}}\right)\hfill \\ & +{\displaystyle {\displaystyle \sum }_{{\mathcal{p}}_{\mathcal{b}}{\mathcal{p}}_{\mathcal{y}}\in {\mathbb{E}}_{\mathfrak{i}}^{\prime },{\mathcal{o}}_{\mathcal{a}}={\mathcal{o}}_{\mathcal{x}},{\mathcal{q}}_{\mathcal{c}}={\mathcal{q}}_{\mathcal{z}}}}{\mathsf{\gamma }}_{{{\mathfrak{W}}_{\mathfrak{i}}}^{\prime }}\left({\mathcal{p}}_{\mathcal{b}}{\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\gamma }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathcal{x}}\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_{\mathcal{z}}\right)\hfill \\ & +{\displaystyle {\displaystyle \sum }_{{\mathcal{q}}_{\mathcal{c}}{\mathcal{q}}_{\mathcal{z}}\in {\mathbb{E}}_{\mathfrak{i}}^{″},{\mathcal{p}}_{\mathcal{b}}={\mathcal{p}}_{\mathcal{y}},{\mathcal{o}}_{\mathcal{a}}={\mathcal{o}}_{\mathcal{x}}}}{\mathsf{\gamma }}_{{{\mathfrak{W}}_{\mathfrak{i}}}^{″}}\left({\mathcal{q}}_{\mathcal{c}}{\mathcal{q}}_{\mathcal{z}}\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_{\mathcal{y}}\right)\vee {\mathsf{\gamma }}_{\mathfrak{L}}\left({\mathcal{o}}_{\mathcal{x}}\right)+{\mathsf{\gamma }}_{{\mathfrak{L}}^{\ast }}\left({\mathcal{o}}_{\mathcal{x}},{\mathcal{p}}_{\mathcal{y}},{\mathcal{q}}_{\mathcal{z}}\right)\hfill \end{array}$$

The degree and total degree of each vertex present in the maximal product ${\mathfrak{G}}_{\mathfrak{c}}$ (Figure 4 and Figure 5) are computed as follows:

$$\begin{gathered} {\mathfrak{D}}_{\alpha }\left({\mathcal{o}}_1,{\mathcal{p}}_1,{\mathcal{q}}_1\right)={\alpha }_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{o}}_1{\mathcal{o}}_2\right)\vee {\alpha }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_1\right)\vee {\alpha }_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_1\right)+{\alpha }_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{p}}_1{\mathcal{p}}_2\right)\vee {\alpha }_{\mathfrak{L}}\left({\mathcal{o}}_1\right)\vee {\alpha }_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_1\right) \\ +{\alpha }_{{{\mathfrak{W}}^{″}}_{\mathfrak{i}}}\left({\mathcal{q}}_1{\mathcal{q}}_2\right)\vee {\alpha }_{\mathfrak{L}}\left({\mathcal{o}}_1\right)\vee {\alpha }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_1\right) \\ =(0.35\vee 0.45\vee 0.20)+\left(0.30\vee 0.35\vee 0.20\right)+\left(0.70\vee 0.35\vee 0.45\right)=1.5 \end{gathered}$$

$$\begin{gathered} {\mathfrak{D}}_{\beta }\left({\mathcal{o}}_1,{\mathcal{p}}_1,{\mathcal{q}}_1\right)={\beta }_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{o}}_1{\mathcal{o}}_2\right)\vee {\beta }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_1\right)\vee {\beta }_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_1\right)+{\beta }_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{p}}_1{\mathcal{p}}_2\right)\vee {\beta }_{\mathfrak{L}}\left({\mathcal{o}}_1\right)\vee {\beta }_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_1\right) \\ +{\beta }_{{{\mathfrak{W}}^{″}}_{\mathfrak{i}}}\left({\mathcal{q}}_1{\mathcal{q}}_2\right)\vee {\beta }_{\mathfrak{L}}\left({\mathcal{o}}_1\right)\vee {\beta }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_1\right) \\ =(0.55\vee 0.80\vee 0.70)+\left(0.60\vee 0.50\vee 0.70\right)+\left(0.80\vee 0.80\vee 0.50\right)=2.3 \end{gathered}$$

