A Comparative Study of Fuzzy Domination and Fuzzy Coloring in an Optimal Approach

Abstract

An optimal network refers to a computer or communication network designed, configured, and managed to maximize efficiency, performance, and effectiveness while minimizing cost and resource utilization. In a network design and management context, optimal typically implies achieving the best possible outcomes between various factors. This research investigated the use of fuzzy graph edge coloring for various fuzzy graph operations, and it focused on the efficacy and efficiency of the fuzzy network product using the minimal spanning tree and the chromatic index of the fuzzy network product. As a network made of nodes and vertices, measurement with vertices is a parameter for domination, and edge measurement is a parameter for edge coloring, so we used these two parameters in the algorithm. This paper aims to identify an optimal network that can be established using product outcomes. This study shows a way to find an optimal fuzzy network based on comparative optimal parameter domination and edge coloring, which can be elaborated with applications. An algorithm was generated using an optimal approach, which was subsequently implemented in the form of applications.

1. Introduction

A mathematical tool known as graph theory plays a vital role in numerous branches of research and technology. A graph typically depicts a real and relevant problem graphically. A graph is a collection of sets (K, L), where K is a collection of non-empty vertices, and L is an edge set. Kaufman (1973) presented the concept of fuzzy graphs, and further, Rosenfeld (1975) interpreted it. Samanta and Pal (2015, 2013) defined various forms of fuzzy graphs. In the literature, there are many ways to color graphs. Fuzzy set theory and fuzzy graph theory have made it possible to model most real-world situations more precisely and adaptably than their classical counterparts [1,2,3,4,5,6,7]. More study is being conducted on fuzzy graphs [8,9,10]. The usual graph model of a network has a collection of nodes joined by edges or connections. The network provides a flexible framework for locating and observing complex systems [11]. The idea of studying complex networks is essential and crosses many academic fields. Real-world problems contain a variety of data that can be represented using a variety of graph types, including fuzzy graphs, intuitionistic fuzzy graphs, and neutrosophic graphs [12,13,14,15,16,17]. Arif introduced the concepts of the soft overset and the soft over graph [18]. Samanta and others provided an explanation for several recently created ideas about intuitionistic fuzzy graphs (IFGs), as well as a few key concepts that had already been established [19]. Using the concepts of intuitionistic fuzzy sets, intuitionistic fuzzy relations, and index matrices as a foundation, a new generalization of IFGs has been presented [20]. Many authors have studied various types of dominations [21,22,23,24,25,26,27,28] and developed this field of study. The crisp-graph coloring technique has been used to color the α-cuts of these fuzzy graphs. As a result, many crisp graphs are colored for different values of α, and for identical fuzzy graphs, the chromatic index changes depending on the value of α [29]. Additionally, Bershtein and Bozhenuk suggested using the minimax criterion to define the ideal center allocation in fuzzy transportation networks [30]. One study defined the concept of a fuzzy graph and determined the minimum number of colors based on the value of the separation degree [31,32,33,34,35,36,37,38]. A new coloring technique was employed to color a political map and to address a brand-new traffic light coloring issue [39]. A colored vertex on fuzzy graphs was used to color maps. To overcome radio frequency issues, Mahapatra et al. extended the coloring approach to radio fuzzy graphs [40,41,42]. The relationship (edges) can be more meaningful than the individual (nodes) at times. For instance, links rather than nodes are crucial in fuzzy social networks. An associated concept known as edge coloring is crucial for issues based on uncertainty. Regarding the application of graph coloring to communication systems based on utilizing ring-splits, in addition to comparing the peak throughput of NoCs for circulant and mesh topologies using deadlock-free routing algorithms, the results of high-level modeling were presented in [43]. The suggested method used fewer hardware resources while still achieving minimal transmission delay and enhanced thermal efficiency [44].

The motivation of this research work is to create a new platform to identify an optimal network in order to develop an effective optical network by utilizing various fuzzy graph operations based on edge color and domination parameters, including the operations of residue products, symmetric differences, max products, and lexicographic products. Using comparative studies on domination and edge coloring, our research and analysis aimed to assess the network’s strength. We have provided an algorithm to examine the effectiveness and efficiency of the constructed networks, which is the main framework of this research finding. Furthermore, we have developed applications for the product operation of these fuzzy networks.

2. Preliminaries

Definition 1

([12]). A Fuzzy graph ($H^{\ast }{}_{FG}=(V_{FG}^{\ast },E_{FG}^{\ast })$) is a pair of functions (${\sigma }_{V_{FG}^{\ast }}:V_{FG}^{\ast }\to [0,1]$ and ${\mu }_{V_{FG}^{\ast }}:V_{FG}^{\ast }\times V_{FG}^{\ast }\to [0,1]$) where ${\mu }_{V_{FG}^{\ast }}(b_1,b_2)\le \min \{{\sigma }_{V_{FG}^{\ast }}(b_1{}^{\ast }),{\sigma }_{V_{FG}^{\ast }}(b_2{}^{\ast })\}$ for $b_1{}^{\ast },b_2{}^{\ast }\in V{*}_{FG}$.

Definition 2

([12]). The underlying graph of a fuzzy graph is in the form ($H^{\ast }{}_{FG}=\hspace{0.17em}(V_{FG}^{\ast },E_{FG}^{\ast })$), where $V_{FG}^{\ast }=\{a_1{}^{\ast }\in V_{FG}^{\ast }:{\sigma }_{V_{FG}^{\ast }}(a_1{}^{\ast })>0\}$ and $E_{FG}^{\ast }=\{(a_1{}^{\ast },a_2{}^{\ast })\in V_{FG}^{\ast }\times V_{FG}^{\ast }:{\mu }_{V_{FG}^{\ast }}(a_1{}^{\ast },a_2\ast )>0\}$.

Definition 3

([12]). A subset ($T_F^{\ast }$) of $V_{FG}^{\ast }$ is said to be a dominating set of a fuzzy graph if every vertex in $V_{FG}^{\ast }$ − $T_{FG}^{\ast }$ is dominated by at least one vertex of $V_{FG}^{\ast }$. The dominating set ($T_{FG}^{\ast }$) is said to be minimal if no proper subset of $T_{FG}^{\ast }$ is a dominating set.

