A comparative study of fuzzy domination and fuzzy coloring in an optimal approach

Citation: Meenakshi, A.; Kannan, A.; Mahdal, M.; Karthik, K.; Guras, R. A Comparative Study of Fuzzy Domination and Fuzzy Coloring in an Optimal Approach. Mathematics 2023,11, 4019. https://doi.org/ 10.3390/math11184019 Academic Editor: Konstantin Kozlov Received: 24 July 2023 Revised: 10 September 2023 Accepted: 11 September 2023 Published: 21 September 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article A Comparative Study of Fuzzy Domination and Fuzzy Coloring in an Optimal Approach Annamalai Meenakshi 1, Adhimoolam Kannan 1,2 , Miroslav Mahdal 3, Krishnasamy Karthik 4,* and Radek Guras 3 1Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai 600062, India; [email protected] (A.M.); [email protected] (A.K.) 2Department of Mathematics, Vel Tech Multi Tech Dr. Rangarajan Dr. Sakunthala Engineering College, Chennai 600062, India 3Department of Control Systems and Instrumentation, Faculty of Mechanical Engineering, VSB-Technical University of Ostrava, 17. Listopadu 2172/15, 70800 Ostrava, Czech Republic; miroslav[email protected] (M.M.); [email protected] (R.G.) 4Department of Mechanical Engineering, Vel Tech Rangarajan Dr. Sagunthala R&D Institute of Science and Technology, Chennai 600062, India *Correspondence: [email protected] 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. Keywords: fuzzy coloring; minimum spanning tree; domination number; optimal network MSC: 05C15; 05C76 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 Kis a collection of non-empty vertices, and Lis 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 – 7 ]. More study is being conducted on fuzzy graphs [ 8 – 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 Mathematics 2023,11, 4019. https://doi.org/10.3390/math11184019 https://www.mdpi.com/journal/mathematics Mathematics 2023,11, 4019 2 of 16 neutrosophic graphs [ 12 – 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 – 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 – 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 – 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∗FG = (V∗ FG , E∗ FG) ) is a pair of functions ( σV∗ FG :V∗ FG →[0, 1] and µV∗ FG :V∗ FG ×V∗ FG →[0, 1] ) where µV∗ FG (b1 , b2)≤min{σV∗ FG (b1∗) , σV∗ FG (b2∗)} for b1∗,b2∗∈V∗FG. Definition 2 ([ 12 ]) . The underlying graph of a fuzzy graph is in the form ( H∗FG = (V∗ FG , E∗ FG) ), where V∗ FG =na1∗∈V∗ FG :σV∗ FG (a1∗)>0o and E∗ FG ={(a1∗ , a2∗)∈V∗ FG ×V∗ FG : µV∗ FG (a1∗,a2∗)>0}. Definition 3 ([ 12 ]) . A subset ( T∗ F ) of V∗ FG is said to be a dominating set of a fuzzy graph if every vertex in V∗ FG −T∗ FG is dominated by at least one vertex of V∗ FG . The dominating set ( T∗ FG ) is said to be minimal if no proper subset of T∗ FG is a dominating set. Definition 4 ([ 12 ]) . An arc ( a1∗ −a2∗ ) is said to be strong if the value of degree of an edge membership of an arc (a1∗ −a2∗) 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. Mathematics 2023,11, 