$$\begin{gathered} {\mathfrak{D}}_{\gamma }\left({\mathcal{o}}_1,{\mathcal{p}}_1,{\mathcal{q}}_1\right)={\gamma }_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{o}}_1{\mathcal{o}}_2\right)\vee {\gamma }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_1\right)\vee {\gamma }_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_1\right)+{\gamma }_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{p}}_1{\mathcal{p}}_2\right)\vee {\gamma }_{\mathfrak{L}}\left({\mathcal{o}}_1\right)\vee {\gamma }_{{\mathfrak{L}}^{″}}\left({\mathcal{q}}_1\right) \\ +{\gamma }_{{{\mathfrak{W}}^{″}}_{\mathfrak{i}}}\left({\mathcal{q}}_1{\mathcal{q}}_2\right)\vee {\gamma }_{\mathfrak{L}}\left({\mathcal{o}}_1\right)\vee {\gamma }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{p}}_1\right) \\ =(0.75\vee 0.55\vee 0.50)+\left(0.45\vee 0.65\vee 0.50\right)+\left(0.95\vee 0.65\vee 0.55\right)=2.35 \end{gathered}$$

Therefore, the degree of the vertex $\left({\mathcal{o}}_1,{\mathcal{p}}_1,{\mathcal{q}}_1\right) \mathrm{is}$ ${\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_1,{\mathcal{p}}_1,{\mathcal{q}}_1\right)$ = ⟨[1.5, 2.3], 2.35⟩.

Similarly, ${\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_1,{\mathcal{p}}_1,{\mathcal{q}}_2\right)$ = ⟨[1.5, 2.35], 2.4⟩, ${\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_1,{\mathcal{p}}_2,{\mathcal{q}}_1\right)$ = ⟨[1.4, 2.2], 2.35⟩, ${\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_1,{\mathcal{p}}_2,{\mathcal{q}}_2\right)$ = ⟨[1.4, 2.4], 2.35⟩, ${\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_2,{\mathcal{p}}_1,{\mathcal{q}}_1\right)$ = ⟨[1.7, 2.4], 2.4⟩, ${\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_2,{\mathcal{p}}_1,{\mathcal{q}}_2\right)$ = ⟨[1.7, 2.4], 2.4⟩, ${\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_2,{\mathcal{p}}_2,{\mathcal{q}}_1\right)$ = ⟨[1.6, 2.3], 2.4⟩, ${\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_2,{\mathcal{p}}_2,{\mathcal{q}}_2\right)$ = ⟨[1.6, 2.4], 2.4⟩.

Also, the total degree of each vertex is as follows:

$$\begin{gathered} \mathcal{T}{\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_1,{\mathcal{p}}_1,{\mathcal{q}}_1\right)=⟨[1.95, 3.15], 2.9⟩, \mathcal{T}{\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_1,{\mathcal{p}}_1,{\mathcal{q}}_2\right)=⟨[1.85, 3.15], 3.05⟩, \\ \mathcal{T}{\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_1,{\mathcal{p}}_2,{\mathcal{q}}_1\right)=⟨[1.65, 2.9], 2.85⟩, \mathcal{T}{\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_1,{\mathcal{p}}_2,{\mathcal{q}}_2\right)=⟨[1.75, 3.2], 3⟩, \\ \mathcal{T}{\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_2,{\mathcal{p}}_1,{\mathcal{q}}_1\right)=⟨[2.25, 3.2], 3.1⟩, \mathcal{T}{\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_2,{\mathcal{p}}_1,{\mathcal{q}}_2\right)=⟨[2.05, 3.2], 3.05⟩, \\ \mathcal{T}{\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_2,{\mathcal{p}}_2,{\mathcal{q}}_1\right)=⟨[2.15, 3.1], 3.1⟩, \mathcal{T}{\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{o}}_2,{\mathcal{p}}_2,{\mathcal{q}}_2\right)=⟨[1.95, 3.2], 3.05⟩ \end{gathered}$$