Definition 4

([12]). An arc ($a_1*-a_2*$) is said to be strong if the value of degree of an edge membership of an arc ($a_1*-a_2*$) is equal to strength of connectedness between a₁* and a₂*.

Definition 5

([12]). A vertex a₁* dominates a₂* if there is a strong arc between them.

Definition 6

([34]). A fuzzy graph ${\eta }_{ec}=\left(V_{ec}, {\sigma }_{ec}, {\mu }_{ec}\right)$ is a set that is not empty, together with a pair of functions ${\sigma }_{ec}: V_{ec}\to [0, 1]$ and ${\mu }_{ec}:V_{ec}\times V_{ec} \to [0, 1]$, such that $x, y ϵ V_{ec}$, ${\mu }_{ec}\left(c,d\right)\le {\sigma }_{ec}\left(c\right)Ʌ{\sigma }_{ec}\left(d\right)$, where ${\sigma }_{ec}\left(c\right)$ and ${\mu }_{ec}(c,d)$ represent the vetex membership values and the edge membership values, respectively.

Definition 7

[35]). A fuzzy graph ${\eta }_{ec}=(V_{ec}, {\sigma }_{ec}, {\mu }_{ec})$ is complete if ${\mu }_{ec}\left(p, q\right)=min\{{\sigma }_{ec}\left(p\right), {\sigma }_{ec}\left(q\right)\}$ for all p, q ∈ Vec, where (p, q) represents the edges between the vertices p and q.

Definition 8

([35]). A fuzzy graph ${\eta }_{ec}=\left(V_{ec}, {\sigma }_{ec}, {\mu }_{ec}\right)$ is said to be bipartite if vertex set $V_{ec}$ is divided into two nonempty sets $V_{{ec}_1}$ and $V_{{ec}_2}$, such that ${\mu }_{ec}\left(V_{{ec}_1}, V_{{ec}_2}\right)=0$ if $v_{{ec}_1}, v_{{ec}_2}ϵ V_{{ec}_1}$ or $v_{{ec}_1}, v_{{ec}_2}ϵ V_{{ec}_2}$. Further, if ${\mu }_{ec}\left(V_{{ec}_1}, V_{{ec}_2}\right)=min\{{\sigma }_{ec}\left(v_{{ec}_1}\right), {\sigma }_{ec}\left(v_{{ec}_2}\right)\}$ for all $v_{{ec}_1}ϵ V_{{ec}_1}and v_{{ec}_2}ϵ V_{{ec}_2}$, then ${\eta }_{ec}$ is called a fuzzy complete bipartite graph.

Definition 9

([29]). Let H = {h₁, h₂, …, h_λ}, λ ≥ 1 be the collection of neutral hues. Then, fuzzy set (H, k), where k: H → (0, 1), is known as a collection of fuzzy colors, and 0 < k(h_i) ≤ 1; the color’s membership value is the quantity of each element of the combination of h_i with the color white. Hence, the color (h_i, k(h_i)) is referred to as the fuzzy color that matches the fundamental color h_i. Thus, the k(h_i) [≤1] amount of h_i is mixed with 1 − k(h_i) to determine how much white color is needed to create the fuzzy color (h_i, k(h_i)). As stated in the definition above, the basic color is the building block from which all other colors are created. For example, green is a fundamental hue. A “fuzzy green” color can be blended to create other colors with 0.8 units of green and 0.2 units of white. This “fuzzy green” is denoted by a green value of 0.8. Similarly, another fuzzy red color (red, 0.6) may be formed by mixing 0.6 units of red with 0.4 units of white, and so on.

Definition 10

([29]). Let ${\eta }_{ec}=(V_{ec}, {\sigma }_{ec}, {\mu }_{ec})$ be a connected fuzzy graph and $c_{ec}=(c_{{ec}_1, }{c}_{{ec}_2}, \dots , c_{{ec}_k})$ be a set of basic colors. Now, two edges are only given two fuzzy colors whose basic colors differ if they are adjacent to one another; otherwise, they may be given fuzzy colors whose basic colors are the same. If the color of any edge is ${(c}_{{ec}_i}, f_{{ec}_j}\left(c_{{ec}_i}\right)),$ then $C_{{ec}_i}$ is the basic color of edge $e_{{ec}_j}$ = (p, q) and $f_{{ec}_j}\left(c_{{ec}_i}\right)$ is its membership value, which is calculated as $k_{{ec}_j}\left(c_{{ec}_i}\right)=\frac{{\mu }_{ec}(p.q)}{{\sigma }_{ec}\left(p\right)\wedge {\sigma }_{ec\left(q\right)}}$, where ${\sigma }_{ec}\left(p\right) and {\sigma }_{ec}\left(q\right)$ are the membership values of vertices p and q, respectively. Finally, ${\mu }_{ec}(p, q)$ is the membership value of the edge $e_{{ec}_j}$ , i.e., (p, q) in the fuzzy graph ${\eta }_{ec}$.

Definition 11

([29]). The fuzzy chromatic index of a fuzzy graph is the minimal set of fundamental colors required to color a fuzzy graph. Suppose there are M basic hues at the minimal level, the strengths of edges cannot be described by this chromatic index. For example, when two fuzzy graphs have identical chromatic indices, these graphs cannot be compared using this chromatic index. Hence, there is some weight assigned to the chromatic index. The weight is denoted by Sec, which is defined by

$$S_{ec}={\sum }_{i=1}^M\left\{{Maxf}_{{ec}_i}\right(c_{{ec}_i}\left)\right\}$$

where the basic color $c_{{ec}_i}$ is used to color edge $e_{{ec}_j}$ for some j and the depth of color is $k_{{ec}_i}\left(c_{{ec}_i}\right)$. Thus, S is the total of each basic color’s maximum membership values. Now, the chromatic index of a fuzzy graph is denoted by (M, S), where M is the minimum number of basic colors to color a graph and S is its weight. We generally follow the operations on fuzzy graph definitions from [13].