4019 3 of 16 Definition 6 ([ 34 ]) . A fuzzy graph ηec =(Vec,σec,µec) is a set that is not empty, together with a pair of functions σec :Vec →[0, 1] and µec :Vec ×Vec →[0, 1] , such that x , yeVec , µec(c,d)≤σec(c)Λσec(d) , where σec(c) and µec(c,d) represent the vetex membership values and the edge membership values, respectively. Definition 7 [ 35 ]) . A fuzzy graph ηec =(Vec,σec,µec) is complete if µec(p,q)=min{σec(p) , σec(q)}for all p, q ∈Vec, where (p, q) represents the edges between the vertices p and q. Definition 8 ([ 35 ]) . A fuzzy graph ηec =(Vec,σec,µec) is said to be bipartite if vertex set Vec is divided into two nonempty sets Vec1 and Vec2 , such that µec(Vec1,Vec2)= 0if vec1 , vec2eVec1 or vec1 , vec2eVec2 . Further, if µec(Vec1,Vec2)=min{σec(vec1),σec(vec2)} for all vec1eVec1and vec2eVec2 , then ηec is called a fuzzy complete bipartite graph. Definition 9 ([29]).Let H = {h1, h2,. . ., hλ}, 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 ηec =(Vec,σec,µec) be a connected fuzzy graph and cec =cec1,cec2, . . . , ceck 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 (ceci , fecj(ceci)) ,then Ceci is the basic color of edge eecj = (p, q) and fecj(ceci) is its membership value, which is calculated as kecj(ceci)=µec(p.q) σec(p)∧σec(q) , where σec(p)and σec(q) are the membership values of vertices p and q, respectively. Finally, µec(p,q) is the membership value of the edge eecj , i.e., (p, q) in the fuzzy graph ηec. 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 Sec =∑M i=1nMax f eci(ceci)o where the basic color ceci is used to color edge eecj for some j and the depth of color is keci(ceci) . 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 Mathematics 2023,11, 4019 4 of 16 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 Wmin and is defined as Wmin = M ∑ p=1nmin geq(cp)o 3.1. Residue Product of Two Fuzzy Graphs Let RF1=(σec1,µec1 ) and RF2=(σec2,µec2) be two fuzzy graph networks of crisp graphs GRF1=(Vec1,Eec1)and GRF2=(Vec2,Eec2) , respectively. Then, its residue product RF1·RF2=(σ1·σ2,µ1·µ2)is defined as (i) ∀(a, b) ∈V1×V2, (σ1·σ2)(a,b)=σ1(a)∧σ2(b). (ii) ∀(a, b) ∈E1and c6=w∈V2,(µ1·µ2)((a,c),(b,w)) =µ1(a,b). 3.1.1. Example G 1 =RF 1 and G 2 =RF 2 are the two fuzzy graphs of GRF1=(Vec1,Eec1) and GRF2=(Vec2,Eec2) , depicted in Figures 1and 2, respectively. Then, the residue product of the fuzzy network is denoted by RF1·RF2, as shown in Figure 3. Mathematics 2023, 11, x FOR PEER REVIEW 4 of 20 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 {}  = =M pp c q e gW 1 )(min min 3.1. Residue Product of Two Fuzzy Graphs Let 𝑅𝐹=(𝜎   ,𝜇  ) and 𝑅𝐹=(𝜎   ,𝜇  ) be two fuzzy graph networks of crisp graphs 𝐺  =𝑉   ,𝐸   and 𝐺  =𝑉   ,𝐸  , respectively. Then, its residue product 𝑅𝐹∙𝑅𝐹 =(𝜎 ∙𝜎 ,𝜇∙𝜇) is defined as (i) ∀ (a, b) ∈ V 1 × V 2 , (𝜎∙ 𝜎)(𝑎,𝑏)= 𝜎(𝑎) ∧ 𝜎(𝑏). (ii) ∀ (a, b) ∈ E 1 and c  w ∈ V 2 , (𝜇∙ 𝜇)((𝑎,𝑐),(𝑏,𝑤))= 𝜇 (𝑎,𝑏). 