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First Entire Zagreb Index of Fuzzy Graph and Its Application

First Entire Zagreb Index of Fuzzy Graph and Its Application axioms Article Umapada Jana and Ganesh Ghorai * Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, India; umapada867032@gmail.com * Correspondence: math.ganesh@mail.vidyasagar.ac.in Abstract: The first entire Zagreb index (FEZI) is a graph parameter that has proven to be essential in various real-life scenarios, such as networking businesses and traffic management on roads. In this research paper, the FEZI was explored for a variety of fuzzy graphs, including star, firefly graph, cycle, path, fuzzy subgraph, vertex elimination, and edge elimination. This study presented several results, including determining the relationship between two isomorphic fuzzy graphs and between a path and cycle (connecting both end vertices of the path). This research also deals with the analysis of a-cut fuzzy graphs and establishes bounds for some fuzzy graphs. To apply these findings to modern life problems, the research team utilized the results to identify areas that require more development in internet systems. These results have practical implications for enhancing the efficiency and effectiveness of internet systems. The conclusion drawn from this research can be used to inform future research and aid in the development of more efficient and effective systems in various fields. Keywords: first entire Zagreb index; second entire Zagreb index; first entire Zagreb index of a vertex; isomorphic graph; internet system MSC: 05C09; 05C72 1. Introduction 1.1. Research Background Citation: Jana, U.; Ghorai, G. First Entire Zagreb Index of Fuzzy Graph In the modern day, fuzzy graph (FG) theory is one of the most applicable to regular and Its Application. Axioms 2023, 12, life. So there are many researchers who are implementing FG theory, especially topological 415. https://doi.org/10.3390/ indices of fuzzy graphs. Rosenfeld [1], inspired by Zadeh’s [2] classical set (fuzzy set) in axioms12050415 1975, introduced the fuzziness for a graph, then, it is called a fuzzy graph. Additionally, this time he introduced several connective parameters of an FG and some applications Academic Editors: Xiaohong Zhang, of these parameters by Yeh et al. [3]. In [4,5], Sunitha et al. studied fuzzy block, fuzzy Eunsuk Yang and Hsien-Chung Wu bridge, FSG, CFG, PFSG, fuzzy tree, fuzzy forest, fuzzy cut vertex, etc. The degree of a Received: 24 March 2023 vertex d(v)/deg(v) in an FG is also discussed In [6]. The degree of an edge (d (e)) is also Revised: 18 April 2023 discussed in [7]. Further information on the FG hypothesis is provided in [8–11]. Bipolar Accepted: 19 April 2023 and m-polar fuzzy graphs are discussed in [12,13]. The first Zagreb index is discussed Published: 24 April 2023 in [14] and was inspired by the paper [15]. In a molecular graph of a chemical compound, we can calculate molecular descriptors by finding topological indices of this graph. A graph’s topology is described by these topological indices, which are numerical numbers. The Zagreb index, established in 1972 by Gutman and Trinajstic [16], is a degree-based Copyright: © 2023 by the authors. topological index. The pi-electron energy of a conjugate system is determined using Licensee MDPI, Basel, Switzerland. topological indices. In a crisp graph, many indices are defined but several issues in real life This article is an open access article cannot be handled by these indices. So we can generalize these indices in fuzzy graphs. In distributed under the terms and this piece, we have introduced the FEZI in fuzzy graphs which is a major generalization conditions of the Creative Commons Attribution (CC BY) license (https:// of FEZI for crisp graphs. These indices are investigated from both a theoretical and an creativecommons.org/licenses/by/ application point of view. 4.0/). Axioms 2023, 12, 415. https://doi.org/10.3390/axioms12050415 https://www.mdpi.com/journal/axioms Axioms 2023, 12, 415 2 of 16 1.2. Research Question These questions are covered in this paper: 1. What is the FEZI upper bound for fuzzy graphs? 2. What are the precise values or boundaries of FEZI for the firefly graph, star, path, cycle, etc.? 3. What is the relation between the value of FEZI of a graph and its sub graph? 4. What is the relation between the value of FEZI for two isomorphic graph? 5. What is the relation for several graphs between the first Zagreb index and FEZI ? 6. What are the applications of this index? 1.3. Objective of the Work Various types of topological indices of a graph can be used for a variety of purposes and yield a wide range of outcomes for crisp graphs. However, in numerous applications, a crisp graph is not enough to solve it. We need to define a fuzzy graph to answer this question. In this paper, FEZI is defined and some results relating to sub graphs, paths, stars, firefly graphs, cycles, isomorphic graphs, etc. are given. At the end of this paper, we applied the first entire Zagreb index in internet network systems. 1.4. Structure of the Study The structure of the article is as follows: in Section 2, some definitions are provided which are necessary for this study. In Section 3, we studied the FEZI of a fuzzy graph and provided some results on sub graphs, paths, stars, firefly graphs, cycles, etc. Also some relation between fuzzy graphs are provided. In Section 4, an application of the first entire Zagreb index in development in internet networking system is discussed. 2. Preliminaries Here, we provide some fundamental definitions and theorems which are crucial to developing the later sections. Let U be a universal set. An FS A on U is a mapping s : U ! [0, 1]. Here, s is the membership function of the FS A. A FS generally indicated by A = (u, s). Assume that F(6= 0) is a known finite set. Then the fuzzy graph (FG) is a triplet, G = (U, w, $), where U(6= 0)  F with w : U ! [0, 1] and $ : U  U ! [0, 1] satisfying $(x , x )  w(x )^ w(x ). The set U is the set of vertices and # := ((x , x ) : $(x , x ) > 0) 1 2 1 2 1 2 1 2 is the set of edges of the FG. w(x ) represents the membership value (MV) of the vertex x 1 1 and $ represents the MV of the edge (x , x ) (or x x ). 2 2 1 1 0 0 0 Let G = (U, w, $) be an FG. Then, H = (U , w , $ ) is called the PFSG of the FG G if 0 0 0 0 0 U  U, w (x )  w(x ), $ (x , x )  $(x , x ) for all x , x 2 U . If w (x ) = w(x ) and 1 1 1 2 1 2 1 2 1 1 0 0 $ (x , x )  $(x , x ) for all x , x 2 U then H is called FSG of the graph of the FG G. 1 2 1 2 1 2 For x 2 V, we denote G as an FSG of the FG G = (V, w, $) with w(x ) = 0 and for 1 1 x x 2 #, G represents the FSG of the FG G with $(x x ) = 0. 1 2 x x 1 2 Let x , x , . . . , x all be vertices of an FG G. Then, the collection of vertices P(x , x , . . . , x ) n n 0 1 0 1 is a path in G if $(x , x ) 6= 0 (for all i). The path’s length is n in this case. If $(x , x ) > 0 for 0 n i i+1 the path P(x , x , . . . , x ) then it is called a cycle. 0 1 n Let x , x , . . . , x be distinct vertices of a fuzzy graph G. Then, G = (x , x , . . . , x ) is 0 n 0 n 1 1 called a star if $(x , x ) 6= 0 for i = 1, 2, . . . , n and for all vertex except x there is no edge 0 i 0 between every two vertices, where x is the center of the star. In a graph G if $(x , x )  0 for all vertices x and x , then it is called a complete i j i j graph. In a fuzzy graph G = (V, w, $) if for every two vertex x and y satisfy the condition $(x, y) = w(x)^ w(y) then the graph is called a CFG. Two FG G = (V , w , $ ) and G = (V , w , $ ) are called isomorphic to each other 1 1 1 1 2 2 2 2 if there exists a bijective mapping g : V ! V for any x, y 2 V , w (x) = w (g(x)) and 1 2 1 1 2 $ (g(x), g(y)) = $ (x, y). 2 1 Axioms 2023, 12, 415 3 of 16 Let G = (V, E) be a crisp graph. Then, the first and second ZI are defined by # 2 # M = (deg(x)) and M = deg(x)deg(y) [17]. å å 1 2 x is either adjacent x2V(G)U E(G) or incident to y The total degree of an FG with respect to vertices is indicated by T (G) and is defined by T (G) = d (v) [7]. Additionally, the total degree of an FG with respect to edge is v G v2V(G) indicated by T (G) and is defined by T (G) = d (e). e e G e2E(G) Example 1. Let G be an FG with vertex set V = f p, q, r, sg such that w( p) = 0.8, w(q) = 0.7, w(r) = 0.6, w(s) = 0.5, $( p, q) = 0.4, $( p, r) = 0.5, $(q, r) = 0.5, $(q, s) = 0.3, $(r, s) = 0.4, as shown in Figure 1. Then, d( p) = 0.9, d(q) = 1.2, d(r) = 1.4, d(s) = 0.7, d( p, q) = 1.3, d( p, r) = 1.3, d(q, r) = 1.6, d(q, s) = 1.3, d(r, s) = 1.3 and T (G) = 0.9 + 1.2 + 1.4 + 0.7 = 4.2 and T (G) = 1.3 + 1.3 + 1.3 + 1.6 + 1.3 + 1.3 = 6.8. Figure 1. A fuzzy graph. 