4. Domination on Maximal Product of Cubic Fuzzy Graph

To analyze the dominating sets of the maximized $\mathcal{C}\mathcal{F}\mathcal{G}$, the cardinality of the vertices and edges is calculated to find the minimum domination number of the weighted $\mathcal{C}\mathcal{F}\mathcal{G}$ network (Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10). Muhiuddin et al. [43] proposed the domination in $\mathcal{C}\mathcal{F}\mathcal{G}$s via strong edges using the vertex and edge cardinality of the graphs. The weighted maximized $\mathcal{C}\mathcal{F}\mathcal{G}$ ${\mathfrak{G}}_{\mathfrak{c}}={\mathcal{G}}_{\mathbb{c}}{\ast }_{\mathcal{M}\mathcal{P}}{\mathcal{G}}_{\mathbb{c}}^{\prime }{\ast }_{\mathcal{M}\mathcal{P}}{\mathcal{G}}_{\mathbb{c}}^{″}=\left({\mathfrak{L}}^{\ast },{{\mathfrak{M}}^{\ast }}_1, {{\mathfrak{M}}^{\ast }}_2,\dots , {{\mathfrak{M}}^{\ast }}_{\mathcal{r}}\right)$ is represented in Figure 10, and its minimal dominating sets are as follows:

$${\mathfrak{S}}_1=\left\{{\mathcal{j}}_2{\mathcal{k}}_1{\mathcal{s}}_2,{\mathcal{j}}_1{\mathcal{k}}_2{\mathcal{s}}_1,{\mathcal{j}}_2{\mathcal{k}}_2{\mathcal{s}}_2,{\mathcal{j}}_1{\mathcal{k}}_3{\mathcal{s}}_1\right\}$$

$${\mathfrak{S}}_2=\left\{{\mathcal{j}}_2{\mathcal{k}}_1{\mathcal{s}}_2,{\mathcal{j}}_2{\mathcal{k}}_1{\mathcal{s}}_1,{\mathcal{j}}_2{\mathcal{k}}_2{\mathcal{s}}_1,{\mathcal{j}}_1{\mathcal{k}}_3{\mathcal{s}}_2\right\}$$

$${\mathfrak{S}}_3=\left\{{\mathcal{j}}_1{\mathcal{k}}_1{\mathcal{s}}_1,{\mathcal{j}}_2{\mathcal{k}}_1{\mathcal{s}}_2,{\mathcal{j}}_1{\mathcal{k}}_2{\mathcal{s}}_2,{\mathcal{j}}_1{\mathcal{k}}_3{\mathcal{s}}_1,{\mathcal{j}}_1{\mathcal{k}}_3{\mathcal{s}}_2\right\}$$

The vertex cardinality of the members of the above dominating sets are

$$\begin{gathered} {\mathcal{j}}_2{\mathcal{k}}_1{\mathcal{s}}_2=1.9, {\mathcal{j}}_1{\mathcal{k}}_2{\mathcal{s}}_1=1.9, {\mathcal{j}}_2{\mathcal{k}}_2{\mathcal{s}}_2=2.2, {\mathcal{j}}_1{\mathcal{k}}_3{\mathcal{s}}_1=1.9, {\mathcal{j}}_1{\mathcal{k}}_3{\mathcal{s}}_2=1.9, \\ {\mathcal{j}}_2{\mathcal{k}}_2{\mathcal{s}}_1=1.9, {\mathcal{j}}_1{\mathcal{k}}_3{\mathcal{s}}_1=1.9, {\mathcal{j}}_2{\mathcal{k}}_1{\mathcal{s}}_1=1.9, {\mathcal{j}}_1{\mathcal{k}}_1{\mathcal{s}}_1=1.9, {\mathcal{j}}_1{\mathcal{k}}_2{\mathcal{s}}_2=1.9 \end{gathered}$$