3.1.1. Example G 1 = RF 1 and G 2 = RF 2 are the two fuzzy graphs of 𝐺  =𝑉   ,𝐸   and 𝐺  = 𝑉  ,𝐸  , depicted in Figure 1 and Figure 2, respectively. Then, the residue product of the fuzzy network is denoted by RF 1 ·RF 2 , as shown in Figure 3. Figure 1. Fuzzy Graph G 1 . Figure 2. Fuzzy Graph G 2 . Using Definition 10, the edge membership values of the above constructed RF 1 ·RF 2 are calculated and shown in Figure 3. Figure 1. Fuzzy Graph G1. Mathematics 2023, 11, x FOR PEER REVIEW 4 of 20 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 {}  = =M pp c q e gW 1 )(min min 3.1. Residue Product of Two Fuzzy Graphs Let 𝑅𝐹=(𝜎   ,𝜇  ) and 𝑅𝐹=(𝜎   ,𝜇  ) be two fuzzy graph networks of crisp graphs 𝐺  =𝑉   ,𝐸   and 𝐺  =𝑉   ,𝐸  , respectively. Then, its residue product 𝑅𝐹∙𝑅𝐹 =(𝜎 ∙𝜎 ,𝜇∙𝜇) is defined as (i) ∀ (a, b) ∈ V 1 × V 2 , (𝜎∙ 𝜎)(𝑎,𝑏)= 𝜎(𝑎) ∧ 𝜎(𝑏). (ii) ∀ (a, b) ∈ E 1 and c  w ∈ V 2 , (𝜇∙ 𝜇)((𝑎,𝑐),(𝑏,𝑤))= 𝜇 (𝑎,𝑏). 3.1.1. Example G 1 = RF 1 and G 2 = RF 2 are the two fuzzy graphs of 𝐺  =𝑉   ,𝐸   and 𝐺  = 𝑉  ,𝐸  , depicted in Figure 1 and Figure 2, respectively. Then, the residue product of the fuzzy network is denoted by RF 1 ·RF 2 , as shown in Figure 3. Figure 1. Fuzzy Graph G 1 . Figure 2. Fuzzy Graph G 2 . Using Definition 10, the edge membership values of the above constructed RF 1 ·RF 2 are calculated and shown in Figure 3. Figure 2. Fuzzy Graph G2. Mathematics 2023, 11, x FOR PEER REVIEW 5 of 20 Figure 3. Edge membership value of RF1·RF2. 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 Wmin 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 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 cybz, 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). Figure 3. Edge membership value of RF1·RF2. Mathematics 2023,11, 4019 5 of 16 Using Definition 10, the edge membership values of the above constructed RF 1· RF 2 are calculated and shown in Figure 3. From Figure 3, the minimum number of basic colors used in RF 1· RF 2 is 4. The weight minimum number of basic colors used in constructed residue product RF 1· RF 2 = 0.16 + 0.2 + 0.16 + 0.16 = 0.68. Thus, the Wmin of RF1·RF2is 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 2shows the weight of the graph, and Table 1sorts 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 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 Mathematics 2023, 11, x FOR PEER REVIEW 6 of 20 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 Figure 4. Minimum Spanning Tree of RF1·RF2. 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 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 =(𝜎 ,𝜇) be the fuzzy graph of the crisp graph GDN = (VDN, EDN), and let DNrDNDN SSS ,...,, 21 be the minimal dominating set of GDN. Finally, let the corresponding dominating number be denoted by r DNDNDNDN γ γ γ γ ,..., 3 , 2 , 1. Among this, the lowest cardinality of the domination number is called lower domination number and is denoted by . LDN γ Let 21212121 .)4(.)3(.)2(.)1( ,, RFRFRFRFRFRFRFRF SandSSS be the sum of the minimal dominating set of RF1.RF2 (refer to Figure 3) },,,{ 2 . 1 )1( dycybyayS RFRF =; },,,{ 2 . 