3. First Entire Zagreb Index of Fuzzy Graphs First entire Zagreb index (FEZI) have an important role for finding strength of vertices. The strength of vertices is important in fuzzy graph theory. Thus, in this section the FEZI of a fuzzy graph is initiated. Various properties and an application of FEZI for fuzzy graphs is given. Definition 1. Let G be an FG. Then, the FEZI of G is indicated by M and is defined by : z 2 2 M := (w(x)d(x)) + ($(e)d(e)) . å å x2V(G) e2E(G) Example 2. Let G be an FG with vertex set V = f p, q, r, sg such that w( p) = 0.8, w(q) = 0.7, w(r) = 0.6, w(s) = 0.5, $( p, q) = 0.4, $( p, r) = 0.5, $(q, r) = 0.5, $(q, s) = 0.3 and $(r, s) = 0.4, shown in Figure 1. Then d( p) = 0.9, d(q) = 1.2, d(r) = 1.4, d(s) = 0.7, d( p, q) = 1.3, d( p, r) = 1.3, d(q, r) = 1.6, d(q, s) = 1.3, d(r, s) = 1.3. z 2 2 Now, M = å (w(x)d(x)) + å ($(e)d(e)) xeV(G) eeE(G) 2 2 2 2 2 2 = (0.8 0.9) + (0.7 1.2) + (0.6 1.4) + (0.5 0.7) + (0.4 1.3) + (0.5 1.3) + 2 2 2 (0.5 1.6) + (0.3 1.3) + (0.4 1.3) = 3.7135. Definition 2. Let (G, w, $) be an FG. Then, the second entire Zagreb index of G is indicated by e z M and is defined by M := w(u)d(u)w(v)d(v) + w(v)d(v)$(e)d(e), å å 2 2 u,vare adjacent to v is incident to e each other where w is the MV of a vertex and $ is the MV of an edge. Example 3. Let G be an FG with vertex set V = f p, q, r, sg such that, w( p) = 0.6, w(q) = 0.5, w(r) = 0.4, w(s) = 0.8, $( p, r) = 0.4, $(q, r) = 0.4, $(r, s) = 0.3 shown in Figure 2. Then Axioms 2023, 12, 415 4 of 16 d( p) = 0.4, d(q) = 0.4, d(r) = 1.1, d(s) = 0.3, d( p, r) = 0.7, d(q, r) = 0.7, d(r, s) = 0.8 Now, M = w(u)d(u)w(v)d(v) + w(v)d(v)w(e)d(e) å å u,vare adjacent v is incident to e to each other = (0.4 0.5 0.4 1.1) + (0.4 0.6 0.4 1.1) + (0.3 0.8 0.4 1.1) + (0.4 0.7 0.4 0.7) + (0.4 0.3 0.7 0.8) + (0.4 0.3 0.7 0.8) + (0.4 0.6 0.4 0.7) + (0.4 0.5 0.4 0.7) + (0.8 0.3 0.3 0.8) + (0.4 0.7 0.4 1.1) + (0.4 0.7 0.4 1.1) + (0.4 0.3 0.8 1.1) = 0.088 + 0.1056 + 0.1056 + 0.0784 + 0.0672 + 0.0672 + 0.0672 + 0.056 + 0.0576 + 0.1232 + 0.1232 + 0.1056 = 1.0448. Figure 2. A fuzzy graph G with M = 1.0448. z 2 2 2 2 Theorem 1. Let the FG G have n vertices and m edges, then M  n T (G) + m T (G), where v e 2 2 T (G) and T (G) represent the total degree with respect to vertex and total degree with respect to v e edge. Proof. Now, the FEZI of G is given by 2 2 M = w(x)d(x) + $(e)d(e) . å å xeV(G) eeE(G) 2 2 2 Since ( pq)  p q , we have å å å 2 2 2 2 w(x)d(x) + $(e)d(e)  (w(x)) (d(x)) + å å å å xeV(G) eeE(G) xeV(G) xeV(G) 2 2 ($(e)) (d(e)) å å eeE(G) eeE(G) 2 2 2 2 w(x) d(x) + $(e) d(e) å å å å xeV(G) xeV(G) eeE(G) eeE(G) 2 2 2 2 w(x) T (G) + $(e) T (G). å å v e xeV(G) eeE(G) z 2 2 2 2 Since 0  w(x)  1 and 0  $(e)  1, therefore M  n T (G) + m T (G). v e Definition 3. Let G = (V, w, $) be an FG. Then FEZI at a vertex of G is indicated by M (v) and z z z is defined by M (v) := M (G) M (G ) where G = V(G) v, w, $ . v v 1 1 1 Example 4. Let G be an FG with vertex set V = f p, q, r, sg such that w( p) = 0.8, w(q) = 0.7, w(r) = 0.5, w(s) = 0.6, $( p, q) = 0.4, $( p, r) = 0.5, $(q, r) = 0.5, $(q, s) = 0.3 and $(r, s) = 0.4 shown in Figure 3. Then, d( p) = 0.9, d(q) = 1.2, d(r) = 1.4, d(s) = 0.7, d( p, q) = 1.3, d( p, r) = 1.3, d(q, r) = 1.6, d(q, s) = 1.3, d(r, s) = 1.3. 2 2 Now, M = å w(x)d(x) + å $(e)d(e) xeV(G) eeE(G) 2 2 2 2 2 2 = (0.8 0.9) + (0.7 1.2) + (0.5 1.4) + (0.6 0.7) + (0.4 1.3) + (0.5 1.3) + 2 2 2 (0.5 1.6) + (0.3 1.3) + (0.4 1.3) So M (G) = 3.6458. 1 Axioms 2023, 12, 415 5 of 16 Now the vertex p is removed from the graph G, then the graph of G is shown in Figure 4. From the figure, d(q) = 0.6, d(r) = 0.7, d(s) = 0.7, d(q, r) = 0.7, d(q, s) = 0.7, d(r, s) = 0.6. Figure 3. A fuzzy graph G with M (G) = 3.6458. Figure 4. The FG G obtained by deleting the vertex p from the above graph. 2 2 Now, M (G ) = w(x)d(x) + $(e)d(e) p å å xeV(G ) eeE(G ) p p 2 2 2 2 2 2 = (0.7 0.6) + (0.5 0.7) + (0.6 0.7) + (0.3 0.7) + (0.6 0.4) + (0.3 0.7) . There- fore, M (G ) = 0.6211. Then, the FEZI at p is M ( p) = 3.6458 0.6211 = 3.0247. Theorem 2. Suppose that H is an FG that is created by removing an edge from G. Then, M ( H) M (G). 0 0 0 Proof. Since G = (V, w, $) is an FG and H = (V , w , $ ) is a graph that is created by removing an edge from G so the MV of a vertex is the same in both graphs and the MV of edges are the same if it contains both E and E . Then, the relation between the membership values of G and H is w(x)  w (x) for all vertices x and $(e)  $ (e) for all edges. 0 0 This shows that d(x)  d (x) for all vertices x. Additionally, d(e)  d (e) for all edge e, where d and d represent the degree of G and H. 2 2 Now, M (G) = w(x)d(x) + $(e)d(e) å å xeV(G) eeE(G) 2 2 0 0 0 0 w (x)d (x) + $ (e)d (e) å å xeV(G) eeE(G) 2 2 0 0 0 0 = w (x)d (x) + $ (e)d (e) å å xeV( H) eeE( H) = M ( H). z z Hence, M (G)  M ( H). 1 1 Theorem 3. If H is an FG that is obtained from G by deleting a vertex from G. Then, M ( H) M (G). 1 Axioms 2023, 12, 415 6 of 16 0 0 0 Proof. Since G = (V, w, $) is an FG and H = (V , w , $ ) is a graph that is created by removing a vertex from G. 0 0 0 0 So, w(x) = w (x) if x 2 V \ V otherwise w(x) > w (x). Additionally, $(e) = $ (e) if 0 0 e 2 E\ E otherwise $(e) > $ (e). Then the relation between the membership values of 0 0 0 0 0 G = (V, w, $) and H = (V , w , $ ) is w(x)  w (x) for all vertices x and $(e)  $ (e) for 0 0 all edges e. This shows that d(x)  d (x) for all vertices x. Additionally, d(e)  d (e) for all edges e. Here, d and d represent the degrees of G and H. 2 2 Now, M (G) = å w(x)d(x) + å $(e)d(e) xeV(G) eeE(G) 2 2 0 0 0 0 w (x)d (x) + $ (e)d (e) å å xeV(G) eeE(G) 2 2 0 0 0 0 = å w (x)d (x) + å $ (e)d (e) xeV( H) eeE( H) z z z = M ( H). So, M (G)  M ( H). 1 1 1 Example 5. Let G be an FG with vertex set V = f p, q, r, sg such that w( p) = 0.8, w(q) = 0.7, w(r) = 0.5, w(s) = 0.6, $( p, q) = 0.4, $( p, r) = 0.5, $(q, r) = 0.5, $(q, s) = 0.3, $(r, s) = 0.4 shown in Figure 3. Then d( p) = 0.9, d(q) = 1.2, d(r) = 1.4, d(s) = 0.7, d( p, q) = 1.3, d( p, r) = 1.3, d(q, r) = 1.6, d(q, s) = 1.3 and d(r, s) = 1.3. 2 2 Now, M (G) = w(x)d(x) + $(e)d(e) . å å xeV(G) eeE(G) 2 2 2 2 2 2 = (0.8 0.9) + (0.7 1.2) + (0.5 1.4) + (0.6 0.7) + (0.4 1.3) + (0.5 1.3) + 2 2 2 (0.5 1.6) + (0.3 1.3) + (0.4 1.3) = 3.6458. The graph of H where the vertex p is deleted from G shown in Figure 4. From the figure, d(q) = 0.6, d(r) = 0.7, d(s) = 0.7, d(q, r) = 0.7, d(q, s) = 0.7, d(r, s) = 0.6. z 2 2 Now, M ( H) = (w(x)d(x)) + ($(e)d(e)) å å xeV( H) eeE( H) 2 2 2 2 2 2 = (0.7 0.6) + (0.5 0.7) + (0.6 0.7) + (0.3 0.7) + (0.6 0.4) + (0.3 0.7) = 0.6211. The graph of K where the edge pr is deleted from G is shown in Figure 5. Figure 5. The FG obtained by deleting an edge from the graph in Figure 3. From this figure, d( p) = 0.4, d(q) = 1, d(r) = 0.7, d(s) = 0.7, d( p, q) = 0.6, d(q, r) = 1.1, d(q, s) = 1.1, d(r, s) = 1 z 2 2 Now, M (K) = (w(x)d(x)) + ($(e)d(e)) å å xeV(K) eeE(K) 2 2 2 2 2 2 = (0.8 0.4) + (0.7 1) + (0.5 0.7) + (0.6 0.7) + (0.6 0.4) + (0.3 1.1) + (0.3 2 2 1.1) + (0.4 1) = 1.3267. This shows that z z 1. M (G)  M ( H) 1 1 z z 2. M (G)  M (K). 1 1 z z Theorem 4. Let G be an FG and H be an FSG of G, then M (G)  M ( H). 1 1 Axioms 2023, 12, 415 7 of 16 0 0 0 Proof. Since H = (V , w , $ ) is an FSG of G = (V, w, $), therefore 0 0 w(x)  w (x) for all vertices x and $(e)  $ (e) for all edges e. This sows that d(x)  d (x) for all vertices x. 0 0 Additionally, d(e)  d (e) for all edges e, where d and d represent the degrees of G 2 2 and H. Now, M (G) = w(x)d(x) + $(e)d(e) å å xeV(G) eeE(G) 2 2 0 0 0 0 w (x)d (x) + $ (e)d (e) å å xeV(G) eeE(G) 2 2 0 0 0 0 = w (x)d (x) + $ (e)d (e) å å xeV( H) eeE( H) = M ( H) z z Hence, M (G)  M ( H). 1 1 Theorem 5. Let G = (V, w, $) be an FG and F = (V, w, $ ) be the corresponding MST of G then z z M (G)  M (F). 1 1 Proof. Since F = (V, w, $ ) is an MST of G = (V, w, $) therefore F is an FSG of G. Then we z z can say from the above Theorem, M (G)  M (F). 1 1 a 0 0 0 Theorem 6. Let G be an FG and let G = (V , w , $ ) be an a cut FG of G = (V, w, $). Then, z z a a 0 0 M (G)  M (G ) where the FG G is defined as V = veV : w(v)  a and w (v) = 1 1 , 0 w(v),$ (u, v) = $(u, v) for all u, veV . a z z a Proof. Since G is an FSG of the FG G, then by the above Theorem, M (G)  M (G ). 1 1 z p z p Theorem 7. Let G be an FG and let 0  p  p  1 Then M (G )  M (G ). 1 2 1 1 p p 2 1 Proof. Since 0  p  p  1, therefore G is a PFSG of G . Then, by the above Theorem, 1 2 z p z p M (G )  M (G ). 