3. Operations on Fuzzy Graphs Using Edge Coloring

Mahabathra et al. [29] introduced Definition 11 to determine the weight of colors in a fuzzy graph. By focusing on the optimality of the fuzzy network, we can compare the formula given in Definition 11 with the sum of the minimal membership value of each color used in the fuzzy network. Hence, this research defines and modifies the formula in terms of the sum of the minimum value of the edge membership values of each color according to the fuzzy network, which yields the optimum value of the created network and helps us determine how effective it is. The chromatic number of a fuzzy graph represents the minimum number of colors required to color the graph. Let us assume that M_min is the minimum number of colors used to color the graph. The degree of membership of such crisp graphs is not sufficient to determine the strength of edges, and hence, some weight is associated with the chromatic number. These weights of the edges can influence the coloring process by indicating the strength of association of an edge with a particular color. The weighted minimum of basic colors used is denoted as W_min and is defined as

$$W_{\min }={\displaystyle \sum _{p=1}^M\left\{\min \hspace{0.17em}{g}_{e_q}(c_p)\right\}}$$

3.1. Residue Product of Two Fuzzy Graphs

Let ${RF}_1=({\sigma }_{{ec}_1},{\mu }_{{ec}_1}$) and ${RF}_2=({\sigma }_{{ec}_2},{\mu }_{{ec}_2})$ be two fuzzy graph networks of crisp graphs $G_{{RF}_1}=\left(V_{{ec}_1},E_{{ec}_1}\right) \mathrm{a}\mathrm{n}\mathrm{d} G_{{RF}_2}=\left(V_{{ec}_2},E_{{ec}_2}\right),$ respectively. Then, its residue product ${RF}_1·{RF}_2=({\sigma }_1·{\sigma }_2,{\mu }_1·{\mu }_2)$ is defined as

(i)

∀ (a, b) ∈ V₁ × V₂, (${\sigma }_1· {\sigma }_2)\left(a, b\right)={\sigma }_1\left(a\right) \wedge {\sigma }_2(b)$.

(ii)

∀ (a, b) ∈ E₁ and c $\ne$ w ∈ V₂,${ (\mu }_1· {\mu }_2\left)\right(\left(a, c\right), \left(b, w\right))={\mu }_1(a, b)$.

3.1.1. Example

G₁ = RF₁ and G₂ = RF₂ are the two fuzzy graphs of $G_{{RF}_1}=\left(V_{{ec}_1}, E_{{ec}_1}\right) \mathrm{a}\mathrm{n}\mathrm{d} G_{{RF}_2}=\left(V_{{ec}_2}, E_{{ec}_2}\right)$, depicted in Figure 1 and Figure 2, respectively. Then, the residue product of the fuzzy network is denoted by RF₁·RF₂, as shown in Figure 3.

Using Definition 10, the edge membership values of the above constructed RF₁·RF₂ are calculated and shown in Figure 3.

From Figure 3, the minimum number of basic colors used in RF₁·RF₂ is 4. The weight minimum number of basic colors used in constructed residue product RF₁·RF₂ = 0.16 + 0.2 + 0.16 + 0.16 = 0.68. Thus, the W_min of RF₁·RF₂ is 0.68.

3.1.2. Find the Weight of the Minimal Spanning Tree Using Kruskal’s Algorithm

To find the minimum spanning tree (MST) using the given set of edges, we follow the following steps:

Table 1 shows the weight of the graph, and

Table 2. The edges, sorted by weight in ascending order.

Edge	ay-bz	by-az	by-cz	cy-bz	cy-dz	dy-cz	ax-by	cx-by	bx-cy	dx-cy	ax-bz
Weight	0.16	0.16	0.16	0.16	0.2	0.2	0.28	0.28	0.33	0.33	0.42

sorts the edges in ascending order based on their weight. We begin by adding the edge ay-bz with a specific weight to the MST. Next, we add the edge by-az to the MST with a weight of 0.16. This edge does not create a cycle within the MST. Moving on, we include the edge by-cz with a weight of 0.16 in the MST. This edge also does not create any cycles. We continue by adding the edge cy-bz, with a weight of 0.16, to the MST. Once again, this edge maintains the property of not creating a cycle. Another edge, cy-dz, with a weight of 0.2, is added to the MST without causing any cycles. The edge dy-cz, with a weight of 0.2, is added to the MST, ensuring that no cycles are formed. We proceed by adding the edge ax-bzy, weight 0.28, to the MST. The inclusion of this edge does not result in any cycles. The edge cx-by, having a weight of 0.28, is included in the MST without creating cycles. Moving forward, we add the edge bx-cy with a weight of 0.33 to the MST; no cycles are introduced by this inclusion. Similarly, the edge dx-cy with weight 0.33 is integrated into the MST without creating any cycles. Lastly, we come across the edge ax-bz with weight 0.42. However, adding this edge would create a cycle within the MST. Thus, we discard it. Having visited all the nodes and ensuring that the number of edges is fewer than the number of nodes, we can conclude that the algorithm can now be stopped.

Table 1. The weight of a given graph (Figure 4).

Edge	ax-by	ax-bz	bx-cy	cx-by	dx-cy	ay-bz	by-az	by-cz	cy-bz	cy-dz	dy-cz
Weight	0.28	0.42	0.33	0.28	0.33	0.16	0.16	0.16	0.16	0.2	0.2

Table 1. The weight of a given graph (Figure 4).

Edge	ax-by	ax-bz	bx-cy	cx-by	dx-cy	ay-bz	by-az	by-cz	cy-bz	cy-dz	dy-cz
Weight	0.28	0.42	0.33	0.28	0.33	0.16	0.16	0.16	0.16	0.2	0.2

Using Kruskal’s algorithm, the weight of the minimal spanning tree of RF₁··RF₂ is shown to be 2.68.

3.1.3. Lower Domination Number of RF₁.RF₂

It is possible for there to be more than one minimal dominating set in an established network. The set with the lowest domination number among all minimal dominating sets is considered to be the created network’s lowest domination number, which allows us to test the network’s optimality. Let $G_{DN}=({\sigma }_{DN}, {\mu }_{DN}$) be the fuzzy graph of the crisp graph G_DN = (V_DN, E_DN), and let $S_{D{N}_1},S_{D{N}_2},\dots ,S_{DNr}$ be the minimal dominating set of G_DN. Finally, let the corresponding dominating number be denoted by ${\gamma }_{D{N}_1},\hspace{0.17em}{\gamma }_{D{N}_2},{\gamma }_{D{N}_3},\dots ,{\gamma }_{D{N}_r}$. Among this, the lowest cardinality of the domination number is called lower domination number and is denoted by ${\gamma }_{LD{N}_{.}}$