1 )2( dxcxbxaxS RFRF =; Figure 4. Minimum Spanning Tree of RF1·RF2. Mathematics 2023,11, 4019 6 of 16 Table 2. 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 RF 1·· RF 2 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 GDN =(σDN,µDN ) be the fuzzy graph of the crisp graph G DN = (V DN ,E DN ), and let SDN1 , SDN2 , . . . , SDNr be the minimal dominating set of G DN . Finally, let the corresponding dominating number be denoted by γDN1 , γDN2 , γDN3 , . . . , γDNr . Among this, the lowest cardinality of the domination number is called lower domination number and is denoted by γLDN. Let S(1)RF1.RF2 , S(2)RF1.RF2 , S(3)RF1.RF2and S(4)RF1.RF2 be the sum of the minimal dominating set of RF1.RF2(refer to Figure 3) S(1)RF1.RF2={ay,by,cy,dy};S(2)RF1.RF2={ax,bx,cx,dx}; S(3)RF1.RF2={az,bz,cz,dz} γ(1)RF1.RF2of S(1)RF1.RF2=0.3 +0.6 +0.6 +0.3 =1.8 γ(2)RF1.RF2of S(2)RF1.RF2=2.6. γ(2)RF1.RF2of S(3)RF1.RF2=2.1. γLDN−RF1.RF2of RF1RF2is 1.8. 3.2. Symmetric Difference of Two Fuzzy Graphs Let SDF 1 = (σec1,µec1) and SDF 2 = (σec2,µec2) be two fuzzy graphs of crisp graphs GSDF1=(Vec1,Eec1)andGSDF2=(Vec2,Eec2) , respectively. Then, the symmetric difference between SDF 1 and SDF 2 is denoted by SDF 1⊕ SDF 2 = ( σec1⊕σec2 , µec1⊕µec2 ) and is defined as follows 1. ∀(a,b)∈V1×V2, σSDF1⊕σSDF2(a,b)=σSDF1(a)∧σSDF2(b). 2. ∀a∈V1and (b,c)∈E2(µSDF1⊕µSDF2(a,b),(a,c)=σSDF1(a)∧µSDF2(b,c). 3. ∀a∈V2and (b,c)∈E1(µSDF1⊕µSDF2((b,a),(c,a)) =µSDF1(b,c)∧σSDF2(a). 4. ∀(a,b)/∈E1and (c,w)∈E2 , (µSDF1⊕µSDF2((a,c),(b,w)) =min{σSDF1(a) , σSDF1(b),µSDF2(c,w)}. 5. ∀(a,b)∈E1and (c,w)/∈E2 , (µSDF1⊕µSDF2((a,c),(b,w)) =min{µSDF1(a,b) , σSDF2(c),σSDF2(w)}. 3.2.1. Example Let G 1 =SDF 1 and G 2 =SDF 2 be the two fuzzy graphs of crisp graphs GSDF1=(Vec1,Eec1)and GSDF2=(Vec2,Eec2), depicted in Figures 1and 2, respectively. The symmetric difference of fuzzy network SDF1⊕SDF2is shown in Figure 5. Mathematics 2023,11, 4019 7 of 16 Mathematics 2023, 11, x FOR PEER REVIEW 8 of 20 Figure 5. Edge Membership value of SDF1 ⊕ SDF2. 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 Wmin 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 21 )4( 21 )3( , 21 )2( , 21 )1( SDFSDF Sand SDFSDF S SDFSDF S SDFSDF S⊕⊕⊕⊕ be some of the minimal dominating set of SDF1 ⊕ SDF2 (see Figure 5) },{ 21 )1( ayax SDFSDF S= ⊕; },{ 21 )2( ayaz SDFSDF S= ⊕; },{ 21 )3( dydx SDFSDF S= ⊕; }.,{ 21 )4( dzdy SDFSDF S= ⊕ 2 . 1 )1( SDFSDF ⊕ γ of = ⊕21 )1( SDFSDF S 2.2 2 . 1 )2( SDFSDF ⊕ γ of .6.1 21 )2( = ⊕SDFSDF S 2 . 1 )3( SDFSDF ⊕ γ of .2.2 21 )3( = ⊕SDFSDF S 2 . 1 )4( SDFSDF ⊕ γ of .9.1 21 )4( = ⊕SDFSDF S Figure 5. Edge Membership value of SDF1⊕SDF2. Using Definition 10, the edge membership values of the above constructed SDF1⊕SDF2are calculated, as shown in Figure 5. As shown in Figure 5, the minimum number of basic colors used in SDF 1⊕ SDF 2 is 7. The weight minimum number of basic colors used in the constructed symmetric difference network SDF 1⊕ SDF 2 = 0.5 + 0.5 + 0.5 + 0.5 + 0.5 + 0.75 + 0.5 = 3.75. Thus, the W min of SDF1⊕SDF2is 3.75. Using Kruskal’s algorithm, the weight of the minimal spanning tree of SDF 1⊕ SDF 2 was found to be 7. 