1 1 Corollary 1. Let G be an FG and let 0  p  p  p  . . .  p  1. 1 2 3 n z p z p z p z p n1 2 1 Then, M (G )  M (G )  . . .  M (G )  M (G ). 1 1 1 1 Theorem 8. Let G = P(v , v , v , . . . , v ) be a path. Then, 0 1 2 n i=n1 2 2 2 z 2 1. M (G) = ZF (G) + $(e )$(e ) + $(e )$(e ) + $ (e ) $(e ) + $(e ) . 1 1 2 n n1 i i1 i+1 i=2 2. M (G)  8(n 1). Proof. Given that G = P(v , v , v , . . . , v ) is a path, there are (n + 1) vertices and n edges. 0 2 n 1. Now, 2 2 M (G) = w(x)d(x) + $(e)d(e) (1) å å xeV(G) eeE(G) Here, the degree of each vertex v , except v and v , is $(e ) + $(e ) and the degree i 0 n i i+1 of v is $(e ) and degree of v is $(e ). n n 0 1 Additionally, the degree of each edge e , except e and e , is $(e ) + $(e ) and i 1 n i1 i+1 the degree of e is e , the degree of e is e . 1 2 n1 Using this result, we have from (1), i=n1 2 2 z 2 2 2 M (G) = (w $ ) + (w $ ) + å w (v ) $(e ) + $(e ) + $(e )$(e ) + 0 n n 2 1 i i i+1 1 i=1 i=n1 2 2 $(e )$(e ) + $ (e ) $(e ) + $(e ) n n1 i i1 i+1 i=2 i=n1 2 2 2 = ZF (G) + $(e )$(e ) + $(e )$(e ) + $ (e ) $(e ) + $(e ) (2) 1 1 2 n n1 å i i1 i+1 i=2 Axioms 2023, 12, 415 8 of 16 2. From Equation (2), we get i=n1 2 2 2 z 2 M (G) = ZF (G) + $(e )$(e ) + $(e )$(e ) + $ (e ) $(e ) + $(e ) . n å 1 1 2 n1 i i1 i+1 i=2 As 0  $(e )  1 and 0  w(v )  1, i i M (G)  2(2n 1) + 1 + 1 + 4(n 2) (where ZF (G)  2(2n 1)) = 4n 2 + 2 + 4n 8 = 8n 8. So, M (G)  8(n 1). Example 6. Let A = P( p, q, r, s, e) be an FG with vertex set V = p, q, r, s, e such that w( p) = 0.8, w(q) = 0.7, w(r) = 0.9, w(s) = 0.5, w(e) = 0.6, $( p, q) = 0.6, $(q, r) = 0.6, $(r, s) = 0.4, $(s, e) = 0.5 shown in Figure 6. Then, d( p) = 0.6, d(q) = 1.2, d(r) = 1.0, d(s) = 0.9, d(e) = 0.5, d( p, q) = 0.6, d(q, r) = 1.0, d(r, s) = 1.1 and d(s, e) = 0.4. 2 2 Now, M ( A) = w(x)d(x) + $(e)d(e) å å xeV( A) eeE( A) 2 2 2 2 2 2 = (0.8 0.6) + (0.7 1.2) + (0.9 1.0) + (0.5 0.9) + (0.6 0.5) + (0.6 0.6) + 2 2 2 (0.6 1.0) + (0.4 1.1) + (0.5 0.4) = 0.2304 + 0.7056 + 0.81 + 0.2025 + 0.09 + 0.1296 + 0.36 + 0.1936 + 0.04 = 2.7617. Figure 6. A fuzzy path G with M ( A) = 2.7617. Theorem 9. Let G = P(v , v , v , . . . , v ) be a path. If we take v = v , it becomes a cycle n n 0 1 2 0 z z H = C(v , v , v , . . . , v ). Then M (G)  M ( H). 0 1 2 n 1 1 Proof. Let d and d denote the degree of a vertex/edge in G and H respectively. Here, d(v) = d (v) for all vertices except v and v . Now, d(v ) = e and d(v ) = e and 0 n 0 1 n n 0 0 d (v ) = e + e . Additionally, d(e) = d (e) for all edges except e and e . Now, d(e ) = e n n 0 1 1 1 2 0 0 and d(e ) = e and d (e ) = e + e , d (e ) = e + e . n n1 1 2 n n 1 n1 2 2 Now, M (G) = w(v)d(v) + $(e)d(e) å å veV(G) eeE(G) 2 2 2 2 2 = w(v )d(v ) + w(v )d(v ) + å w(v)d(v) + $(e )d(e ) + $(e )d(e ) + 0 0 n n 1 1 n n veV(G)v ,v 0 n $(e)d(e) eeE(G)e ,e 1 n 2 2 2 2 0 0 0 0 0 0 = (w (v )e ) + w (v )e + w (v)d (v) + $ (e )e + $ (e )e + 0 1 n n 1 2 n n1 veV( H)v 0 0 å $ (e)d (e) eeE( H)e ,e 1 n 2 2 2 2 2 0 0 0 0 0 0 = w (v )e + w (v )e + w (v)d (v) + $ (e )e + $ (e )e + n å n 0 1 0 1 2 n1 veV( H)v 0 0 $ (e)d (e) eeE( H)e ,e 1 n 2 2 2 0 2 2 0 0 0 0 = w (v ) (e ) + (e ) + å w (v)d (v) + $ (e )e + $ (e )e + 0 n 2 n 1 1 n1 veV( H)v 0 0 $ (e)d (e) eeE( H)e ,e 1 n Axioms 2023, 12, 415 9 of 16 2 2 2 2 0 0 0 0 0 w (v ) (e ) + (e ) + w (v)d (v) + $ (e )(e + e ) + $ (e )(e + e ) + 0 1 n 1 2 n n n1 1 veV( H)v 0 0 $ (e)d (e) eeE( H)e ,e 1 n = M ( H). z z So, M (G)  M ( H). 1 1 Example 7. Let G be a path with vertex set V = p, q, r, s, e such that w( p) = 0.8, w(q) = 0.7, w(r) = 0.9, w(s) = 0.5, w(e) = 0.8, $( p, q) = 0.6, $(q, r) = 0.6, $(r, s) = 0.4 and $(s, e) = 0.5 shown in Figure 7. Figure 7. A fuzzy path G with M (G) = 2.8317. Then, d( p) = 0.6, d(q) = 1.2, d(r) = 1.0, d(s) = 0.9, d(e) = 0.5, d( p, q) = 0.6, d(q, r) = 1.0, d(r, s) = 1.1, d(s, e) = 0.4 2 2 Now, M (G) = w(x)d(x) + $(e)d(e) å å xeV(G) eeE(G) 2 2 2 2 2 2 = (0.8 0.6) + (0.7 1.2) + (0.9 1.0) + (0.5 0.9) + (0.8 0.5) + (0.6 0.6) + 2 2 2 (0.6 1.0) + (0.4 1.1) + (0.5 0.4) = 0.2304 + 0.7056 + 0.81 + 0.2025 + 0.16 + 0.1296 + 0.36 + 0.1936 + 0.04 = 2.8317. Marge two vertex a and e in G, as shown in Figure 8. Figure 8. A fuzzy graph H with M ( H) = 3.7317. Then, d( p) = 1.1, d(q) = 1.2, d(r) = 1.0, d(s) = 0.9, d( p, q) = 1.1, d(q, r) = 1.0, d(r, s) = 1.1, d(s, p) = 1.0. 2 2 Now, M ( H) = w(x)d(x) + $(e)d(e) å å xeV( H) eeE( H) 2 2 2 2 2 2 = (0.8 1.1) + (0.7 1.2) + (0.9 1.0) + (0.5 0.9) + (0.6 1.1) + (0.6 1.0) + 2 2 (0.4 1.1) + (0.5 1.0) = 0.7744 + 0.7056 + 0.81 + 0.2025 + 0.4356 + 0.36 + 0.1936 + 0.25 = 3.7317. Theorem 10. Let G = C(v , v , . . . , v ) be a cycle. Then, 0 1 i=n 2 2 2 2 1. M (G) = ZF (G) + $(e ) $(e ) + $(e ) + $(e ) $(e ) + $(e ) n å 1 1 2 i i1 i+1 i=2 Axioms 2023, 12, 415 10 of 16 2. M (G)  8(n + 1) 4. Proof. Similar proof to Theorem 8. Theorem 11. Let G and G be two fuzzy graphs and they are isomorphic to each other. Then, 1 2 z z M (G ) = M (G ). 1 2 1 1 Proof. As G and G are isomorphic to each other, then there exists a bijective mapping 1 2 f : V ! V and for all u, veV then w (v) = w (f(v)) and $ (uv) = $ f(u)f(v) 1 2 1 1 2 1 2 Then, d (v) = å $ (uv) = å $ f(u)f(v) = d f(v) . 1 2 G G 1 2 u2V f(u)2V 2 2 Now, M (G) = w(v)d (v) + $(uv)d (uv) å å G G 1 1 1 v2V(G ) uv2E(G ) 1 1 2 2 = w f(v) d f(v) + $ f(u)f(v) d f(u)f(v) å å G G 2 2 v2V(G ) f(u)f(v)2E(G ) 2 2 = M (G ). z z So, M (G ) = M (G ). 1 2 1 1 Theorem 12. If G = K (see Figure 9) is a star and satisfies the condition w(o)  w(v), 1, p1 where o is the center of the star, then the value of the FEZI is z 2 2 2 M (G) = (w(o)) ( w(v) + w(o) ( p 1)( p 3 p + 3)). veV(G)o Figure 9. Graph of K . 1, p1 Proof. Given that w(o)  w(v), where o is the center of the star, so $(e) = w(o) for all edges. Now, deg(v) for all veV o is given by $(ov) = w(o) and deg(o) = $(ov) = veVo ( p 1)w(o) also deg(e) = ( p 2)w(o). 2 2 Now, M (G) = å w(v)deg(v) + å $(e)deg(e) veV(G) eeE(G) 2 2 2 = w(v)w(o) + w(o)( p 1)w(o) + w(o)( p 2)w(o) å å veV(G)o eeE(G) 2 2 2 = (w(o)) ( å w(v) + w(o) ( p 1)( p 3 p + 3)). veV(G)o z 2 2 2 This shows that M (G) = (w(o)) w(v) + w(o) ( p 1)( p 3 p + 3) . veV(G)o Axioms 2023, 12, 415 11 of 16 Theorem 13. If G = S(a, b) is a double star, c is the center of the first star and c is the center 1 2 of the second star and satisfies the condition w(c ) = w(c )  w(v) for all veV(G), then 1 2 z 3 3 2 M (G) = ZF G + w(c ) a + b + (a + b) . 1 1 Proof. Since w(c ) = w(c )  w(v) for all veV(G), 1 2 $(c u) = w(c ), $(c v) = w(c ) = w(c ). 1 1 2 2 1 Additionally, deg(u) = w(c ), deg(v) = w(c ) = w(c ), 1 2 1 deg(c ) = (a + 1)w(c ), deg(c ) = (b + 1)w(c ) = (b + 1)w(c ), 1 1 2 2 1 deg(uc ) = aw(c ), deg(uc ) = bw(c ) = bw(c ), deg(c c ) = (a + b)w(c ). 1 1 2 2 1 1 2 1 2 2 Now, M = w(x)d(x) + $(e)d(e) å å xeV(G) eeE(G) 2 2 = ZF (G) + $(e)deg(e) + $(e)deg(e) + $(c c )deg(c c ) å å 1 1 2 1 2 eeE(G ) eeE(G ) 1 2 2 2 2 = ZF (G) + w(c )aw(c ) + w(c )bw(c ) + w(c )(a + b)w(c ) å å 1 1 1 1 1 1 1 eeE(G ) eeE(G ) 3 3 2 = ZF (G) + w(c ) a + b + (a + b) 1 1 z 3 3 2 M (G) = ZF G + w(c ) a + b + (a + b) . 1 1 Theorem 14. For a firefly graph F , if the MV of each vertex as well as the edge is one, s,t, p2s2t1 z 2 2 then the value of the FEZI is M (G) = p + 10s + 4t 1 + ( p t 1) + ( p t 1)( p t 2) . Proof. Since the MV of each vertex and each edge is one, the number of vertices with degree 1 is p 2s 2t 1 + t = p 2s t 1, the number of vertices with degree 2 is 2s + t, and deg(c) = 2( p + s 1) ( p 2s t 1) 2(2s + t) = p t 1, the number of edges with degree 1 is t, the number of edges with degree 2 is s, the remaining (p-t-1) edges have degree ( p t 2). 2 2 Then, the FEZI is M (G) = w(x)d(x) + $(e)d(e) å å xeV(G) eeE(G) 2 2 = ( p 2s t 1) + 4(2s + t) + ( p t 1) + t + 4s + ( p t 1)( p t 2) = p + 10s + 2 2 4t 1 + ( p t 1) + ( p t 1)( p t 2) . Corollary 2. If s = t = 0, then for a firefly graph, z 2 2 2 M (G) = (w(c)) w(v) + w(c) (n 1)(n 3n + 3) . veV(G)c If we put s = t = 0 in F , then the graph is a star similar to K s,t,n2s2t1 1,n1 z z 2 2 2 So, M (G) = M (K ) = (w(c)) w(v) + w(c) (n 1)(n 3n + 3) . 