Let $S_{(1)R{F}_1\hspace{0.17em}.\hspace{0.17em}R{F}_2},\hspace{0.17em}{S}_{(2)R{F}_1.R{F}_2},\hspace{0.17em}{S}_{(3)R{F}_1\hspace{0.17em}.\hspace{0.17em}R{F}_2}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{S}_{(4)R{F}_1\hspace{0.17em}.\hspace{0.17em}R{F}_2}$ be the sum of the minimal dominating set of RF_1.RF₂ (refer to Figure 3)

$$S_{(1)R{F}_1.R{F}_2}=\{ay,by,cy,dy\}; S_{(2)R{F}_1.R{F}_2}=\{ax,bx,cx,dx\};$$

$$S_{(3)R{F}_1.R{F}_2}=\{az,bz,cz,dz\}$$

$${\gamma }_{{}_{(1)R{F}_1.R{F}_2}} \mathrm{of} S_{(1)R{F}_1.R{F}_2}=\hspace{0.17em}0.3+0.6+0.6+0.3=1.8$$

$${\gamma }_{{}_{(2)R{F}_1.R{F}_2}} \mathrm{of} S_{(2)R{F}_1.R{F}_2}=2.6.$$

$${\gamma }_{{}_{(2)R{F}_1.R{F}_2}} \mathrm{of} S_{(3)R{F}_1.R{F}_2}=\hspace{0.17em}\hspace{0.17em}2.1.$$

$${\gamma }_{{}_{LDN-R{F}_1.R{F}_2}} \mathrm{of} R{F}_1 R{F}_2 \mathrm{is} 1.8.$$

3.2. Symmetric Difference of Two Fuzzy Graphs

Let SDF₁ = $({\sigma }_{{ec}_1},{\mu }_{{ec}_1})$ and SDF₂ = $({\sigma }_{{ec}_2},{\mu }_{{ec}_2})$ be two fuzzy graphs of crisp graphs $G_{{SDF}_1}=\left(V_{{ec}_1},E_{{ec}_1}\right) and{G}_{{SDF}_2}=(V_{{ec}_2},E_{{ec}_2})$, respectively. Then, the symmetric difference between SDF₁ and SDF₂ is denoted by SDF₁ ⊕ SDF₂ = (${\sigma }_{{ec}_1}$⊕${ \sigma }_{{ec}_2}$, ${\mu }_{{ec}_1}$⊕${\mu }_{{ec}_2}$) and is defined as follows

• $\forall \left(a,b\right)\in V_1\times V_{2,}$
  ${\sigma }_{{SDF}_1}\oplus {\sigma }_{{SDF}_2}\left(a,b\right)={\sigma }_{{SDF}_1}\left(a\right)\wedge {\sigma }_{{SDF}_2}\left(b\right)$.
• $\forall a\in V_1 and \left(b,c\right)\in E_2$
  ${(\mu }_{{SDF}_1}\oplus { \mu }_{{SDF}_2}\left)\right(\left(a,b\right),\left(a,c\right)={\sigma }_{{SDF}_1}\left(a\right)\wedge {\mu }_{{SDF}_2}(b,c)$.
• $\forall a\in V_2 and \left(b,c\right)\in E_1$
  ${(\mu }_{{SDF}_1}\oplus { \mu }_{{SDF}_2}\left)\right(\left(b,a\right),(c,a))={\mu }_{{SDF}_1}(b,c)\wedge {\sigma }_{{SDF}_2}(a)$.
• $\forall \left(a,b\right)\notin E_1 and \left(c,w\right)\in E_2$,
  ${(\mu }_{{SDF}_1}\oplus { \mu }_{{SDF}_2}\left)\right(\left(a,c\right),(b,w))=\mathrm{m}\mathrm{i}\mathrm{n}\{{\sigma }_{{SDF}_1}\left(a\right),{\sigma }_{{SDF}_1}\left(b\right),{\mu }_{{SDF}_2}(c,w)\}$.
• $\forall \left(a,b\right)\in E_1 and \left(c,w\right)\notin E_2$,
  ${(\mu }_{{SDF}_1}\oplus { \mu }_{{SDF}_2}\left)\right(\left(a,c\right),(b,w))=\mathrm{m}\mathrm{i}\mathrm{n}\{{\mu }_{{SDF}_1}\left(a,b\right),{\sigma }_{{SDF}_2}\left(c\right),{\sigma }_{{SDF}_2}\left(w\right)\}$.

3.2.1. Example

Let G₁ = SDF₁ and G₂ = SDF₂ be the two fuzzy graphs of crisp graphs $G_{{SDF}_1}=\left(V_{{ec}_1},E_{{ec}_1}\right) and G_{{SDF}_2}=\left(V_{{ec}_2},E_{{ec}_2}\right)$, depicted in Figure 1 and Figure 2, respectively.

The symmetric difference of fuzzy network SDF₁$\oplus$SDF₂ is shown in Figure 5.

Using Definition 10, the edge membership values of the above constructed SDF₁ ⊕ SDF₂ are calculated, as shown in Figure 5.

As shown in Figure 5, the minimum number of basic colors used in SDF₁ ⊕ SDF₂ is 7. The weight minimum number of basic colors used in the constructed symmetric difference network SDF₁ ⊕ SDF₂ = 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.75 + 0.5 = 3.75. Thus, the W_min of SDF₁ ⊕ SDF₂ is 3.75.

Using Kruskal’s algorithm, the weight of the minimal spanning tree of SDF₁ ⊕ SDF₂ was found to be 7.