The cardinality of the dominating sets ${\mathfrak{S}}_1, {\mathfrak{S}}_2, {\mathfrak{S}}_3$ are

$${\mathfrak{S}}_1\left({\mathcal{G}}_{\mathbb{c}}{\ast }_{\mathcal{M}\mathcal{P}}{{\mathcal{G}}^{\prime }}_{\mathbb{c}}{\ast }_{\mathcal{M}\mathcal{P}}{{\mathcal{G}}^{″}}_{\mathbb{c}}\right)=1.9+1.9+2.2+1.9=7.9$$

$${\mathfrak{S}}_2\left({\mathcal{G}}_{\mathbb{c}}{\ast }_{\mathcal{M}\mathcal{P}}{{\mathcal{G}}^{\prime }}_{\mathbb{c}}{\ast }_{\mathcal{M}\mathcal{P}}{{\mathcal{G}}^{″}}_{\mathbb{c}}\right)=1.9+1.9+1.9+1.9=7.6$$

$${\mathfrak{S}}_3\left({\mathcal{G}}_{\mathbb{c}}{\ast }_{\mathcal{M}\mathcal{P}}{{\mathcal{G}}^{\prime }}_{\mathbb{c}}{\ast }_{\mathcal{M}\mathcal{P}}{{\mathcal{G}}^{″}}_{\mathbb{c}}\right)=1.9+1.9+1.9+1.9+1.9=9.5 \mathrm{respectively}.$$

The domination number of ${\mathcal{G}}_{\mathbb{c}}{\ast }_{\mathcal{M}\mathcal{P}}{{\mathcal{G}}^{\prime }}_{\mathbb{c}}{\ast }_{\mathcal{M}\mathcal{P}}{{\mathcal{G}}^{″}}_{\mathbb{c}}$ is 7.6, which is obtained from the dominating set ${\mathfrak{S}}_2$.

The degree and total degree of each vertex present in the maximal product ${\mathfrak{G}}_{\mathfrak{c}}$ (Figure 9) are computed as follows:

$$\begin{gathered} {\mathfrak{D}}_{\alpha }\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)={\alpha }_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{j}}_1{\mathcal{j}}_2\right)\vee {\alpha }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\alpha }_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+{\alpha }_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{k}}_1{\mathcal{k}}_2\right)\vee {\alpha }_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\alpha }_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+ \\ {\alpha }_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{k}}_2{\mathcal{k}}_3\right)\vee {\alpha }_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\alpha }_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+{\alpha }_{{{\mathfrak{W}}^{″}}_{\mathfrak{i}}}\left({\mathcal{s}}_1{\mathcal{s}}_2\right)\vee {\alpha }_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\alpha }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right) \\ =(0.5\vee 0.3\vee 0.6)+(0.1\vee 0.2\vee 0.6)+(0.2\vee 0.2\vee 0.6)+(0.5\vee 0.2\vee 0.3)=2.3 \end{gathered}$$

$$\begin{gathered} {\mathfrak{D}}_{\beta }\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)={\beta }_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{j}}_1{\mathcal{j}}_2\right)\vee {\beta }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\beta }_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+{\beta }_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{k}}_1{\mathcal{k}}_2\right)\vee {\beta }_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\beta }_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+ \\ {\beta }_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{k}}_2{\mathcal{k}}_3\right)\vee {\beta }_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\beta }_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+{\beta }_{{{\mathfrak{W}}^{″}}_{\mathfrak{i}}}\left({\mathcal{s}}_1{\mathcal{s}}_2\right)\vee {\beta }_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\beta }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right) \\ =(0.7\vee 0.6\vee 0.8)+(0.6\vee 0.6\vee 0.8)+(0.5\vee 0.6\vee 0.8)+(0.6\vee 0.6\vee 0.6)=3.0 \end{gathered}$$