1,n1 1 1 veV(G)c 4. Application of FEZI for Fuzzy Graphs to Find out the State Which Require More Development for Internet System 4.1. Model Construction In the modern day, the internet is the most important part of our regular life. Here in this paper, we analyzed the Reliance Jio infocomm Ltd internet system in India. The data of Reliance Jio infocomm Ltd internet users are given in Table 1. These data were taken from https://dot.gov.in/sites/default/files/2022, accessed on 15 December 2022 . Then, we constructed a Reliance Jio infocomm Ltd internet system graph (see Figure 10). Here, the whole graph is similar to a star where Reliance Jio infocomm Ltd (C) is the center of the star and each state is a pendent vertex of the star. Axioms 2023, 12, 415 12 of 16 Figure 10. Fuzzy graph of internet network of Reliance jio. Table 1. Data of internet users for all states. Internet Users (in Total Population (in Population Percent- State Million) Million) age of the State Andhra Pradesh (A.P) 56.06 52.883 4 Assam (AS) 14.14 35.4 2.6 Bihar (BI) 48.11 125.1 9.2 Delhi (DE) 38.89 31.2 2.3 Gujrat (GU) 43.68 70.7 5.2 Haryana (HA) 16.74 29.9 2.2 Himachal Pradesh (H.P) 5.89 7.45 0.50 Jammu and Kashmir (J.K) 7.55 13.6 1.0 Karnataka (KA) 43.68 67.3 4.9 Kerala (KE) 24.92 35.4 2.6 Madhya Pradesh (M.P) 47.78 85.6 6.3 Maharashtra (MA) 61.12 125.5 9.3 Odisha (OD) 19.02 44.2 3.3 Punjab (PU) 25.1 30.6 2.2 Rajasthan (RA) 41.75 80.2 5.9 Tamil Nadu (T.N) 49.17 76.7 5.6 Uttar Pradesh (U.P) 91.35 233.4 17.2 West Bengal (W.B) 49.1 98.7 7.3 4.2. Representation of Membership Values Now, the MV of a vertex is denoted as w(S) according to the formula below: Total internet users in the state ^ {1, }. Total po pul ation in the state Here we see that w(S) 2 [0, 1], since there is an edge between center C and a state S. The MV of this edge is denoted as $(CS) and is defined by the following formula: Po pul ation persentage in this state S Total Jio internet users in this state S ^{1, + }. Total po pul ation in this state S 100 Here, we see that $(CS) 2 [0, 1]. The membership values are given in the Tables 2 and 3. Axioms 2023, 12, 415 13 of 16 Table 2. Some values with respect to internet users. Jio Internet Users population percentage Total internet users Jio internet users Jio internet users State + Total population Total population Total population 100 (Million) A.P 28.9 1.06 0.54 0.58 AS 7.3 0.4 0.21 0.236 BI 24.82 0.38 0.19 0.282 DE 20.06 1.25 0.64 0.663 GU 22.53 0.62 0.32 0.372 HA 8.64 0.56 0.29 0.312 H.P 3.03 0.79 0.41 0.415 J.K 3.89 0.56 0.29 0.30 KA 22.53 0.65 0.34 0.389 KE 12.86 0.70 0.36 0.386 M.P 24.65 0.56 0.29 0.353 MA 31.54 0.49 0.25 0.343 OD 9.81 0.43 0.22 0.253 PU 12.95 0.82 0.42 0.442 RA 21.54 0.52 0.27 0.369 T.N 25.37 0.64 0.33 0.386 U.P 47.13 0.39 0.20 0.372 W.B 25.34 0.5 0.26 0.333 Then the value of the FEZI is given by 2 2 M (G) = w(v)d(v) + $(e)d(e) = 112.02. å å veV(G) eeE(G) The FEZI of states (vertex) are given in the Table 4, and they are calculated by the formula z z z M (State) = M (G) M (G ). State 1 1 1 4.3. Decision Making z z z z z From the Table 4, we have M ( AS) = M (B I) = M (OD) < M (U.P) < M ( H A) = 1 1 1 1 1 z z z z z z z M ( J.K) = M ( M A) = M (W.B) < M ( M.P) = M (R A) < M (GU) < M (K A) = 1 1 1 1 1 1 1 z z z z z z M (T N) < M (KE) < M ( H.P) < M (PU) < M ( A.P) < M (DE). 1 1 1 1 1 1 Now, the least FEZI of a vertex indicates that the vertex is most crucial for the devel- opment of internet network systems. Here, the first entire Zagreb index of two or more states is equal. In this situation, we will find the population percentage of these states who cannot use the internet. Then, the state having the largest population percentage that does not use the internet becomes the first to develop an internet network system. Here, the percentages of the population who do not use the internet (PP) of the states are as follows: PP(AS) = 21.26, PP(BI) = 25.18, PP(OD) = 76.99, PP(HA) = 13.16, PP(J.K) = 6.05, PP(MA) = 64.38, PP(W.B) = 49.6, PP(M.P) = 37.82, PP(RA) = 38.45, PP(KA) = 23.62, PP(T.N) = 27.53. Then we can order the states as follows, and needing more development: Odisha, Bihar, Assam, Uttar Pradesh, Maharashtra, West Bengal, Haryana, Jammu and Kashmir, Rajasthan, Madhya Pradesh, Gujrat, Tamil Nadu, Karnataka, Kerala, Himachal Pradesh, Punjab, Andhra Pradesh, Delhi. Axioms 2023, 12, 415 14 of 16 Table 3. Membership values and degrees of the FG of Figure 10. Degree of the State (Ver- Degree of an Edge between State MV of the State (Vertex) tex) = MV of the Edge State and Center of the Star A.P 1 0.58 6.060 AS 0.4 0.236 6.550 BI 0.38 0.282 6.504 DE 1 0.663 6.123 GU 0.62 0.372 6.414 HA 0.56 0.312 6.474 H.P 0.79 0.415 6.371 J.K 0.56 0.30 6.486 KA 0.65 0.389 6.397 KE 0.70 0.386 6.400 M.P 0.56 0.353 6.433 MA 0.49 0.343 6.443 OD 0.43 0.253 6.533 PU 0.82 0.442 6.344 RA 0.52 0.369 6.417 T.N 0.64 0.386 6.400 U.P 0.39 0.372 6.414 W.B 0.5 0.333 6.453 Table 4. Value of first entire Zagreb index for all states. z z State M (G State) M (State) 1 1 A.P 111.68 0.34 AS 112.01 0.01 BI 112.01 0.01 DE 111.58 0.44 GU 111.97 0.05 HA 111.99 0.03 H.P 111.91 0.11 J.K 111.99 0.03 KA 111.96 0.06 KE 111.95 0.07 M.P 111.98 0.04 MA 111.99 0.03 OD 112.01 0.01 PU 111.89 0.13 RA 111.98 0.04 T.N 111.96 0.06 U.P 112.00 0.02 W.B 111.99 0.03 Axioms 2023, 12, 415 15 of 16 5. Conclusions In this article, the FEZI (first entire Zagreb index) was introduced as a graph parameter to quantify the structural characteristics of a graph. This study introduced several results and established relationships between various isomorphic graphs and a-cut fuzzy graphs. Additionally, the paper provided bounds for some fuzzy graphs and applied these results to real-life problems in the field of internet system development. The precise values or boundaries of FEZI with regards to graphs such as the firefly graph, star, path, cycle, and others, were explored in this study. The relationship between the value of the FEZI of a graph and its subgraph, two isomorphic graphs, and the first Zagreb index and FEZI were also described here. To analyze the Reliance Jio infocomm Ltd. internet system in India, this paper constructed an internet system graph. In this graph, the least FEZI of a vertex indicates that the vertex is most crucial for the development of the internet network system, and the first entire Zagreb index of two or more states is equal. According to this paper, the state with the highest proportion of people who do not use the internet is the first to develop an internet network system. The first entire Zagreb index is also important in biochemistry, chemical graph theory, spectral graph theory, etc. Author Contributions: Conceptualization, U.J. and G.G.; methodology, U.J. and G.G.; validation, U.J. and G.G. ; formal analysis, U.J. and G.G.; investigation, U.J. and G.G.; data curation, U.J. and G.G.; writing—original draft preparation, U.J. and G.G.; writing—review and editing, U.J. and G.G.; visualization, U.J. and G.G.; supervision, U.J. and G.G.; project administration, U.J. and G.G.; funding acquisition. All authors have read and agreed to the published version of the manuscript. Funding: The UGC of the Government of India is appreciative of the financial assistance under UGC-Ref. No.:201610294524 (CSIR-UGC NET NOVEMBER 2020) dated 01/04/2021. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data sets used and/or analyzed during the current study are available from the corresponding author on reasonable request. Acknowledgments: The authors would like to express their sincere gratitude to the anonymous referees for valuable suggestions, which led to great deal of improvement of the original manuscript. Conflicts of Interest: The authors declare no conflict of interest. References 1. Rosenfeld, A. Fuzzy Graph. In Fuzzy Sets and Their Applications; Zadeh, L.A., Fu, K.S., Shimura, M., Eds.; Academic Press: New York, NY, USA, 1975; pp. 77–95. 2. Zadeh, L.A. Fuzzy Sets. Inf. Control 1965, 8, 338–353. [CrossRef] 3. Yeh, R.T.; Bang, S.Y. Fuzzy relations. In Fuzzy Sets and Their Applications to Cognitive and Decision Process; Academic Press: New York, NY, USA, 1975; pp. 125–149. 4. Sunitha, M.S.; Vijayakumar, A. A characterisation of fuzzy trees. Inf. Sci. 1999, 113, 293–300. [CrossRef] 5. Sunitha, M.S.; Vijayakumar, A. Blocks in fuzzy graphs. J. Fuzzy Math. 2005, 13, 13–23. 6. Mordeson, J.N.; Mathew, S. Advanced Topics in Fuzzy Graph Theory; Springer: Berlin, Germany, 2019; Volume 375. 7. Al Mutab, H.M. Fuzzy Graphs. J. Adv. Math. 2019, 17, 77–95. [CrossRef] 8. Bhutani, K.R. On automorphisms of fuzzy graphs. Pattern Recognit Lett. 1989, 9, 159–162. [CrossRef] 9. Binu, M.; Mathew, S.; Mordeson, J.N. Wiener index of a fuzzy graph and application to illegal immigration networks. Fuzzy Sets Syst. 2020, 384, 132–147. 10. De, N.; Nayeem, S.M.A.; Pal, A. Reformulated First Zagreb index of Some Graph Operations. Mathematics 2015, 3, 945–960. [CrossRef] 11. Hosamani, S.M.; Basavanagoud, B. New upper bounds for the first Zagreb index. MATCH Commun. Math. Comput. Chem. 2015, 74, 97–101. 12. Ghorai, G. Characterization of regular bipolar fuzzy graphs. Afr. Mat. 2021, 32, 1043–1057. [CrossRef] 13. Ghorai, G.; Pal, M. Novel concepts of strongly edge irregular m-polar fuzzy graphs. Int. J. Appl. Comput. Math. 2016, 3, 3321–3332. [CrossRef] 14. Islam, S.R.; Pal, M. First Zagreb index on a fuzzy graph and its application. J. Intell. Fuzzy Syst. 2021, 40, 10575–10587. [CrossRef] 15. Poulik, S.; Ghorai, G. Determination of journeys order based on graphs Wiener absolute index with bipolar fuzzy information. Inf. Sci. 2021, 545, 608–619. [CrossRef] Axioms 2023, 12, 415 16 of 16 16. Gutman, I.; Trinajstic, N. Graph theory and molecular orbitals. Total f-electron energy of alternate hydrocarbons. Chem. Phys. Lett. 1972, 17, 535–538. [CrossRef] 17. Alwardi, A.; Alqesmah, A.; Rangarajan, R.; Cangul, I.N. Entire Zagreb indices of graphs. Discret. Math. Algorithms Appl. 2018, 10, 1850037. 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First Entire Zagreb Index of Fuzzy Graph and Its Application