3.2.2. Lower Domination Number of SDF₁ ⊕ SDF₂

Let $S_{(1)SD{F}_1\oplus SD{F}_2},\hspace{0.17em}{S}_{(2)SD{F}_1\oplus SD{F}_2},S_{(3)SD{F}_1\oplus SD{F}_2}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{S}_{(4)SD{F}_1\oplus SD{F}_2}$ be some of the minimal dominating set of SDF₁ ⊕ SDF₂ (see Figure 5)

$$S_{(1)SD{F}_1\oplus SD{F}_2}=\{ax,ay\}; S_{(2)SD{F}_1\oplus SD{F}_2}=\{az,ay\};$$

$$S_{(3)SD{F}_1\oplus SD{F}_2}=\{dx,dy\}; S_{(4)SD{F}_1\oplus SD{F}_2}=\{dy,dz\}.$$

$${\gamma }_{{}_{(1)SD{F}_1.\oplus SD{F}_2}} \mathrm{of} S_{(1)SD{F}_1\oplus SD{F}_2}=2.2$$

$${\gamma }_{{}_{(2)SD{F}_1.\oplus SD{F}_2}} \mathrm{of} S_{(2)SD{F}_1\oplus SD{F}_2}=1.6.$$

$${\gamma }_{{}_{(3)SD{F}_1.\oplus SD{F}_2}} \mathrm{of} S_{(3)SD{F}_1\oplus SD{F}_2}\hspace{0.17em}=2.2.$$

$${\gamma }_{{}_{(4)SD{F}_1.\oplus SD{F}_2}} \mathrm{of} S_{(4)SD{F}_1\oplus SD{F}_2}=1.9.$$

$${\gamma }_{{}_{LDN-SD{F}_1\oplus .SD{F}_2}} \mathrm{of} SD{F}_1\oplus SD{F}_2 \mathrm{is} 1.6.$$

3.3. Max Product of Two Fuzzy Graphs

Let MF₁ = ${(\sigma }_{{mf}_1},{\mu }_{{mf}_1}) \mathrm{a}\mathrm{n}\mathrm{d} {MF}_2=({\sigma }_{{mf}_2},{\mu }_{{mf}_2})$ be two fuzzy networks of crisp graphs $G_{{MF}_1}=\left(V_{{mf}_1},E_{{mf}_1}\right) \mathrm{a}\mathrm{n}\mathrm{d} G_{{MF}_2}=\left(V_{{mf}_2},E_{{mf}_2}\right),$ respectively. The maximal product of fuzzy graphs MF₁ and MF₂ is represented by MF₁ * MF₂ = ${(\sigma }_{{mf}_1}$ * ${\sigma }_{{mf}_2}$, ${\mu }_{{mf}_1}$*, ${\mu }_{{mf}_2}$) and is defined as:

(i)

∀ (a, b) ∈ $V_{{mf}_1}$ × $V_{{mf}_2}$,

$\left({\sigma }_{{mf}_1}\mathrm{*}{\sigma }_{{mf}_2}\right)\left(a,b\right)={\sigma }_{{mf}_1}\left(a\right)\vee {\sigma }_{{mf}_2}\left(b\right)$.

(ii)

∀ a ∈$V_{{mf}_1}$ and (b, c) ∈$E_{{mf}_2}$,

${(\mu }_{{mf}_1}\mathrm{*}{\mu }_{{mf}_2}\left)\right(\left(a,b\right),(a,c))={\sigma }_{{mf}_1}(a)\vee {\mu }_{{mf}_2}(b,c)$.

(iii)

∀ a ∈$V_{{mf}_2}$ and (b, c) ∈$E_{{mf}_1}$,

$\left({\mu }_{{mf}_1}\mathrm{*}{\mu }_{{mf}_2}\right)\left(\left(b,a\right),\left(c,a\right)\right)={\mu }_{{mf}_1}\left(b,c\right)\vee {\sigma }_{{mf}_2}\left(a\right)$.

3.3.1. Example

G₁ = MF₁ and G₂ = MF₂ defines the two fuzzy graphs of crisp graphs $G_{{MF}_1}=\left(V_{{ec}_1}, E_{{ec}_1}\right) \mathrm{a}\mathrm{n}\mathrm{d} G_{{MF}_2}=\left(V_{{ec}_2}, E_{{ec}_2}\right),$ depicted in Figure 1 and Figure 2, respectively. The max product of the fuzzy network MF₁ * MF₂ is shown in Figure 6.

Using Definition 10, the edge membership values of the above constructed MF₁ * MF₂ were calculated, as shown in Figure 6.

As shown in Figure 6, the minimum number of basic colors used in MF₁ * MF₂ is 4. The weight minimum number of basic colors used in the constructed maximal product network MF₁ * MF₂ = 0.37 + 1 + 0.83 + 0.83 = 3.03. Thus, the W_min of MF₁ * MF₂ is 3.03.

Using Kruskal’s algorithm, the weight of the minimal spanning tree of MF₁ * MF₂ was found to be 9.29.

3.3.2. Lower Domination Number of MF₁ * MF₂

Let $S_{(1)M{F}_1*M{F}_2},\hspace{0.17em}{S}_{(2)M{F}_1*M{F}_2},S_{(3)M{F}_1*M{F}_2}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}{S}_{(4)M{F}_1*M{F}_2}$ be some of the minimal dominating set of MF₁ * MF₂ (see Figure 6)

$$S_{(1)M{F}_1*M{F}_2}=\{ay,cx,cz,dy\}{S}_{(2)M{F}_1*M{F}_2}=\{ay,bx,by,dy\};$$

$$S_{(3)M{F}_1*M{F}_2}=\{bx,cx,ay,dy\}; S_{(4)M{F}_1*M{F}_2}=\{ax,bz,cx,dz\}.$$

$${\gamma }_{{}_{(1)M{F}_1.*M{F}_2}}\mathrm{of}{S}_{(1)M{F}_1*M{F}_2}=\hspace{0.17em}\hspace{0.17em}6.5;{\gamma }_{{}_{(2)M{F}_1.*M{F}_2}} \mathrm{of} S_{(2)M{F}_1*M{F}_2}=\hspace{0.17em}\hspace{0.17em}6.7;$$

$${\gamma }_{{}_{(3)SM{F}_1.*M{F}_2}}\mathrm{of}{S}_{(3)M{F}_1*M{F}_2}=\hspace{0.17em}\hspace{0.17em}7.2;{\gamma }_{{}_{(4)M{F}_1.*M{F}_2}} \mathrm{of} S_{(4)M{F}_1*M{F}_2}=\hspace{0.17em}\hspace{0.17em}5.7;$$

$${\gamma }_{{}_{LDN-M{F}_1.*M{F}_2}} \mathrm{of} M{F}_1*M{F}_2 \mathrm{is} 5.7.$$

3.4. Lexicographic Product of Two Fuzzy Graphs

Let LF₁ = (M₁, P₁) and LF₂ = (M₂, P₂) be two fuzzy graphs of the crisp graphs $G_{{LF}_1}=(V_{{ec}_1},E_{{ec}_1})$ and $G_{{LF}_2}=(V_{{ec}_2},E_{{ec}_2})$ respectively.