3.2.2. Lower Domination Number of SDF1⊕SDF2 Let S(1)SDF1⊕SDF2 , S(2)SDF1⊕SDF2 , S(3)SDF1⊕SDF2and S(4)SDF1⊕SDF2 be some of the minimal dominating set of SDF1⊕SDF2(see Figure 5) S(1)SDF1⊕SDF2={ax,ay};S(2)SDF1⊕SDF2={az,ay}; S(3)SDF1⊕SDF2={dx,dy};S(4)SDF1⊕SDF2={dy,dz}. γ(1)SDF1.⊕SDF2of S(1)SDF1⊕SDF2=2.2 γ(2)SDF1.⊕SDF2of S(2)SDF1⊕SDF2=1.6. γ(3)SDF1.⊕SDF2of S(3)SDF1⊕SDF2=2.2. γ(4)SDF1.⊕SDF2of S(4)SDF1⊕SDF2=1.9. γLDN−SDF1⊕.SDF2of SDF1⊕SDF2is 1.6. Mathematics 2023,11, 4019 8 of 16 3.3. Max Product of Two Fuzzy Graphs Let MF 1 = (σm f 1,µm f 1and MF2=σm f 2,µm f 2 be two fuzzy networks of crisp graphs GMF1=Vm f 1,Em f 1and GMF2=Vm f 2,Em f 2 , respectively. The maximal product of fuzzy graphs MF 1 and MF 2 is represented by MF 1 * MF 2 = (σm f 1 * σm f 2 , µm f 1 *, µm f 2 ) and is defined as: (i) ∀(a,b)∈Vm f 1×Vm f 2,σm f 1*σm f 2(a,b)=σm f 1(a)∨σm f 2(b). (ii) ∀a∈Vm f 1and (b,c)∈Em f 2,(µm f 1*µm f 2((a,b),(a,c)) =σm f 1(a)∨µm f 2(b,c). (iii) ∀a∈Vm f 2and (b,c)∈Em f 1,µm f 1*µm f 2((b,a),(c,a)) =µm f 1(b,c)∨σm f 2(a). 3.3.1. Example G 1 =MF 1 and G 2 =MF 2 defines the two fuzzy graphs of crisp graphs GMF1=(Vec1,Eec1)and GMF2=(Vec2,Eec2) , depicted in Figures 1and 2, respectively. The max product of the fuzzy network MF1* MF2is shown in Figure 6. Mathematics 2023, 11, x FOR PEER REVIEW 9 of 20 2 . 1SDFSDFLDN ⊕− γ of SDF1 ⊕ SDF2 is 1.6. 3.3. Max Product of Two Fuzzy Graphs Let MF1 = (𝜎,𝜇) and 𝑀𝐹=(𝜎 ,𝜇) be two fuzzy networks of crisp graphs 𝐺=𝑉 ,𝐸 and 𝐺=𝑉 ,𝐸, respectively. The maximal product of fuzzy graphs MF1 and MF2 is represented by MF1 * MF2 = (𝜎 * 𝜎, 𝜇*,𝜇) and is defined as: (i) ∀ (a, b) ∈ 𝑉 × 𝑉, 𝜎∗ 𝜎(𝑎,𝑏)= 𝜎(𝑎) ∨ 𝜎(𝑏). (ii) ∀ a ∈𝑉 and (b, c) ∈𝐸, (𝜇∗ 𝜇)((𝑎,𝑏),(𝑎,𝑐))= 𝜎(𝑎) ∨ 𝜇(𝑏,𝑐). (iii) ∀ a ∈𝑉 and (b, c) ∈𝐸, 𝜇∗ 𝜇(𝑏,𝑎),(𝑐,𝑎)= 𝜇(𝑏,𝑐)∨ 𝜎(𝑎). 3.3.1. Example G1 = MF1 and G2 = MF2 defines the two fuzzy graphs of crisp graphs 𝐺= 𝑉,𝐸 and 𝐺=𝑉 ,𝐸, 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. Figure 6. Edge Membership value of MF1 * MF2. 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 Wmin 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. Figure 6. Edge Membership value of MF1*MF2. Using Definition 10, the edge membership values of the above constructed MF 1 *MF 2 were calculated, as shown in Figure 6. As shown in Figure 6, the minimum number of basic colors used in MF 1 * MF 2 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 Wmin of MF1* MF2is 3.03. Using Kruskal’s algorithm, the weight of the minimal spanning tree of MF 1 * MF 2 was found to be 9.29. 3.3.2. Lower Domination Number of MF1*MF2 Let S(1)MF1∗MF2 , S(2)MF1∗MF2 , S(3)MF1∗MF2and S(4)MF1∗MF2 be some of the minimal dominating set of MF1* MF2(see Figure 6) S(1)MF1∗MF2={ay,cx,cz,dy}S(2)MF1∗MF2={ay,bx,by,dy}; Mathematics 2023,11, 4019 9 of 16 S(3)MF1∗MF2={bx,cx,ay,dy};S(4)MF1∗MF2={ax,bz,cx,dz}. γ(1)MF1.∗MF2ofS(1)MF1∗MF2=6.5;γ(2)MF1.∗MF2of S(2)MF1∗MF2=6.7; γ(3)SMF1.∗MF2ofS(3)MF1∗MF2=7.2;γ(4)MF1.∗MF2of S(4)MF1∗MF2=5.7; γLDN−MF1.∗MF2of MF1∗MF2is 5.7. 