Jana, Umapada; Ghorai, Ganesh
Axioms , Volume 12 (5) – Apr 24, 2023
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axioms Article Umapada Jana and Ganesh Ghorai * Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, India; umapada867032@gmail.com * Correspondence: math.ganesh@mail.vidyasagar.ac.in Abstract: The first entire Zagreb index (FEZI) is a graph parameter that has proven to be essential in various real-life scenarios, such as networking businesses and traffic management on roads. In this research paper, the FEZI was explored for a variety of fuzzy graphs, including star, firefly graph, cycle, path, fuzzy subgraph, vertex elimination, and edge elimination. This study presented several results, including determining the relationship between two isomorphic fuzzy graphs and between a path and cycle (connecting both end vertices of the path). This research also deals with the analysis of a-cut fuzzy graphs and establishes bounds for some fuzzy graphs. To apply these findings to modern life problems, the research team utilized the results to identify areas that require more development in internet systems. These results have practical implications for enhancing the efficiency and effectiveness of internet systems. The conclusion drawn from this research can be used to inform future research and aid in the development of more efficient and effective systems in various fields. Keywords: first entire Zagreb index; second entire Zagreb index; first entire Zagreb index of a vertex; isomorphic graph; internet system MSC: 05C09; 05C72 1. Introduction 1.1. Research Background Citation: Jana, U.; Ghorai, G. First Entire Zagreb Index of Fuzzy Graph In the modern day, fuzzy graph (FG) theory is one of the most applicable to regular and Its Application. Axioms 2023, 12, life. So there are many researchers who are implementing FG theory, especially topological 415. https://doi.org/10.3390/ indices of fuzzy graphs. Rosenfeld [1], inspired by Zadeh’s [2] classical set (fuzzy set) in axioms12050415 1975, introduced the fuzziness for a graph, then, it is called a fuzzy graph. Additionally, this time he introduced several connective parameters of an FG and some applications Academic Editors: Xiaohong Zhang, of these parameters by Yeh et al. [3]. In [4,5], Sunitha et al. studied fuzzy block, fuzzy Eunsuk Yang and Hsien-Chung Wu bridge, FSG, CFG, PFSG, fuzzy tree, fuzzy forest, fuzzy cut vertex, etc. The degree of a Received: 24 March 2023 vertex d(v)/deg(v) in an FG is also discussed In [6]. The degree of an edge (d (e)) is also Revised: 18 April 2023 discussed in [7]. Further information on the FG hypothesis is provided in [8–11]. Bipolar Accepted: 19 April 2023 and m-polar fuzzy graphs are discussed in [12,13]. The first Zagreb index is discussed Published: 24 April 2023 in [14] and was inspired by the paper [15]. In a molecular graph of a chemical compound, we can calculate molecular descriptors by finding topological indices of this graph. A graph’s topology is described by these topological indices, which are numerical numbers. The Zagreb index, established in 1972 by Gutman and Trinajstic [16], is a degree-based Copyright: © 2023 by the authors. topological index. The pi-electron energy of a conjugate system is determined using Licensee MDPI, Basel, Switzerland. topological indices. In a crisp graph, many indices are defined but several issues in real life This article is an open access article cannot be handled by these indices. So we can generalize these indices in fuzzy graphs. In distributed under the terms and this piece, we have introduced the FEZI in fuzzy graphs which is a major generalization conditions of the Creative Commons Attribution (CC BY) license (https:// of FEZI for crisp graphs. These indices are investigated from both a theoretical and an creativecommons.org/licenses/by/ application point of view. 4.0/). Axioms 2023, 12, 415. https://doi.org/10.3390/axioms12050415 https://www.mdpi.com/journal/axioms Axioms 2023, 12, 415 2 of 16 1.2. Research Question These questions are covered in this paper: 1. What is the FEZI upper bound for fuzzy graphs? 2. What are the precise values or boundaries of FEZI for the firefly graph, star, path, cycle, etc.? 3. What is the relation between the value of FEZI of a graph and its sub graph? 4. What is the relation between the value of FEZI for two isomorphic graph? 5. What is the relation for several graphs between the first Zagreb index and FEZI ? 6. What are the applications of this index? 1.3. Objective of the Work Various types of topological indices of a graph can be used for a variety of purposes and yield a wide range of outcomes for crisp graphs. However, in numerous applications, a crisp graph is not enough to solve it. We need to define a fuzzy graph to answer this question. In this paper, FEZI is defined and some results relating to sub graphs, paths, stars, firefly graphs, cycles, isomorphic graphs, etc. are given. At the end of this paper, we applied the first entire Zagreb index in internet network systems. 1.4. Structure of the Study The structure of the article is as follows: in Section 2, some definitions are provided which are necessary for this study. In Section 3, we studied the FEZI of a fuzzy graph and provided some results on sub graphs, paths, stars, firefly graphs, cycles, etc. Also some relation between fuzzy graphs are provided. In Section 4, an application of the first entire Zagreb index in development in internet networking system is discussed. 2. Preliminaries Here, we provide some fundamental definitions and theorems which are crucial to developing the later sections. Let U be a universal set. An FS A on U is a mapping s : U ! [0, 1]. Here, s is the membership function of the FS A. A FS generally indicated by A = (u, s). Assume that F(6= 0) is a known finite set. Then the fuzzy graph (FG) is a triplet, G = (U, w, $), where U(6= 0)  F with w : U ! [0, 1] and $ : U  U ! [0, 1] satisfying $(x , x )  w(x )^ w(x ). The set U is the set of vertices and # := ((x , x ) : $(x , x ) > 0) 1 2 1 2 1 2 1 2 is the set of edges of the FG. w(x ) represents the membership value (MV) of the vertex x 1 1 and $ represents the MV of the edge (x , x ) (or x x ). 2 2 1 1 0 0 0 Let G = (U, w, $) be an FG. Then, H = (U , w , $ ) is called the PFSG of the FG G if 0 0 0 0 0 U  U, w (x )  w(x ), $ (x , x )  $(x , x ) for all x , x 2 U . If w (x ) = w(x ) and 1 1 1 2 1 2 1 2 1 1 0 0 $ (x , x )  $(x , x ) for all x , x 2 U then H is called FSG of the graph of the FG G. 1 2 1 2 1 2 For x 2 V, we denote G as an FSG of the FG G = (V, w, $) with w(x ) = 0 and for 1 1 x x 2 #, G represents the FSG of the FG G with $(x x ) = 0. 1 2 x x 1 2 Let x , x , . . . , x all be vertices of an FG G. Then, the collection of vertices P(x , x , . . . , x ) n n 0 1 0 1 is a path in G if $(x , x ) 6= 0 (for all i). The path’s length is n in this case. If $(x , x ) > 0 for 0 n i i+1 the path P(x , x , . . . , x ) then it is called a cycle. 0 1 n Let x , x , . . . , x be distinct vertices of a fuzzy graph G. Then, G = (x , x , . . . , x ) is 0 n 0 n 1 1 called a star if $(x , x ) 6= 0 for i = 1, 2, . . . , n and for all vertex except x there is no edge 0 i 0 between every two vertices, where x is the center of the star. In a graph G if $(x , x )  0 for all vertices x and x , then it is called a complete i j i j graph. In a fuzzy graph G = (V, w, $) if for every two vertex x and y satisfy the condition $(x, y) = w(x)^ w(y) then the graph is called a CFG. Two FG G = (V , w , $ ) and G = (V , w , $ ) are called isomorphic to each other 1 1 1 1 2 2 2 2 if there exists a bijective mapping g : V ! V for any x, y 2 V , w (x) = w (g(x)) and 1 2 1 1 2 $ (g(x), g(y)) = $ (x, y). 