The lexicographic product of the two graphs is denoted as LF₁·LF₂ in fuzzy graph pair (M, P), such that

(i)

$M\left(a_1,b_2\right)=\mathrm{m}\mathrm{i}\mathrm{n}(M_1\left(a_1\right),M_2\left(b_2\right))$, $\forall \left(a_1,b_2\right)\in M_1\times M_2$.

(ii)

$P(\left(x,b_2\right)\left(x,d_2\right)=\mathrm{m}\mathrm{i}\mathrm{n}(M_1\left(x\right),M_2\left(b_2{d}_2\right)$, $\forall x\in M_1,b_2{d}_2\in P_2$.

(iii)

$P\left(\left(a_1,b_2\right)\left(c_1,d_2\right)\right)=\min \left(P_1\left(a_1{c}_1\right),P_2\left(b_2{d}_2\right)\right),\forall a_1{b}_1\in P_1 \mathrm{a}\mathrm{n}\mathrm{d} b_2{d}_2\in P_2$.

3.4.1. Example

Let G₁ = LF₁ and G₂ = LF₂ be two fuzzy graphs of crisp graphs $G_{{LF}_1}=\left(V_{{ec}_1}, E_{{ec}_1}\right) \mathrm{a}\mathrm{n}\mathrm{d} G_{{LF}_2}=\left(V_{{ec}_2}, E_{{ec}_2}\right)$, depicted in Figure 1 and Figure 2, respectively. The lexicographic product of the fuzzy network is denoted by LF₁·LF₂, as shown in Figure 7.

Using Definition 10, the edge membership values of the above constructed LF₁·LF₂ were calculated, as shown in Figure 7.

As shown in Figure 7, the minimum number of basic colors used in LF₁·LF₂ is 6. The weight minimum number of basic colors used in the constructed lexicographic product of network LF₁·LF₂ = 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.5 = 3. Thus, the W_min of LF₁·LF₂ is 3.

Using Kruskal’s algorithm, the weight of the minimal spanning tree of LF₁·LF₂ was found to be 5.5.

3.4.2. Lower Domination Number of LF₁·LF₂

Let $S_{L{F}_1*L{F}_2}$ be the minimal dominating set (only one dominating set) of LF₁·LF₂ (see Figure 7)

$$S_{L{F}_1*L{F}_2}=\hspace{0.17em}\hspace{0.17em}\{ay,by,cy,dy\}.{\gamma }_{{}_{LDN-L{F}_1.*L{F}_2}} \mathrm{of} S_{L{F}_1*L{F}_2}\hspace{0.17em}is\hspace{0.17em}3.$$

Algorithm 1 provides comparative studies of fuzzy domination and fuzzy coloring in operations of fuzzy networks.

4. Algorithm to Find an Optimal Network

4.1. Algorithm

This algorithm explores how to determine an optimal network using the product operations of a fuzzy network.

Algorithm 1: Find an optimal network using the operations of a fuzzy network

Input: $\mathrm{Two} \mathrm{fuzzy} \mathrm{networks} {FG}_1=({\sigma }_{c_1}, {\mu }_{c_1}$)$\mathrm{and} {FG}_2=({\sigma }_{c_2}, {\mu }_{c_2})$$\mathrm{of} \mathrm{crisp} \mathrm{graphs} G_{F_1}=\left(V_{c_1}, E_{c_1}\right) and G_{F_2}=\left(V_{c_2}, E_{c_2}\right),$ respectively. Output: Optimal fuzzy network Begin Step 1: Construct a collection of finite networks, e.g., $N_1, N_2, \dots , N_r$, by performing separate operations on a fuzzy network with vertex sets $V=V_{c_1}\times V_{c_2}$. Step 2: Calculate $k_{{ec}_j}\left(c_{{ec}_i}\right)=\frac{{\mu }_{ec}(p.q)}{{\sigma }_{ec}\left(p\right)\wedge {\sigma }_{ec\left(q\right)}}$ of all edges. Vertices are to be labeled as 1, 2, 3,…, n Step 3:$\mathrm{Find} \mathrm{the} \mathrm{membership} \mathrm{function} \mathrm{value} \mathrm{for} \mathrm{each} \mathrm{node} \mathrm{and} \mathrm{edge} \mathrm{of} N_1, N_2, \dots , N_r$ using the operations applied to the constructed network. Steps 4: In the constructed network, e.g., N1, identify vertex ‘1′ of the maximum degree and color all its incident edges such that no two incident edges receive the same color. Step 5: Next, focus the direct neighbors of vertex 1, color the incident edges of the neighboring vertex, and label them as 12, 13, …, 1m, if there are m neighbors. If the number of neighbors is less than m, then use the minimum number of the same color used for vertex ‘1′. Step 6: Proceed with Step 5 again until all the edges receive the colors. From the above step, we get the minimum number of colors used to color the given network. Steps 4, 5, and 6 will continue for the other constructed networks $N_2, \dots , N_r$ to find the minimum number of colors used to color the edges of the networks. Step 7: Find the minimal edge coloring set with the minimum number of colors used to color the established network and the corresponding optimal weight of the colors. Step 8:$\mathrm{Find} \mathrm{the} \mathrm{lower} \mathrm{domination} \mathrm{number} \mathrm{of} \mathrm{established} \mathrm{network} N_1, N_2, \dots , N_r$ and let it be ${\gamma }_{LD{N}_1},\hspace{0.17em}{\gamma }_{LD{N}_2},{\gamma }_{LD{N}_3},\dots ,{\gamma }_{LD{N}_r}$, respectively. Step 9: $\mathrm{Find} \mathrm{the} \mathrm{minimum} \mathrm{number} \mathrm{of} \mathrm{colors} \mathrm{used} \mathrm{to} \mathrm{color} \mathrm{the} \mathrm{established} \mathrm{network}, \mathrm{e}.\mathrm{g}., S_{{ec}_1}, S_{{ec}_2}\dots S_{{ec}_r}$$, \mathrm{and} \mathrm{its} \mathrm{corresponding} \mathrm{weight}, \mathrm{e}.\mathrm{g}., {WS}_{{ec}_{1, }},{WS}_{{ec}_2},\dots ,{WS}_{{ec}_r}$$\mathrm{of} N_1, N_2, \dots , N_r$, respectively. Step 10: Determine the minimal spanning tree of the constructed networks, e.g., ${ST}_1, {ST}_2,\dots , {ST}_r$ of $N_1, N_2, \dots , N_r$, respectively, and let its corresponding minimum weight of ${ST}_1, {ST}_2,\dots , {ST}_r$ be $M_{{ST}_1, }{M}_{{ST}_{2, }}...,M_{{ST}_{r, }}$ Step 11: Let $O_{D{N}_i}$ denote the sum of the lower domination number of established network Ni and the minimum weight of spanning tree Ni, that is, $O_{D{N}_i}={\gamma }_{LD{N}_i}+M_{S{T}_i}$$. O_{N_{i }}$ is the sum of the weight of the basic colors used in establised network Ni and the minimum weight of spanning tree Ni, that is