$$\begin{gathered} {\mathfrak{D}}_{\gamma }\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)={\gamma }_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{j}}_1{\mathcal{j}}_2\right)\vee {\gamma }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\gamma }_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+{\gamma }_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{k}}_1{\mathcal{k}}_2\right)\vee {\gamma }_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\gamma }_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+ \\ {\gamma }_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{k}}_2{\mathcal{k}}_3\right)\vee {\gamma }_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\gamma }_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+{\gamma }_{{{\mathfrak{W}}^{″}}_{\mathfrak{i}}}\left({\mathcal{s}}_1{\mathcal{s}}_2\right)\vee {\gamma }_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\gamma }_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right) \\ =(0.8\vee 0.5\vee 0.7)+(0.5\vee 0.7\vee 0.7)+(0.7\vee 0.7\vee 0.7)+(0.5\vee 0.5\vee 0.7)=2.9 \end{gathered}$$

Therefore, the degree of the vertex $\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)$ is ${\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)$ = ⟨[2.3, 2.9], 2.9⟩.

Similarly, ${\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}$ (${\mathcal{j}}_1,{\mathcal{k}}_2,{\mathcal{s}}_1$) = ⟨[2.3, 2.9], 2.9⟩, ${\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}}$ (${\mathcal{j}}_1,{\mathcal{k}}_3,{\mathcal{s}}_1$) = ⟨[2.4, 3.0], 3.0⟩

$$\begin{gathered} {\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}} ({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_2)=⟨[1.9, 2.7], 2.9⟩, {\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}} ({\mathcal{j}}_1,{\mathcal{k}}_2,{\mathcal{s}}_2)=⟨[1.9, 2.7], 2.9⟩ \\ {\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}} ({\mathcal{j}}_1,{\mathcal{k}}_3,{\mathcal{s}}_2)=⟨[2.0, 2.8], 3.0⟩, {\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}} ({\mathcal{j}}_2,{\mathcal{k}}_1,{\mathcal{s}}_1)=⟨[2.3, 3.1], 2.9⟩ {\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}} ({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_1)=⟨[2.3, 3.1], 2.9⟩, \\ {\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}} ({\mathcal{j}}_2,{\mathcal{k}}_3,{\mathcal{s}}_1)=⟨[2.4, 3.1], 3.0⟩, {\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}} ({\mathcal{j}}_2,{\mathcal{k}}_1,{\mathcal{s}}_2)=⟨[1.8, 3.1], 2.8⟩, {\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}} ({\mathcal{j}}_2,{\mathcal{k}}_2,{\mathcal{s}}_2)=⟨[1.8, 3.1], 2.8⟩, {\mathfrak{D}}_{{\mathfrak{G}}_{\mathfrak{c}}} ({\mathcal{j}}_2,{\mathcal{k}}_3,{\mathcal{s}}_2)=⟨[2.0, 3.1], 3.0⟩, \end{gathered}$$

Also, the total degree of each vertex is as follows:

$$\begin{gathered} \mathcal{T}{\mathfrak{D}}_{\mathsf{\alpha }}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)={\mathsf{\alpha }}_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{j}}_1{\mathcal{j}}_2\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+{\mathsf{\alpha }}_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{k}}_1{\mathcal{k}}_2\right)\vee {\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+ \\ {\mathsf{\alpha }}_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{k}}_2{\mathcal{k}}_3\right)\vee {\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+{ \mathsf{\alpha }}_{{{\mathfrak{W}}^{″}}_{\mathfrak{i}}}\left({\mathcal{s}}_1{\mathcal{s}}_2\right)\vee {\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\alpha }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)+{\mathsf{\alpha }}_{\mathfrak{L}}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right) \\ =(0.5\vee 0.3\vee 0.6)+(0.1\vee 0.2\vee 0.6)+(0.2\vee 0.2\vee 0.6)+(0.5\vee 0.2\vee 0.3)+0.6=2.9 \end{gathered}$$