2 1 Axioms 2023, 12, 415 3 of 16 Let G = (V, E) be a crisp graph. Then, the first and second ZI are defined by # 2 # M = (deg(x)) and M = deg(x)deg(y) [17]. å å 1 2 x is either adjacent x2V(G)U E(G) or incident to y The total degree of an FG with respect to vertices is indicated by T (G) and is defined by T (G) = d (v) [7]. Additionally, the total degree of an FG with respect to edge is v G v2V(G) indicated by T (G) and is defined by T (G) = d (e). e e G e2E(G) Example 1. Let G be an FG with vertex set V = f p, q, r, sg such that w( p) = 0.8, w(q) = 0.7, w(r) = 0.6, w(s) = 0.5, $( p, q) = 0.4, $( p, r) = 0.5, $(q, r) = 0.5, $(q, s) = 0.3, $(r, s) = 0.4, as shown in Figure 1. Then, d( p) = 0.9, d(q) = 1.2, d(r) = 1.4, d(s) = 0.7, d( p, q) = 1.3, d( p, r) = 1.3, d(q, r) = 1.6, d(q, s) = 1.3, d(r, s) = 1.3 and T (G) = 0.9 + 1.2 + 1.4 + 0.7 = 4.2 and T (G) = 1.3 + 1.3 + 1.3 + 1.6 + 1.3 + 1.3 = 6.8. Figure 1. A fuzzy graph. 3. First Entire Zagreb Index of Fuzzy Graphs First entire Zagreb index (FEZI) have an important role for finding strength of vertices. The strength of vertices is important in fuzzy graph theory. Thus, in this section the FEZI of a fuzzy graph is initiated. Various properties and an application of FEZI for fuzzy graphs is given. Definition 1. Let G be an FG. Then, the FEZI of G is indicated by M and is defined by : z 2 2 M := (w(x)d(x)) + ($(e)d(e)) . å å x2V(G) e2E(G) Example 2. Let G be an FG with vertex set V = f p, q, r, sg such that w( p) = 0.8, w(q) = 0.7, w(r) = 0.6, w(s) = 0.5, $( p, q) = 0.4, $( p, r) = 0.5, $(q, r) = 0.5, $(q, s) = 0.3 and $(r, s) = 0.4, shown in Figure 1. Then d( p) = 0.9, d(q) = 1.2, d(r) = 1.4, d(s) = 0.7, d( p, q) = 1.3, d( p, r) = 1.3, d(q, r) = 1.6, d(q, s) = 1.3, d(r, s) = 1.3. z 2 2 Now, M = å (w(x)d(x)) + å ($(e)d(e)) xeV(G) eeE(G) 2 2 2 2 2 2 = (0.8 0.9) + (0.7 1.2) + (0.6 1.4) + (0.5 0.7) + (0.4 1.3) + (0.5 1.3) + 2 2 2 (0.5 1.6) + (0.3 1.3) + (0.4 1.3) = 3.7135. Definition 2. Let (G, w, $) be an FG. Then, the second entire Zagreb index of G is indicated by e z M and is defined by M := w(u)d(u)w(v)d(v) + w(v)d(v)$(e)d(e), å å 2 2 u,vare adjacent to v is incident to e each other where w is the MV of a vertex and $ is the MV of an edge. Example 3. Let G be an FG with vertex set V = f p, q, r, sg such that, w( p) = 0.6, w(q) = 0.5, w(r) = 0.4, w(s) = 0.8, $( p, r) = 0.4, $(q, r) = 0.4, $(r, s) = 0.3 shown in Figure 2. Then Axioms 2023, 12, 415 4 of 16 d( p) = 0.4, d(q) = 0.4, d(r) = 1.1, d(s) = 0.3, d( p, r) = 0.7, d(q, r) = 0.7, d(r, s) = 0.8 Now, M = w(u)d(u)w(v)d(v) + w(v)d(v)w(e)d(e) å å u,vare adjacent v is incident to e to each other = (0.4 0.5 0.4 1.1) + (0.4 0.6 0.4 1.1) + (0.3 0.8 0.4 1.1) + (0.4 0.7 0.4 0.7) + (0.4 0.3 0.7 0.8) + (0.4 0.3 0.7 0.8) + (0.4 0.6 0.4 0.7) + (0.4 0.5 0.4 0.7) + (0.8 0.3 0.3 0.8) + (0.4 0.7 0.4 1.1) + (0.4 0.7 0.4 1.1) + (0.4 0.3 0.8 1.1) = 0.088 + 0.1056 + 0.1056 + 0.0784 + 0.0672 + 0.0672 + 0.0672 + 0.056 + 0.0576 + 0.1232 + 0.1232 + 0.1056 = 1.0448. Figure 2. A fuzzy graph G with M = 1.0448. z 2 2 2 2 Theorem 1. Let the FG G have n vertices and m edges, then M  n T (G) + m T (G), where v e 2 2 T (G) and T (G) represent the total degree with respect to vertex and total degree with respect to v e edge. Proof. Now, the FEZI of G is given by 2 2 M = w(x)d(x) + $(e)d(e) . å å xeV(G) eeE(G) 2 2 2 Since ( pq)  p q , we have å å å 2 2 2 2 w(x)d(x) + $(e)d(e)  (w(x)) (d(x)) + å å å å xeV(G) eeE(G) xeV(G) xeV(G) 2 2 ($(e)) (d(e)) å å eeE(G) eeE(G) 2 2 2 2 w(x) d(x) + $(e) d(e) å å å å xeV(G) xeV(G) eeE(G) eeE(G) 2 2 2 2 w(x) T (G) + $(e) T (G). å å v e xeV(G) eeE(G) z 2 2 2 2 Since 0  w(x)  1 and 0  $(e)  1, therefore M  n T (G) + m T (G). v e Definition 3. Let G = (V, w, $) be an FG. Then FEZI at a vertex of G is indicated by M (v) and z z z is defined by M (v) := M (G) M (G ) where G = V(G) v, w, $ . v v 1 1 1 Example 4. Let G be an FG with vertex set V = f p, q, r, sg such that w( p) = 0.8, w(q) = 0.7, w(r) = 0.5, w(s) = 0.6, $( p, q) = 0.4, $( p, r) = 0.5, $(q, r) = 0.5, $(q, s) = 0.3 and $(r, s) = 0.4 shown in Figure 3. Then, d( p) = 0.9, d(q) = 1.2, d(r) = 1.4, d(s) = 0.7, d( p, q) = 1.3, d( p, r) = 1.3, d(q, r) = 1.6, d(q, s) = 1.3, d(r, s) = 1.3. 2 2 Now, M = å w(x)d(x) + å $(e)d(e) xeV(G) eeE(G) 2 2 2 2 2 2 = (0.8 0.9) + (0.7 1.2) + (0.5 1.4) + (0.6 0.7) + (0.4 1.3) + (0.5 1.3) + 2 2 2 (0.5 1.6) + (0.3 1.3) + (0.4 1.3) So M (G) = 3.6458. 1 Axioms 2023, 12, 415 5 of 16 Now the vertex p is removed from the graph G, then the graph of G is shown in Figure 4. From the figure, d(q) = 0.6, d(r) = 0.7, d(s) = 0.7, d(q, r) = 0.7, d(q, s) = 0.7, d(r, s) = 0.6. Figure 3. A fuzzy graph G with M (G) = 3.6458. Figure 4. The FG G obtained by deleting the vertex p from the above graph. 2 2 Now, M (G ) = w(x)d(x) + $(e)d(e) p å å xeV(G ) eeE(G ) p p 2 2 2 2 2 2 = (0.7 0.6) + (0.5 0.7) + (0.6 0.7) + (0.3 0.7) + (0.6 0.4) + (0.3 0.7) . There- fore, M (G ) = 0.6211. Then, the FEZI at p is M ( p) = 3.6458 0.6211 = 3.0247. Theorem 2. Suppose that H is an FG that is created by removing an edge from G. Then, M ( H) M (G). 0 0 0 Proof. Since G = (V, w, $) is an FG and H = (V , w , $ ) is a graph that is created by removing an edge from G so the MV of a vertex is the same in both graphs and the MV of edges are the same if it contains both E and E . Then, the relation between the membership values of G and H is w(x)  w (x) for all vertices x and $(e)  $ (e) for all edges. 0 0 This shows that d(x)  d (x) for all vertices x. Additionally, d(e)  d (e) for all edge e, where d and d represent the degree of G and H. 2 2 Now, M (G) = w(x)d(x) + $(e)d(e) å å xeV(G) eeE(G) 2 2 0 0 0 0 w (x)d (x) + $ (e)d (e) å å xeV(G) eeE(G) 2 2 0 0 0 0 = w (x)d (x) + $ (e)d (e) å å xeV( H) eeE( H) = M ( H). z z Hence, M (G)  M ( H). 1 1 Theorem 3. If H is an FG that is obtained from G by deleting a vertex from G. Then, M ( H) M (G). 1 Axioms 2023, 12, 415 6 of 16 0 0 0 Proof. Since G = (V, w, $) is an FG and H = (V , w , $ ) is a graph that is created by removing a vertex from G. 0 0 0 0 So, w(x) = w (x) if x 2 V \ V otherwise w(x) > w (x). Additionally, $(e) = $ (e) if 0 0 e 2 E\ E otherwise $(e) > $ (e). Then the relation between the membership values of 0 0 0 0 0 G = (V, w, $) and H = (V , w , $ ) is w(x)  w (x) for all vertices x and $(e)  $ (e) for 0 0 all edges e. This shows that d(x)  d (x) for all vertices x. Additionally, d(e)  d (e) for all edges e. Here, d and d represent the degrees of G and H. 2 2 Now, M (G) = å w(x)d(x) + å $(e)d(e) xeV(G) eeE(G) 2 2 0 0 0 0 w (x)d (x) + $ (e)d (e) å å xeV(G) eeE(G) 2 2 0 0 0 0 = å w (x)d (x) + å $ (e)d (e) xeV( H) eeE( H) z z z = M ( H). So, M (G)  M ( H). 1 1 1 Example 5. Let G be an FG with vertex set V = f p, q, r, sg such that w( p) = 0.8, w(q) = 0.7, w(r) = 0.5, w(s) = 0.6, $( p, q) = 0.4, $( p, r) = 0.5, $(q, r) = 0.5, $(q, s) = 0.3, $(r, s) = 0.4 shown in Figure 3. Then d( p) = 0.9, d(q) = 1.2, d(r) = 1.4, d(s) = 0.7, d( p, q) = 1.3, d( p, r) = 1.3, d(q, r) = 1.6, d(q, s) = 1.3 and d(r, s) = 1.3. 2 2 Now, M (G) = w(x)d(x) + $(e)d(e) . å å xeV(G) eeE(G) 2 2 2 2 2 2 = (0.8 0.9) + (0.7 1.2) + (0.5 1.4) + (0.6 0.7) + (0.4 1.3) + (0.5 1.3) + 2 2 2 (0.5 1.6) + (0.3 1.3) + (0.4 1.3) = 3.6458. The graph of H where the vertex p is deleted from G shown in Figure 4. From the figure, d(q) = 0.6, d(r) = 0.7, d(s) = 0.7, d(q, r) = 0.7, d(q, s) = 0.7, d(r, s) = 0.6. z 2 2 Now, M ( H) = (w(x)d(x)) + ($(e)d(e)) å å xeV( H) eeE( H) 2 2 2 2 2 2 = (0.7 0.6) + (0.5 0.7) + (0.6 0.7) + (0.3 0.7) + (0.6 0.4) + (0.3 0.7) = 0.6211. The graph of K where the edge pr is deleted from G is shown in Figure 5. Figure 5. The FG obtained by deleting an edge from the graph in Figure 3. From this figure, d( p) = 0.4, d(q) = 1, d(r) = 0.7, d(s) = 0.7, d( p, q) = 0.6, d(q, r) = 1.1, d(q, s) = 1.1, d(r, s) = 1 z 2 2 Now, M (K) = (w(x)d(x)) + ($(e)d(e)) å å xeV(K) eeE(K) 2 2 2 2 2 2 = (0.8 0.4) + (0.7 1) + (0.5 0.7) + (0.6 0.7) + (0.6 0.4) + (0.3 1.1) + (0.3 2 2 1.1) + (0.4 1) = 1.3267. This shows that z z 1. M (G)  M ( H) 1 1 z z 2. M (G)  M (K). 1 1 z z Theorem 4. Let G be an FG and H be an FSG of G, then M (G)  M ( H). 1 1 Axioms 2023, 12, 415 7 of 16 0 0 0 Proof. Since H = (V , w , $ ) is an FSG of G = (V, w, $), therefore 0 0 w(x)  w (x) for all vertices x and $(e)  $ (e) for all edges e. This sows that d(x)  d (x) for all vertices x. 0 0 Additionally, d(e)  d (e) for all edges e, where d and d represent the degrees of G 2 2 and H. Now, M (G) = w(x)d(x) + $(e)d(e) å å xeV(G) eeE(G) 2 2 0 0 0 0 w (x)d (x) + $ (e)d (e) å å xeV(G) eeE(G) 2 2 0 0 0 0 = w (x)d (x) + $ (e)d (e) å å xeV( H) eeE( H) = M ( H) z z Hence, M (G)  M ( H). 