3.4. Lexicographic Product of Two Fuzzy Graphs Let LF 1 = (M 1 , P 1 ) and LF 2 = (M 2 , P 2 ) be two fuzzy graphs of the crisp graphs GLF1=(Vec1,Eec1)and GLF2=(Vec2,Eec2)respectively. The lexicographic product of the two graphs is denoted as LF 1· LF 2 in fuzzy graph pair (M,P), such that (i) M(a1,b2)=min(M1(a1),M2(b2)),∀(a1,b2)∈M1×M2. (ii) P((x,b2)(x,d2)=min(M1(x),M2(b2d2),∀x∈M1,b2d2∈P2. (iii) P((a1,b2)(c1,d2)) =min(P1(a1c1),P2(b2d2)),∀a1b1∈P1and b2d2∈P2. 3.4.1. Example Let G 1 =LF 1 and G 2 =LF 2 be two fuzzy graphs of crisp graphs GLF1=(Vec1,Eec1) and GLF2=(Vec2,Eec2) , depicted in Figures 1and 2, respectively. The lexicographic product of the fuzzy network is denoted by LF1·LF2, as shown in Figure 7. Mathematics 2023, 11, x FOR PEER REVIEW 11 of 20 Figure 7. Edge Membership value of LF1·LF2. 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 Wmin 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 2 * 1LFLF S be the minimal dominating set (only one dominating set) of LF1·LF2 (see Figure 7) }.,,,{ 2 * 1 dycybyay LFLF S=2 *. 1LFLFLDN − γ of .3 2 * 1is LFLF S Figure 7. Edge Membership value of LF1·LF2. Using Definition 10, the edge membership values of the above constructed LF 1· LF 2 were calculated, as shown in Figure 7. As shown in Figure 7, the minimum number of basic colors used in LF 1· LF 2 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 Wmin of LF1·LF2is 3. Mathematics 2023,11, 4019 16 of 16 13. Meenakshi, A.; Mythreyi, O.; Bramila, M.; Kannan, A.; Senbagamalar, J. Application of neutrosophic optimal network using operations. J. Intell. Fuzzy Syst. 2023,45, 421–433. [CrossRef] 14. Yang, H.L.; Guo, Z.L.; She, Y.; Liao, X. On single valued neutrosophic relations. J. Intell. Fuzzy Syst. 2016 ,30, 1045–1056. [CrossRef] 15. Atanassov, K.T. On Intuitionistic Fuzzy Sets Theory; Springer: Berlin, Germany, 2012. 16. Broumi, S.; Smarandache, F. New Distance and Similarity Measures of Interval Neutrosophic Sets. In Proceedings of the 17th International Conference on Information Fusion, Salamanca, Spain, 7–10 July 2014. 17. Islam, S.R.; Pal, M. Hyper-Wiener index for fuzzy graph and its application in share market. J. Intell. Fuzzy Syst. 2021 , 41, 2073–2083. [CrossRef] 18. Arif, N.E. Domination (set and Number) in Neutrosophic Soft over Graphs. Wasit J. Pure Sci. 2022,1, 26–43. 19. Rashmanlou, H.; Samanta, S.; Pal, M.; Borzooei, R.A. Intuitionistic fuzzy graphs with categorical properties. Fuzzy Inf. Eng. 2015 , 7, 317–334. [CrossRef] 20. Shannon, A.; Atanassov, K. On a generalization of intuitionistic fuzzy graphs. NIFS 2006,12, 24–29. 21. Bondy, J.A.; Murty, U.S.R. Graph Theory with Applications; Elsevier: New York, NY, USA, 1982. 22. Ore, O. Theory of Graphs; American Mathematical Society Colloquium Publications: Providence, RI, USA, 1962. 23. Haynes, T.W.; Hedetniemi, S.; Slater, P. Fundamentals of in Domination in Graphs; Springer International Publishing: New York, NY, USA, 1998. 24. Swaminathan, V.; Dharmalingam, K.M. Degree equitable domination on graphs. Kragujev. J. Math. 2011,35, 191–197. 