$$\begin{gathered} \mathcal{T}{\mathfrak{D}}_{\mathsf{\beta }}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)={\mathsf{\beta }}_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{j}}_1{\mathcal{j}}_2\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+{\mathsf{\beta }}_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{k}}_1{\mathcal{k}}_2\right)\vee {\mathsf{\beta }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+ \\ {\mathsf{\beta }}_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{k}}_2{\mathcal{k}}_3\right)\vee {\mathsf{\beta }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+{ \mathsf{\beta }}_{{{\mathfrak{W}}^{″}}_{\mathfrak{i}}}\left({\mathcal{s}}_1{\mathcal{s}}_2\right)\vee {\mathsf{\beta }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\beta }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)+{\mathsf{\beta }}_{\mathfrak{L}}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right) \\ =(0.7\vee 0.6\vee 0.8)+(0.6\vee 0.6\vee 0.8)+(0.5\vee 0.6\vee 0.8)+(0.6\vee 0.6\vee 0.6)+0.7=3.7 \end{gathered}$$

$$\begin{gathered} \mathcal{T}{\mathfrak{D}}_{\mathsf{\gamma }}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right)={\mathsf{\gamma }}_{{\mathfrak{W}}_{\mathfrak{i}}}\left({\mathcal{j}}_1{\mathcal{j}}_2\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+{\mathsf{\gamma }}_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{k}}_1{\mathcal{k}}_2\right)\vee {\mathsf{\gamma }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+ \\ {\mathsf{\gamma }}_{{{\mathfrak{W}}^{\prime }}_{\mathfrak{i}}}\left({\mathcal{k}}_2{\mathcal{k}}_3\right)\vee {\mathsf{\gamma }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{″}}\left({\mathcal{s}}_1\right)+{ \mathsf{\gamma }}_{{{\mathfrak{W}}^{″}}_{\mathfrak{i}}}\left({\mathcal{s}}_1{\mathcal{s}}_2\right)\vee {\mathsf{\gamma }}_{\mathfrak{L}}\left({\mathcal{j}}_1\right)\vee {\mathsf{\gamma }}_{{\mathfrak{L}}^{\prime }}\left({\mathcal{k}}_1\right)+{\mathsf{\gamma }}_{\mathfrak{L}}\left({\mathcal{j}}_1,{\mathcal{k}}_1,{\mathcal{s}}_1\right) \\ =(0.8\vee 0.5\vee 0.7)+(0.5\vee 0.7\vee 0.7)+(0.7\vee 0.7\vee 0.7)+(0.5\vee 0.5\vee 0.7)+0.8=3.7 \end{gathered}$$

5. Applications

5.1. Telecommunication Network

Telecommunications network businesses frequently combine or form partnerships, necessitating the integration of separate networks to create a unified service network. For example, if three mobile carriers merge, their distinct cellular networks (each depicted as a graph with nodes as cell towers and edges as communication lines) are joined to provide seamless service coverage and greater redundancy, and additional linkages may be established between the nearest or most strategic towers of the various networks. This approach includes determining each connection’s capacity to handle the projected load and updating infrastructure as needed.

5.2. Emergency Services Coordination

Integrating communication networks for police, fire, and medical services may significantly improve response times and coordination in emergency situations. Each service may have its own dispatch and communication network, and integrating these can improve information and resource-sharing, particularly during large-scale crises.

5.3. Research Collaboration Network

In academia and research, connecting the networks of various research institutes and databases enables scientists to interact effectively across boundaries. This involves exchanging research data, computing resources, and publishing databases in order to facilitate innovative research using a collaborative and integrated approach.

5.4. Traffic Management

Cities with complicated transportation networks benefit from combining traffic management, public transit, and emergency response systems. This type of connection enables real-time traffic monitoring, adaptive traffic signal regulation, and priority emergency vehicle routing, which greatly improves the traffic flow and reduces the emergency response time.

5.5. Financial Networks

When banks or financial service companies merge, it is critical to integrate their separate networks (ATMs, internet banking systems, etc.) to provide smooth service to customers. Each network can be represented as a different graph with nodes representing ATMs, branch locations, or servers and edges reflecting connection and transaction flows. The integration seeks to enhance consumer accessibility while reducing transaction times, maintaining a strong network capable of handling more data, and providing redundancy.