1 1 Theorem 5. Let G = (V, w, $) be an FG and F = (V, w, $ ) be the corresponding MST of G then z z M (G)  M (F). 1 1 Proof. Since F = (V, w, $ ) is an MST of G = (V, w, $) therefore F is an FSG of G. Then we z z can say from the above Theorem, M (G)  M (F). 1 1 a 0 0 0 Theorem 6. Let G be an FG and let G = (V , w , $ ) be an a cut FG of G = (V, w, $). Then, z z a a 0 0 M (G)  M (G ) where the FG G is defined as V = veV : w(v)  a and w (v) = 1 1 , 0 w(v),$ (u, v) = $(u, v) for all u, veV . a z z a Proof. Since G is an FSG of the FG G, then by the above Theorem, M (G)  M (G ). 1 1 z p z p Theorem 7. Let G be an FG and let 0  p  p  1 Then M (G )  M (G ). 1 2 1 1 p p 2 1 Proof. Since 0  p  p  1, therefore G is a PFSG of G . Then, by the above Theorem, 1 2 z p z p M (G )  M (G ). 1 1 Corollary 1. Let G be an FG and let 0  p  p  p  . . .  p  1. 1 2 3 n z p z p z p z p n1 2 1 Then, M (G )  M (G )  . . .  M (G )  M (G ). 1 1 1 1 Theorem 8. Let G = P(v , v , v , . . . , v ) be a path. Then, 0 1 2 n i=n1 2 2 2 z 2 1. M (G) = ZF (G) + $(e )$(e ) + $(e )$(e ) + $ (e ) $(e ) + $(e ) . 1 1 2 n n1 i i1 i+1 i=2 2. M (G)  8(n 1). Proof. Given that G = P(v , v , v , . . . , v ) is a path, there are (n + 1) vertices and n edges. 0 2 n 1. Now, 2 2 M (G) = w(x)d(x) + $(e)d(e) (1) å å xeV(G) eeE(G) Here, the degree of each vertex v , except v and v , is $(e ) + $(e ) and the degree i 0 n i i+1 of v is $(e ) and degree of v is $(e ). n n 0 1 Additionally, the degree of each edge e , except e and e , is $(e ) + $(e ) and i 1 n i1 i+1 the degree of e is e , the degree of e is e . 1 2 n1 Using this result, we have from (1), i=n1 2 2 z 2 2 2 M (G) = (w $ ) + (w $ ) + å w (v ) $(e ) + $(e ) + $(e )$(e ) + 0 n n 2 1 i i i+1 1 i=1 i=n1 2 2 $(e )$(e ) + $ (e ) $(e ) + $(e ) n n1 i i1 i+1 i=2 i=n1 2 2 2 = ZF (G) + $(e )$(e ) + $(e )$(e ) + $ (e ) $(e ) + $(e ) (2) 1 1 2 n n1 å i i1 i+1 i=2 Axioms 2023, 12, 415 8 of 16 2. From Equation (2), we get i=n1 2 2 2 z 2 M (G) = ZF (G) + $(e )$(e ) + $(e )$(e ) + $ (e ) $(e ) + $(e ) . n å 1 1 2 n1 i i1 i+1 i=2 As 0  $(e )  1 and 0  w(v )  1, i i M (G)  2(2n 1) + 1 + 1 + 4(n 2) (where ZF (G)  2(2n 1)) = 4n 2 + 2 + 4n 8 = 8n 8. So, M (G)  8(n 1). Example 6. Let A = P( p, q, r, s, e) be an FG with vertex set V = p, q, r, s, e such that w( p) = 0.8, w(q) = 0.7, w(r) = 0.9, w(s) = 0.5, w(e) = 0.6, $( p, q) = 0.6, $(q, r) = 0.6, $(r, s) = 0.4, $(s, e) = 0.5 shown in Figure 6. Then, d( p) = 0.6, d(q) = 1.2, d(r) = 1.0, d(s) = 0.9, d(e) = 0.5, d( p, q) = 0.6, d(q, r) = 1.0, d(r, s) = 1.1 and d(s, e) = 0.4. 2 2 Now, M ( A) = w(x)d(x) + $(e)d(e) å å xeV( A) eeE( A) 2 2 2 2 2 2 = (0.8 0.6) + (0.7 1.2) + (0.9 1.0) + (0.5 0.9) + (0.6 0.5) + (0.6 0.6) + 2 2 2 (0.6 1.0) + (0.4 1.1) + (0.5 0.4) = 0.2304 + 0.7056 + 0.81 + 0.2025 + 0.09 + 0.1296 + 0.36 + 0.1936 + 0.04 = 2.7617. Figure 6. A fuzzy path G with M ( A) = 2.7617. Theorem 9. Let G = P(v , v , v , . . . , v ) be a path. If we take v = v , it becomes a cycle n n 0 1 2 0 z z H = C(v , v , v , . . . , v ). Then M (G)  M ( H). 0 1 2 n 1 1 Proof. Let d and d denote the degree of a vertex/edge in G and H respectively. Here, d(v) = d (v) for all vertices except v and v . Now, d(v ) = e and d(v ) = e and 0 n 0 1 n n 0 0 d (v ) = e + e . Additionally, d(e) = d (e) for all edges except e and e . Now, d(e ) = e n n 0 1 1 1 2 0 0 and d(e ) = e and d (e ) = e + e , d (e ) = e + e . n n1 1 2 n n 1 n1 2 2 Now, M (G) = w(v)d(v) + $(e)d(e) å å veV(G) eeE(G) 2 2 2 2 2 = w(v )d(v ) + w(v )d(v ) + å w(v)d(v) + $(e )d(e ) + $(e )d(e ) + 0 0 n n 1 1 n n veV(G)v ,v 0 n $(e)d(e) eeE(G)e ,e 1 n 2 2 2 2 0 0 0 0 0 0 = (w (v )e ) + w (v )e + w (v)d (v) + $ (e )e + $ (e )e + 0 1 n n 1 2 n n1 veV( H)v 0 0 å $ (e)d (e) eeE( H)e ,e 1 n 2 2 2 2 2 0 0 0 0 0 0 = w (v )e + w (v )e + w (v)d (v) + $ (e )e + $ (e )e + n å n 0 1 0 1 2 n1 veV( H)v 0 0 $ (e)d (e) eeE( H)e ,e 1 n 2 2 2 0 2 2 0 0 0 0 = w (v ) (e ) + (e ) + å w (v)d (v) + $ (e )e + $ (e )e + 0 n 2 n 1 1 n1 veV( H)v 0 0 $ (e)d (e) eeE( H)e ,e 1 n Axioms 2023, 12, 415 9 of 16 2 2 2 2 0 0 0 0 0 w (v ) (e ) + (e ) + w (v)d (v) + $ (e )(e + e ) + $ (e )(e + e ) + 0 1 n 1 2 n n n1 1 veV( H)v 0 0 $ (e)d (e) eeE( H)e ,e 1 n = M ( H). z z So, M (G)  M ( H). 1 1 Example 7. Let G be a path with vertex set V = p, q, r, s, e such that w( p) = 0.8, w(q) = 0.7, w(r) = 0.9, w(s) = 0.5, w(e) = 0.8, $( p, q) = 0.6, $(q, r) = 0.6, $(r, s) = 0.4 and $(s, e) = 0.5 shown in Figure 7. Figure 7. A fuzzy path G with M (G) = 2.8317. Then, d( p) = 0.6, d(q) = 1.2, d(r) = 1.0, d(s) = 0.9, d(e) = 0.5, d( p, q) = 0.6, d(q, r) = 1.0, d(r, s) = 1.1, d(s, e) = 0.4 2 2 Now, M (G) = w(x)d(x) + $(e)d(e) å å xeV(G) eeE(G) 2 2 2 2 2 2 = (0.8 0.6) + (0.7 1.2) + (0.9 1.0) + (0.5 0.9) + (0.8 0.5) + (0.6 0.6) + 2 2 2 (0.6 1.0) + (0.4 1.1) + (0.5 0.4) = 0.2304 + 0.7056 + 0.81 + 0.2025 + 0.16 + 0.1296 + 0.36 + 0.1936 + 0.04 = 2.8317. Marge two vertex a and e in G, as shown in Figure 8. Figure 8. A fuzzy graph H with M ( H) = 3.7317. Then, d( p) = 1.1, d(q) = 1.2, d(r) = 1.0, d(s) = 0.9, d( p, q) = 1.1, d(q, r) = 1.0, d(r, s) = 1.1, d(s, p) = 1.0. 2 2 Now, M ( H) = w(x)d(x) + $(e)d(e) å å xeV( H) eeE( H) 2 2 2 2 2 2 = (0.8 1.1) + (0.7 1.2) + (0.9 1.0) + (0.5 0.9) + (0.6 1.1) + (0.6 1.0) + 2 2 (0.4 1.1) + (0.5 1.0) = 0.7744 + 0.7056 + 0.81 + 0.2025 + 0.4356 + 0.36 + 0.1936 + 0.25 = 3.7317. Theorem 10. Let G = C(v , v , . . . , v ) be a cycle. Then, 0 1 i=n 2 2 2 2 1. M (G) = ZF (G) + $(e ) $(e ) + $(e ) + $(e ) $(e ) + $(e ) n å 1 1 2 i i1 i+1 i=2 Axioms 2023, 12, 415 10 of 16 2. M (G)  8(n + 1) 4. Proof. Similar proof to Theorem 8. Theorem 11. Let G and G be two fuzzy graphs and they are isomorphic to each other. Then, 1 2 z z M (G ) = M (G ). 1 2 1 1 Proof. As G and G are isomorphic to each other, then there exists a bijective mapping 1 2 f : V ! V and for all u, veV then w (v) = w (f(v)) and $ (uv) = $ f(u)f(v) 1 2 1 1 2 1 2 Then, d (v) = å $ (uv) = å $ f(u)f(v) = d f(v) . 1 2 G G 1 2 u2V f(u)2V 2 2 Now, M (G) = w(v)d (v) + $(uv)d (uv) å å G G 1 1 1 v2V(G ) uv2E(G ) 1 1 2 2 = w f(v) d f(v) + $ f(u)f(v) d f(u)f(v) å å G G 2 2 v2V(G ) f(u)f(v)2E(G ) 2 2 = M (G ). z z So, M (G ) = M (G ). 1 2 1 1 Theorem 12. If G = K (see Figure 9) is a star and satisfies the condition w(o)  w(v), 1, p1 where o is the center of the star, then the value of the FEZI is z 2 2 2 M (G) = (w(o)) ( w(v) + w(o) ( p 1)( p 3 p + 3)). veV(G)o Figure 9. Graph of K . 1, p1 Proof. Given that w(o)  w(v), where o is the center of the star, so $(e) = w(o) for all edges. Now, deg(v) for all veV o is given by $(ov) = w(o) and deg(o) = $(ov) = veVo ( p 1)w(o) also deg(e) = ( p 2)w(o). 2 2 Now, M (G) = å w(v)deg(v) + å $(e)deg(e) veV(G) eeE(G) 2 2 2 = w(v)w(o) + w(o)( p 1)w(o) + w(o)( p 2)w(o) å å veV(G)o eeE(G) 2 2 2 = (w(o)) ( å w(v) + w(o) ( p 1)( p 3 p + 3)). veV(G)o z 2 2 2 This shows that M (G) = (w(o)) w(v) + w(o) ( p 1)( p 3 p + 3) . veV(G)o Axioms 2023, 12, 415 11 of 16 Theorem 13. If G = S(a, b) is a double star, c is the center of the first star and c is the center 1 2 of the second star and satisfies the condition w(c ) = w(c )  w(v) for all veV(G), then 1 2 z 3 3 2 M (G) = ZF G + w(c ) a + b + (a + b) . 1 1 Proof. Since w(c ) = w(c )  w(v) for all veV(G), 1 2 $(c u) = w(c ), $(c v) = w(c ) = w(c ). 1 1 2 2 1 Additionally, deg(u) = w(c ), deg(v) = w(c ) = w(c ), 1 2 1 deg(c ) = (a + 1)w(c ), deg(c ) = (b + 1)w(c ) = (b + 1)w(c ), 1 1 2 2 1 deg(uc ) = aw(c ), deg(uc ) = bw(c ) = bw(c ), deg(c c ) = (a + b)w(c ). 1 1 2 2 1 1 2 1 2 2 Now, M = w(x)d(x) + $(e)d(e) å å xeV(G) eeE(G) 2 2 = ZF (G) + $(e)deg(e) + $(e)deg(e) + $(c c )deg(c c ) å å 1 1 2 1 2 eeE(G ) eeE(G ) 1 2 2 2 2 = ZF (G) + w(c )aw(c ) + w(c )bw(c ) + w(c )(a + b)w(c ) å å 1 1 1 1 1 1 1 eeE(G ) eeE(G ) 3 3 2 = ZF (G) + w(c ) a + b + (a + b) 1 1 z 3 3 2 M (G) = ZF G + w(c ) a + b + (a + b) . 1 1 Theorem 14. For a firefly graph F , if the MV of each vertex as well as the edge is one, s,t, p2s2t1 z 2 2 then the value of the FEZI is M (G) = p + 10s + 4t 1 + ( p t 1) + ( p t 1)( p t 2) . Proof. Since the MV of each vertex and each edge is one, the number of vertices with degree 1 is p 2s 2t 1 + t = p 2s t 1, the number of vertices with degree 2 is 2s + t, and deg(c) = 2( p + s 1) ( p 2s t 1) 2(2s + t) = p t 1, the number of edges with degree 1 is t, the number of edges with degree 2 is s, the remaining (p-t-1) edges have degree ( p t 2). 