$O_{N_i}=W{S}_{e{c}_i}+M_{S{T}_i}$

Step 12: Optimal value of the established network using domination,$O_{opt-DN}=\min \left\{O_{D{N}_i}\right\}\hspace{0.17em}\hspace{0.17em}i\hspace{0.17em}=\hspace{0.17em}1,2,\dots ,r$, and optimal value of the established network using chromatic index, $O_{opt-CI}=\min \left\{O_{N_i}\right\}\hspace{0.17em}\hspace{0.17em}i\hspace{0.17em}=\hspace{0.17em}1,2,\dots ,r$. Step-13: Effective optimal value of the established network,$O_{eff-opt-val}=\min \{O_{opt-DN},O_{opt-CI}\}\hspace{0.17em}\hspace{0.17em}i\hspace{0.17em}=\hspace{0.17em}1,2,\dots ,r$. End

4.2. Flow Chart of the Algorithm

Figure 8, shows the flow chart of algorithm. We derived the following from Section 3.

Let $W{S}_{e{c}_1}$, $W{S}_{e{c}_2}$, $W{S}_{e{c}_3}$, and $W{S}_{e{c}_4}$ be the weight of the minimum number of basic colors used in the residue product, symmetric difference, max product, and lexicographic, respectively, and let its corresponding minimum weight of the spanning tree be $M_{S{T}_1}$, $M_{S{T}_2}$, $M_{S{T}_3}$ and $M_{S{T}_4}$. Let $O_{N_1}, O_{N_2}, O_{N_3}, \mathrm{a}\mathrm{n}\mathrm{d} O_{N_4}$ be the sum of the weight of the minimum number of basic colors used in the established network and the minimum weight of the spanning tree of the residue product, symmetric difference, max product, and lexicographic, respectively.

$$W{S}_{e{c}_1} \mathrm{of} {RF}_1·{RF}_2 \mathrm{is} 4.68, \mathrm{and} M_{S{T}_1} \mathrm{is} 2.68. \mathrm{Hence} O_{N_1}=4.68+2.68=7.36.$$

$$W{S}_{e{c}_2} \mathrm{of} {SDF}_1 \oplus {SDF}_2 \mathrm{is} 10.75 \mathrm{and} M_{S{T}_2} \mathrm{is} 7. \mathrm{Hence} O_{N_2}=10.75+7=17.75.$$

$$W{S}_{e{c}_3}\mathrm{of} {MF}_1*{MF}_2 \mathrm{is} 7.03 \mathrm{and} M_{S{T}_3} \mathrm{is} 9.29. \mathrm{Hence} O_{N_3}=7.03+9.29=16.32.$$

$$W{S}_{e{c}_3} \mathrm{of} {LF}_1·{LF}_2 \mathrm{is} 9, \mathrm{and} M_{S{T}_3} \mathrm{is} 5.5. \mathrm{Hence} O_{N_4}=9+5.5=14.5.$$

Let $O_{{DN}_1}, O_{D{N}_2}, O_{{DN}_3}, and O_{D{N}_4}$ be the sum of the lower domination number and minimum weight of the spanning tree of residue product, symmetric difference, max product, and lexicographic, respectively

$${\gamma }_{{}_{LDN-R{F}_1.R{F}_2}} \mathrm{of} {RF}_1·{RF}_2 \mathrm{is} 1.8, {\gamma }_{{}_{LDN-SD{F}_1\oplus .SD{F}_2}} \mathrm{of} SD{F}_1\oplus SD{F}_2 \mathrm{is} 1.6.$$

$${\gamma }_{{}_{LDN-M{F}_1.*M{F}_2}} \mathrm{of} {MF}_1*{MF}_2 \mathrm{is} 5.7 \mathrm{and} {\gamma }_{{}_{LDN-L{F}_1.*L{F}_2}} \mathrm{of}$$

$$S_{L{F}_1*L{F}_2}is\hspace{0.17em}3.$$

$$\mathrm{Hence}, O_{{DN}_1}=1.8+2.68=4.48, O_{{DN}_2}=1.6+ 7=8.6, O_{{DN}_3}=5.7+9.29=14.99 \mathrm{and}$$

$$O_{{DN}_4}=3+5.5=8.5$$

The optimal value of the established network $:O_{opt-CI}={min}_i\left\{O_{N_i}\right\} ,i=1,2,3,4=3.36.$

The optimal value of the established network using domination:$O_{opt-DN}=\min \left\{O_{D{N}_i}\right\}\hspace{0.17em}\hspace{0.17em}i\hspace{0.17em}=\hspace{0.17em}1,2,3,4=4.48$

The effective optimal value of the established network: $O_{eff-opt-val}=\min \{O_{opt-DN},O_{opt-CI}\}\hspace{0.17em}\hspace{0.17em}i\hspace{0.17em}=\hspace{0.17em}1,2,3,4=3.36.$

Comparison between the optimal value using the weight of the minimum basic colors used and the domination number of the established network, as shown in

Table 3. Comparison of $O_{D{N}_i}$ and $O_{N_i}$.

S. No	Established Network	${\mathit{O}}_{\mathit{D}{\mathit{N}}_{\mathit{i}}}$	${\mathit{O}}_{{\mathit{N}}_{\mathit{i}}}$
1	RF1·RF2	4.48	3.36
2	SDF1 ⊕ SDF2	7.6	7.75
3	MF1* MF2	14.99	12.32
4	LF1·LF2	8.5	8.5

.

Hence the effective optimal value of the established network using edge coloring is more effective than the network using the domination number.