5.6. Airline and Defense Systems

In aerospace and the armed forces, combining communication, surveillance, and operational networks from various systems (satellites, ground stations, and airborne units) is essential in sustaining national security and carrying out coordinated missions. This includes maximizing network connection and redundancy to enable reliable communication and surveillance capabilities in all situations.

5.7. Network Security at Organizations

Data networks from different data centers are frequently integrated by technology organizations to maintain redundancy, balance loads, and improve data availability. This might entail linking network infrastructure across geographically separated data centers (each shown as a graph) via high-speed communication lines. The goal of optimization would be to reduce latency while increasing bandwidth, which is critical in maintaining high-speed access to cloud services and assuring disaster recovery capabilities.

In a corporate environment, a wide range of software and tools intended to foster collaboration, resource-sharing, and interaction within and between computer networks are known as networking applications. Standalone networks such as operations, security, and IT infrastructure networks can significantly enhance efficiency, security, and decision-making. This networking is essential in both entrepreneurial and private environments. By integrating these networks, a company achieves a holistic view of its operations, enabling more agile and informed decision-making. The seamless flow of information enhances operational efficiency, improves security posture, and supports strategic initiatives. This integrated approach not only streamlines internal processes but also enhances the company’s ability to respond to external challenges, such as market shifts or cybersecurity threats, ensuring a competitive edge in its industry. Even though operations, security, and IT infrastructure networks may have distinct objectives and requirements, aligning them can bring about significant benefits in organizational effectiveness and resilience.

Despite these networks possessing various goals, integrating these three networks into a cohesive whole presents a unique set of challenges but also offers significant benefits in terms of operational efficiency, enhanced security, and better decision-making capabilities. Security, core IT support, and operational procedures work together closely to be efficient. The management of these networks faces several challenges, such as preserving high security without compromising operational effectiveness and making adjustments to the quickly changing technical environment. The key to success lies in careful planning, rigorous security measures, and a commitment to continuous improvement. A maximized network structure allows for doing so. Flexibility is one of the main reasons why a network structure is implemented in the first place. The aim is to make the networking process more open, interesting, and effective. The effectiveness of the maximized network is established using the cubic fuzzy membership values.

Let us consider the above networks, ${\mathcal{G}}_{\mathbb{c}}$, as shown in Figure 6, depicting the operations network. ${\mathcal{G}}_{\mathbb{c}}^{\prime }$ is represented in Figure 7 as the security network and ${\mathcal{G}}_{\mathbb{c}}$” in Figure 8 as the IT infrastructure network shown. In these figures, the nodes of the network ${\mathcal{G}}_{\mathbb{c}}$ play the role of managing and carrying out an organization’s main business and operational procedures. This comprises the network that facilitates node ${\mathcal{j}}_1$, representing production, and node ${\mathcal{j}}_2$ for manufacturing purposes, for which the processes, production lines, and machinery are managed via these systems to ensure reliability and efficiency with other systems. ${\mathcal{G}}_{\mathbb{c}}^{\prime }$ is a security network consisting of tools, guidelines, and procedures put in place to guard against damage, intrusion, and unauthorized access to a company’s devices, network infrastructure, and data. Here, the nodes ${\mathcal{k}}_1$ represent an intrusion prevention system that regulates inbound and outgoing network traffic according to an organization’s security guidelines. ${\mathcal{k}}_2$ represents the antivirus system, and ${\mathcal{k}}_3$ is the security information management system for the analysis and logging of security alarms produced via network hardware and applications in real time. The ${\mathcal{G}}_{\mathbb{c}}^{\prime }$ network is crucial in preserving consumer trust, protecting an organization’s assets, and adhering to legal requirements. ${\mathcal{G}}_{\mathbb{c}}″$, as an IT infrastructure network, facilitates the provision of IT services and solutions to partners, clients, and staff. It consists of hardware in nodes ${\mathcal{s}}_1$ and ${\mathcal{s}}_2$, which denote network services. As the backbone of the operations and security networks, the IT infrastructure network supplies the aspects required for networking, computation, and storage.