2 2 Then, the FEZI is M (G) = w(x)d(x) + $(e)d(e) å å xeV(G) eeE(G) 2 2 = ( p 2s t 1) + 4(2s + t) + ( p t 1) + t + 4s + ( p t 1)( p t 2) = p + 10s + 2 2 4t 1 + ( p t 1) + ( p t 1)( p t 2) . Corollary 2. If s = t = 0, then for a firefly graph, z 2 2 2 M (G) = (w(c)) w(v) + w(c) (n 1)(n 3n + 3) . veV(G)c If we put s = t = 0 in F , then the graph is a star similar to K s,t,n2s2t1 1,n1 z z 2 2 2 So, M (G) = M (K ) = (w(c)) w(v) + w(c) (n 1)(n 3n + 3) . 1,n1 1 1 veV(G)c 4. Application of FEZI for Fuzzy Graphs to Find out the State Which Require More Development for Internet System 4.1. Model Construction In the modern day, the internet is the most important part of our regular life. Here in this paper, we analyzed the Reliance Jio infocomm Ltd internet system in India. The data of Reliance Jio infocomm Ltd internet users are given in Table 1. These data were taken from https://dot.gov.in/sites/default/files/2022, accessed on 15 December 2022 . Then, we constructed a Reliance Jio infocomm Ltd internet system graph (see Figure 10). Here, the whole graph is similar to a star where Reliance Jio infocomm Ltd (C) is the center of the star and each state is a pendent vertex of the star. Axioms 2023, 12, 415 12 of 16 Figure 10. Fuzzy graph of internet network of Reliance jio. Table 1. Data of internet users for all states. Internet Users (in Total Population (in Population Percent- State Million) Million) age of the State Andhra Pradesh (A.P) 56.06 52.883 4 Assam (AS) 14.14 35.4 2.6 Bihar (BI) 48.11 125.1 9.2 Delhi (DE) 38.89 31.2 2.3 Gujrat (GU) 43.68 70.7 5.2 Haryana (HA) 16.74 29.9 2.2 Himachal Pradesh (H.P) 5.89 7.45 0.50 Jammu and Kashmir (J.K) 7.55 13.6 1.0 Karnataka (KA) 43.68 67.3 4.9 Kerala (KE) 24.92 35.4 2.6 Madhya Pradesh (M.P) 47.78 85.6 6.3 Maharashtra (MA) 61.12 125.5 9.3 Odisha (OD) 19.02 44.2 3.3 Punjab (PU) 25.1 30.6 2.2 Rajasthan (RA) 41.75 80.2 5.9 Tamil Nadu (T.N) 49.17 76.7 5.6 Uttar Pradesh (U.P) 91.35 233.4 17.2 West Bengal (W.B) 49.1 98.7 7.3 4.2. Representation of Membership Values Now, the MV of a vertex is denoted as w(S) according to the formula below: Total internet users in the state ^ {1, }. Total po pul ation in the state Here we see that w(S) 2 [0, 1], since there is an edge between center C and a state S. The MV of this edge is denoted as $(CS) and is defined by the following formula: Po pul ation persentage in this state S Total Jio internet users in this state S ^{1, + }. Total po pul ation in this state S 100 Here, we see that $(CS) 2 [0, 1]. The membership values are given in the Tables 2 and 3. Axioms 2023, 12, 415 13 of 16 Table 2. Some values with respect to internet users. Jio Internet Users population percentage Total internet users Jio internet users Jio internet users State + Total population Total population Total population 100 (Million) A.P 28.9 1.06 0.54 0.58 AS 7.3 0.4 0.21 0.236 BI 24.82 0.38 0.19 0.282 DE 20.06 1.25 0.64 0.663 GU 22.53 0.62 0.32 0.372 HA 8.64 0.56 0.29 0.312 H.P 3.03 0.79 0.41 0.415 J.K 3.89 0.56 0.29 0.30 KA 22.53 0.65 0.34 0.389 KE 12.86 0.70 0.36 0.386 M.P 24.65 0.56 0.29 0.353 MA 31.54 0.49 0.25 0.343 OD 9.81 0.43 0.22 0.253 PU 12.95 0.82 0.42 0.442 RA 21.54 0.52 0.27 0.369 T.N 25.37 0.64 0.33 0.386 U.P 47.13 0.39 0.20 0.372 W.B 25.34 0.5 0.26 0.333 Then the value of the FEZI is given by 2 2 M (G) = w(v)d(v) + $(e)d(e) = 112.02. å å veV(G) eeE(G) The FEZI of states (vertex) are given in the Table 4, and they are calculated by the formula z z z M (State) = M (G) M (G ). State 1 1 1 4.3. Decision Making z z z z z From the Table 4, we have M ( AS) = M (B I) = M (OD) < M (U.P) < M ( H A) = 1 1 1 1 1 z z z z z z z M ( J.K) = M ( M A) = M (W.B) < M ( M.P) = M (R A) < M (GU) < M (K A) = 1 1 1 1 1 1 1 z z z z z z M (T N) < M (KE) < M ( H.P) < M (PU) < M ( A.P) < M (DE). 1 1 1 1 1 1 Now, the least FEZI of a vertex indicates that the vertex is most crucial for the devel- opment of internet network systems. Here, the first entire Zagreb index of two or more states is equal. In this situation, we will find the population percentage of these states who cannot use the internet. Then, the state having the largest population percentage that does not use the internet becomes the first to develop an internet network system. Here, the percentages of the population who do not use the internet (PP) of the states are as follows: PP(AS) = 21.26, PP(BI) = 25.18, PP(OD) = 76.99, PP(HA) = 13.16, PP(J.K) = 6.05, PP(MA) = 64.38, PP(W.B) = 49.6, PP(M.P) = 37.82, PP(RA) = 38.45, PP(KA) = 23.62, PP(T.N) = 27.53. Then we can order the states as follows, and needing more development: Odisha, Bihar, Assam, Uttar Pradesh, Maharashtra, West Bengal, Haryana, Jammu and Kashmir, Rajasthan, Madhya Pradesh, Gujrat, Tamil Nadu, Karnataka, Kerala, Himachal Pradesh, Punjab, Andhra Pradesh, Delhi. Axioms 2023, 12, 415 14 of 16 Table 3. Membership values and degrees of the FG of Figure 10. Degree of the State (Ver- Degree of an Edge between State MV of the State (Vertex) tex) = MV of the Edge State and Center of the Star A.P 1 0.58 6.060 AS 0.4 0.236 6.550 BI 0.38 0.282 6.504 DE 1 0.663 6.123 GU 0.62 0.372 6.414 HA 0.56 0.312 6.474 H.P 0.79 0.415 6.371 J.K 0.56 0.30 6.486 KA 0.65 0.389 6.397 KE 0.70 0.386 6.400 M.P 0.56 0.353 6.433 MA 0.49 0.343 6.443 OD 0.43 0.253 6.533 PU 0.82 0.442 6.344 RA 0.52 0.369 6.417 T.N 0.64 0.386 6.400 U.P 0.39 0.372 6.414 W.B 0.5 0.333 6.453 Table 4. Value of first entire Zagreb index for all states. z z State M (G State) M (State) 1 1 A.P 111.68 0.34 AS 112.01 0.01 BI 112.01 0.01 DE 111.58 0.44 GU 111.97 0.05 HA 111.99 0.03 H.P 111.91 0.11 J.K 111.99 0.03 KA 111.96 0.06 KE 111.95 0.07 M.P 111.98 0.04 MA 111.99 0.03 OD 112.01 0.01 PU 111.89 0.13 RA 111.98 0.04 T.N 111.96 0.06 U.P 112.00 0.02 W.B 111.99 0.03 Axioms 2023, 12, 415 15 of 16 5. Conclusions In this article, the FEZI (first entire Zagreb index) was introduced as a graph parameter to quantify the structural characteristics of a graph. This study introduced several results and established relationships between various isomorphic graphs and a-cut fuzzy graphs. Additionally, the paper provided bounds for some fuzzy graphs and applied these results to real-life problems in the field of internet system development. The precise values or boundaries of FEZI with regards to graphs such as the firefly graph, star, path, cycle, and others, were explored in this study. The relationship between the value of the FEZI of a graph and its subgraph, two isomorphic graphs, and the first Zagreb index and FEZI were also described here. To analyze the Reliance Jio infocomm Ltd. internet system in India, this paper constructed an internet system graph. In this graph, the least FEZI of a vertex indicates that the vertex is most crucial for the development of the internet network system, and the first entire Zagreb index of two or more states is equal. According to this paper, the state with the highest proportion of people who do not use the internet is the first to develop an internet network system. The first entire Zagreb index is also important in biochemistry, chemical graph theory, spectral graph theory, etc. Author Contributions: Conceptualization, U.J. and G.G.; methodology, U.J. and G.G.; validation, U.J. and G.G. ; formal analysis, U.J. and G.G.; investigation, U.J. and G.G.; data curation, U.J. and G.G.; writing—original draft preparation, U.J. and G.G.; writing—review and editing, U.J. and G.G.; visualization, U.J. and G.G.; supervision, U.J. and G.G.; project administration, U.J. and G.G.; funding acquisition. All authors have read and agreed to the published version of the manuscript. Funding: The UGC of the Government of India is appreciative of the financial assistance under UGC-Ref. No.:201610294524 (CSIR-UGC NET NOVEMBER 2020) dated 01/04/2021. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data sets used and/or analyzed during the current study are available from the corresponding author on reasonable request. Acknowledgments: The authors would like to express their sincere gratitude to the anonymous referees for valuable suggestions, which led to great deal of improvement of the original manuscript. Conflicts of Interest: The authors declare no conflict of interest. References 1. Rosenfeld, A. Fuzzy Graph. In Fuzzy Sets and Their Applications; Zadeh, L.A., Fu, K.S., Shimura, M., Eds.; Academic Press: New York, NY, USA, 1975; pp. 77–95. 2. Zadeh, L.A. Fuzzy Sets. Inf. Control 1965, 8, 338–353. [CrossRef] 3. Yeh, R.T.; Bang, S.Y. Fuzzy relations. 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Axioms – Multidisciplinary Digital Publishing Institute

Published: Apr 24, 2023

Keywords: first entire Zagreb index; second entire Zagreb index; first entire Zagreb index of a vertex; isomorphic graph; internet system

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