4.3. Applications of a Fuzzy Graph in a Social Network

Social networks have been around for a very long time. They apply the simple process of extending the number of individuals you know by connecting with peers and so on. Indeed, many of us now utilize social media platforms like Facebook and Twitter to promote our present and potential enterprises. People who want to connect with other business-related acquaintances frequently go to sites like LinkedIn. LinkedIn is a social media website built especially for professional networking to assist people in finding a job, locating sales leads, and connect with possible business partners. Let us assume the presence of group of people in a network, where $G_3·G_{4 }$ is the network of organizations linked between them. When these two networks collaborate, it is via a new fuzzy network created by lexicographic product operation $G_3· G_4$. Here, the organizations that collaborated with the people, if an applicant’s demands meets the organization’s needs, are represented as vertices, and the information shared between them is represented as edges. Let us consider a, b, c, and d as people with distinct skills and x, y, and z as organizations; see

Table 4. Roles of vertices.

S. No	Vertices	Characteristic of Each Vertex
1	a	Technical skills
2	b	Data-driven skills
3	c	Management skills
4	d	Marketing skills
5	x	Technology Company
6	y	Consulting firms
7	z	Retail and consumer goods company

.

Every applicant satisfies the minimum demand of the organization in terms of their skills, so they are allowed to see what they are working on. Applying lexicographic product operation, LinkedIn provides more personalized networking recommendations.

Above Figure 9 and Figure 10 shows the fuzzy graph G₃ and G₄. When these networks collaborate, they share information, form business-to-business relationships, provide services to find jobs, and share market products in the network. A network of lexicographic products of two graphs was constructed, as shown in Figure 7. For example, if “ax” is a technology organization with one person with technical skills, they may have expertise in the job of data scientist and they may share information about the needs of the organization with “ay “(Big data developer) and” by” (Private Equity Analyst). With experts in the fields of technical and data-driven skills, they collaborate to meet their own needs and, likewise, share information. Using lexicographic products provides a comprehensive view that can potentially reduce the complexity of analyses. This fully connected network makes employees as well the organization more flexible in terms of sharing information that may be depicted using six distinct colors, where each color represents the most efficiently shared information; also, time management is taken into consideration, while the colors of the edges, such as red, green, yellow and blue, provide information regarding the minimum possible weight of the edges. The amount of information shared through edge coloring is effective in networking; the chromatic index of the edges is the minimum that leads to the optimization of the network. Vertices ay, by, cy, and dy seem to have a significant influence and dominate in the network with minimum time management. The information is used to curate relevant collaborations, thereby increasing knowledge sharing as well as the profits benefits for the organizations and making the network more optimal. Hence, using domination in the connected network, i.e., the colored minimum weighted edges, enhances the allocation of resources by considering the timing and coordination of interactions. The minimum weight of the edges contributes to more harmonious network dynamics, ultimately improving the experience of users and achieving the network goals more effectively.

4.4. Applications of Fuzzy Graph Coloring in a Communication Network

Fuzzy graph coloring is used to build fault-tolerant routing schemes with acyclic subnetwork in on-chip networks. Fuzzy graph coloring is a mathematical technique that extends traditional graph coloring by allowing vertices to be partially colored with fractional or fuzzy values, rather than being strictly assigned a single color. This approach has been used in various applications, including fault-tolerant routing schemes in on-chip networks with acyclic subnetworks. Fuzzy graph coloring can be applied in this context in the following ways:

4.4.1. Fault-Tolerant Routing in On-Chip Networks

On-chip networks are essential components of modern microprocessors, enabling communication among different cores and memory banks on a single chip. To ensure reliability in these networks, fault-tolerant routing schemes are employed. These schemes aim to find alternative routes in the presence of faults (e.g., faulty links or routers) to maintain communication.

4.4.2. Acyclic Subnetworks

In some on-chip network topologies, acyclic subnetworks are used. Acyclic networks have no cycles, meaning there are no closed paths or loops in the network. This property simplifies routing but can introduce challenges in fault tolerance, since alternative paths must be carefully chosen to avoid loops.

4.4.3. Fuzzy Graph Coloring

Fuzzy graph coloring can be applied to model and optimize the assignment of communication routes within an on-chip network, taking into account the acyclic subnetwork structure and fault tolerance requirements. It works as follows:

Vertex Coloring

In a traditional graph coloring problem, each vertex (representing a router or node in the network) is assigned a single color, and adjacent vertices cannot have the same color. In fuzzy graph coloring, vertices are assigned fractional colors, allowing multiple vertices to share the same color value to a certain degree.

Edge Weights

In the context of fault-tolerant routing, edge weights can represent the reliability or quality of network links. Faulty or less reliable links are assigned lower weights.

4.4.4. Optimization Objective

The goal is to minimize the total conflict among adjacent vertices while maximizing the reliability of communication paths. This can be formulated as an optimization problem where the fractional coloring of vertices and the selection of routes are jointly optimized.

4.4.5. Fault Tolerance

By allowing vertices to share colors to some extent and optimizing the assignment of routes based on edge weights (reliability), the fuzzy graph coloring approach can help in finding fault-tolerant routing schemes. The algorithm can find alternative routes that avoid faulty components while maintaining the acyclic nature of the subnetwork.

4.4.6. Implementation

Implementing fuzzy graph coloring for fault-tolerant routing schemes in on-chip networks may require specialized algorithms and software tools. This implementation may consider factors like the network topology, fault models, and communication patterns specific to the on-chip network architecture.

Fuzzy graph coloring can be a useful technique for designing fault-tolerant routing schemes in on-chip networks with acyclic subnetworks. It enables the optimization of communication paths while considering fault tolerance and network reliability, ultimately improving the robustness and performance of on-chip communication systems.

5. Conclusions

The implications of this study extend beyond theoretical frameworks and hold practical significance. The insights gained from our research have the potential to inform decision-making processes, optimize calculations, and offer valuable guidance in various real-world applications. This paper has described operations such as residue product, symmetric difference, max product, and lexicographic on fuzzy graphs. Additionally, we have tried to discover some of their characteristics to determine how effective they are, and we have discussed how the lexicographic operation may be used in the real world while using a minimal spanning tree approach, which was developed to perform activities with minimal strength in a social network. The study will eventually cover additional processes and include methods for maximizing the created network’s efficiency. In the future, we will extend our studies to include intuitionistic fuzzy networks and neutrosophic networks.