The maximized cubic fuzzy network ${\mathfrak{G}}_{\mathfrak{c}}$ achieves a holistic view of its operations, enabling more agile and informed decision-making. The seamless flow of information enhances operational efficiency, improves security posture, and supports strategic initiatives. Each node in the maximized network ${\mathfrak{G}}_{\mathfrak{c}}$ fulfils a specific set of responsibilities when it is combined in the maximized network. Here, node ${\mathcal{j}}_1{\mathcal{k}}_1{\mathcal{s}}_1$ supports resource optimization that effectively processes, stores, and retrieves the data required for day-to-day business operations. Node ${\mathcal{j}}_1{\mathcal{k}}_2{\mathcal{s}}_1$ enhances security monitoring throughout the company and the uniform execution of security measures. Node ${\mathcal{j}}_2{\mathcal{k}}_2{\mathcal{s}}_1$ observes the reaction that facilitates broad data collection and analytics, offering a comprehensive picture of the operational and security posture of the company. Node ${\mathcal{j}}_1{\mathcal{k}}_2{\mathcal{s}}_2$ performs data encryption and segmentation to safeguard sensitive data, while node ${\mathcal{j}}_2{\mathcal{k}}_2{\mathcal{s}}_2$ is for planning, resilience, and so on.

The above maximized network includes nodes ${\mathcal{j}}_2{\mathcal{k}}_1{\mathcal{s}}_2, {\mathcal{j}}_2{\mathcal{k}}_1{\mathcal{s}}_1, {\mathcal{j}}_2{\mathcal{k}}_2{\mathcal{s}}_1, {\mathcal{j}}_1{\mathcal{k}}_3{\mathcal{s}}_2$, which form the dominating set with a minimal domination number. Consequently, these nodes collectively serve as a security stronghold due to network connectivity, and businesses can establish a cohesive security approach to enhance decision-making. The smooth data transfer among interconnected networks enhances decision-makers’ data accessibility. Improved operational performance and strategic planning can be achieved through enhanced decision-making, driven by flexibility, reduced downtime expenses, enhanced resilience, and real-time data analytics innovation. Influential nodes that dominate the maximized network play a crucial role in overseeing the network’s capabilities, ensuring alignment with the company’s strategic goals, and maintaining resilience and agility in the face of challenges and disruptions. To gain and retain control over the network’s operations and security, dominating nodes requires a combination of processes, technology, and knowledgeable workers. In a synergistic environment, security is tighter, operations are more efficient, and strategic flexibility is enhanced by connecting operations, security, and IT infrastructure networks. A contemporary, flexible business that can adapt to and prosper in the digital era is built on this integration by maximizing cubic fuzzy graphs.

6. Conclusions

Fuzzy graphs assist in identifying unclear problems. Cubic fuzzy graphs provide greater versatility in solving ambiguous situations. $\mathcal{C}\mathcal{F}\mathcal{G}$s’ utilization of both fuzzy membership values and interval-valued fuzzy membership values allows for an enhanced representation of phenomena, especially when it comes to decision-making and identifying the most effective optimal solution. When modeling and resolving issues in unclear situations, a cubic fuzzy graph ($\mathcal{C}\mathcal{F}\mathcal{G}$) is more flexible than fuzzy and intuitionistic fuzzy graphs. In this paper, the authors have introduced and proposed the maximal product of three cubic fuzzy graphs ($\mathcal{C}\mathcal{F}\mathcal{G}$s). The properties of three cubic fuzzy graphs were explored, and the real-world applications of the maximized network were analyzed by examining how interval-valued fuzzy membership and fuzzy membership, along with the assortment of relations, are simultaneously supported through the aspect of the degree and total degree of a vertex. We also discussed the domination parameter that is used to find the most effective and influential nodes in the weighted $\mathcal{C}\mathcal{F}\mathcal{G}$ network. In the future, this study will be extended to different types of operations with cubic fuzzy graphs in networking.