25. Meenakshi, A. Equitable domination of complement of inflated graph. AIP Conf. Proc. 2019,2112, 1–6. 26. Meenakshi, A.; Babujee, J.B. Paired Equitable domination in graphs. Int. J. Pure Appl. Math. 2016,109, 75–81. 27. Meenakshi, A. Paired Equitable domination in inflated graph. Int. J. Innov. Technol. Explor. Eng. 2019,8, 9–16. 28. Meenakshi, A.; Mythreyi, O. Applications of Neutrosophic social network using max product networks. J. Intell. Fuzzy Syst. 2023 , 45, 407–420. [CrossRef] 29. Mahapatra, R.; Samanta, S.; Pal, M. Applications of edge colouring of fuzzy graphs. Informatica 2020,31, 313–330. [CrossRef] 30. Bershtein, L.; Bozhenyuk, A.; Rozenberg, I. Optimum allocation of centers in transportation networks using fuzzy graph bases. In Proceedings of the 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-13), Milano, Italy, 11–13 September 2013; pp. 270–275. 31. Mahapatra, T.; Ghorai, G.; Pal, M. Fuzzy fractional coloring of fuzzy graph with its application. J. Ambient. Intell. Humaniz. Comput. 2020,11, 5771–5784. [CrossRef] 32. Mahapatra, R.; Samanta, S.; Pal, M.; Lee, J.G.; Khan, S.K.; Naseem, U.; Bhadoria, R.S. Colouring of COVID-19 affected region based on fuzzy directed graphs. Comput. Mater. Contin. 2021,68, 1219–1233. [CrossRef] 33. Lewis, R. A Guide to Graph Colouring; Springer: London, UK, 2016. 34. Malaguti, E.; Toth, P. A survey on vertex coloring problems. Int. Trans. Oper. Res. 2010,17, 1–34. [CrossRef] 35. Galinier, P.; Hamiez, J.P.; Hao, J.K.; Porumbel, D. Recent advances in graph vertex coloring. In Handbook of Optimization: From Classical to Modern Approach; Springer: Berlin, Germany, 2013; pp. 505–528. 36. Lih, K.W. The equitable coloring of graphs. Handb. Comb. Optim. 1998,1, 2015–2038. 37. Leighton, F.T. A graph coloring algorithm for large scheduling problems. J. Res. Natl. Bur. Stand. 1979,84, 489. [CrossRef] 38. Furmanczyk, H.; Kubale, M. Equitable coloring of graphs. Contemp. Math. 2004,352, 35–54. 39. Samanta, S.; Pramanik, T.; Pal, M. Fuzzy colouring of fuzzy graphs. Afr. Mat. 2016,27, 37–50. [CrossRef] 40. Chang, G.J.; Chen, S.H.; Hsu, C.Y.; Hung, C.M.; Lai, H.L. Strong edge-coloring for jellyfish graphs. Discret. Math. 2015 , 338, 2348–2355. [CrossRef] 41. Kishore, A.; Sunitha, M.S. On Injective Coloring of Graphs and Chromaticity of Fuzzy Graphs; LAP Lambert Academic Publishing: Saarbrücken, Germany, 2016. 42. Rosyida, I.; Indrati, C.R.; Sugeng, K.A. A new approach for determining fuzzy chromatic number of fuzzy graph. J. Intell. Fuzzy Syst. 2015,28, 2331–2341. [CrossRef] 43. Romanov, A.Y.; Myachin, N.M.; Lezhnev, E.V.; Ivannikov, A.D.; El-Mesady, A. Ring-Split: Deadlock-Free Routing Algorithm for Circulant Networks-on-Chip. Micromachines 2023,14, 141. [CrossRef] [PubMed] 44. Li, Z.; Shen, R.; Yi, M.; Song, Y.; Wang, X.; Du, G.; Huang, Z. Hotspots Reduction for GALS NoC Using a Low-Latency Multistage Packet Reordering Approach. Micromachines 2023,14, 444. [CrossRef] Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.