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## Original Article

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International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(4): 382-390

Published online December 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.4.382

© The Korean Institute of Intelligent Systems

## The Isomorphism of Total Fuzzy Graphs

Fekadu Tesgera Agama and V. N. Srinivasa Rao Repalle

Department of Mathematics, College of Natural and Computational Sciences, Wollega University, Nekemte, Ethiopia

Correspondence to :

Received: February 3, 2022; Revised: June 6, 2022; Accepted: December 12, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

This manuscript is aimed to deal with the isomorphism properties of total fuzzy graph. In order to achieve the desired objective, we consider two fuzzy graphs and their total fuzzy graphs. For these total fuzzy graphs, the notion of homomorphism of total fuzzy graphs is defined and the existence of a homomorphism between these two total fuzzy graphs is shown by examples. This is achieved by both sketching their graph and verifying the condition of homomorphism. Having a mapping being a homomorphism of the total fuzzy graphs the concept of weak isomorphism and co-weak isomorphism are presented with separate and distinct definition. These definitions are accompanied with supportive examples and the case where a homomorphism between two total fuzzy graphs is not weak-isomorphism is observed and illustrated by supportive example. In addition to these, the notion of co-weak isomorphism is also defined and illustrated by using a typical example. A very great attention is given to the definition of an isomorphism of total fuzzy graph and results arising from this definition. Accordingly, certain theorems related to isomorphic properties of total fuzzy graphs are stated and each of these theorems are thoroughly proved to show the results. Finally the manuscript states the future research works, limitations of the manuscript.

Keywords: Fuzzy graph, Total fuzzy graph, Homomorphism, Isomorphism, Week isomorphism

Fuzzy sets and fuzzy relations are introduced and discussed well by Zadeh [1] in 1965. The emerging of these concepts attracted the attention of many scholars and forced them to associate with many research fields. Hence, notion goes beyond graph theory and took attention in the fields medicine, engineering, statistics, science of management and the like. The distinctions of this fuzzy set is that each element is associated with a membership value chosen from the interval [0, 1]. As a result of these, Rosenfeld [2] well-thought-out fuzzy relations on fuzzy sets and advanced the theory of fuzzy graphs in 1975. The notion of isomorphism of fuzzy graphs also followed the foot step and Perchant [3] familiarized a generic meaning of fuzzy morphism amongst graphs that comprises standard graph related definitions as graph and sub-graph isomorphism. In addition to these, Bhutani in [4] studied different types of isomorphism of fuzzy graphs and some graceful theorems on weak and co-weak isomorphism of m-polar fuzzy graph was studied by S. Satyanarayana & et al. [5].

Currently, the study on fuzzy graph is more inclined to the sub-part of a fuzzy graph called bipolar fuzzy graphs. Accordingly Poulik S. [6] have determined connectivity index and Wiener index in bipolar fuzzy graphs and in [7] the study disclosed that the totally accurate communication between all connected nodes is explained by introducing completely open neighborhood degree and completely closed neighborhood degree of nodes and edges in a bipolar fuzzy graph. Moreover, Soumitra Poulik and Ganesh Ghorai [8] described the empirical results on operations of bipolar fuzzy graphs with their degree. Other studies focus on the isomorphism of fuzzy graphs. Thus, Vijaya M. and Mekala B. [9] studied about weak isomorphism on bipolar total fuzzy graphs and put results related to bipolar TFG. In the book of “Modern Trends in Fuzzy Graphs” [10], the homomorphism products of two intuitionistic fuzzy graphs are presented and results related to these are also elaborated. Isomorphism in generalized fuzzy graphs has been introduced by Samanta [11] to capture the similarity of uncertainties in different networks. Homomorphism, weak isomorphism, co-weak isomorphism and nearly isomorphism are defined and an application of image visualization was described. A new method to explain the homeomorphic between some fuzzy topological graphs which will be applied in smart cities has been presented by Atefa et al. [12]. The idea of an isomorphic picture fuzzy graph as well as its application on a social network have been described by Zuo [13]. The notations of μ-complement, homomorphism, isomorphism, weak isomorphism and co weak isomorphism of regular picture fuzzy graph and mathematical model of communication network and transportation network by using picture fuzzy multigraph is also studies and its application on transportation network/communication network is also presented by Xiao [14]. The concept total fuzzy graph (TFG) was discussed by Nagoor Gani [15]. 2-qusi TFG definition, properties, and its coloring are well studied by Fekadu Tesgera Agama and R.V.N. Srinivasa Rao Repalle [16]. Some related studies on line graph of fuzzy graphs, homeomorphisms and others are also used for this study from [15]. The inverse fuzzy mixed graphs and its application is clearly introduce by Poulik and Ghorai [17]. In their study, they have discussed the isomorphic properties of inverse of mixed fuzzy graphs.

### 1.1 Objective of the Study

In this paper, we give the definition of homomorphism on TFG which is supported by an elaborative example. With this, we define and give examples of weak isomorphism, co-weak isomorphism, and an isomorphism of TFG. In addition, we state some properties regarding the isomorphism of total fuzzy graph and give the proof of these properties.

### 1.2 Organization of the Paper

This manuscript is organized in to four different sections. The first section deals with introduction and the second section discusses some basic terms or definitions required to discuss about the main concept of the manuscript in section three. In section four we present some properties of the isomorphism of total fuzzy graphs and lastly we provide the conclusion to the study.

Under this sub-topic we present definitions of fuzzy graphs and TFG which are necessary for this article.

Definition 1 [18]. A fuzzy graph G: (σ, μ) is a triple consisting of a non-empty set V along with the functions σ: V → [0, 1] and μ: V × V → [0, 1] such that for all u, vV, μ(u, v) ≤ σ(u) ∧ σ(v), where ∧ denotes the minimum.

Definition 2 [18]. Let G: (σ, μ) be a fuzzy graph. The order (Order(G)) of G is defined as:

Order(G)=uVσ(u),

and the size (Size(u)) of G denoted by

Size(u)=u,vVμ(u,v).

Definition 3 [18]. Let G: (σ, μ) be fuzzy graph. Let uV be any vertex of G, then the degree of u is defined as; dG(u) = ∑vu, vV μ(u, v).

Definition 4 [4]. Let g: GG’ be a map g: SS’ which satisfies the following two conditions:

• σ(u) ≤ σ’ (g(u)), ∀ uS and

• μ(u, v) ≤ μ’ (g(u), g(v)), ∀ u, vS where σ and σ’ are membership functions of vertices, μ and μ’ are membership functions of edges, then g is called a homomorphism of fuzzy graphs.

Definition 5 [4]. Suppose g: GG’ is a mapping. If g: SS’ of fuzzy graphs is a bijective homomorphism satisfying σ(u) = σ’ (g(u)), ∀ uS, then g is called a weak-isomorphism.

Definition 6 [4]. Let g: GG’ be a map, g: SS’ which is a bijective homomorphism that satisfies μ(x, y) = μ’ (g(x), g(y)) ∀ x, yS, and then g is called co-weak isomorphism.

Definition 7 [18]. A fuzzy relation μ on a fuzzy subset σ of a set S is said to be a fuzzy equivalence relation if it is reflexive, symmetric, and transitive.

Definition 8 [18]. Let G = (μ, σ) be a fuzzy graph, then μ is called reflexive if μ(u, u) = σ(u) for all uV.

Definition 9 [18]. For a fuzzy graph, G: (σ, μ), μ is said to be symmetric when μ(u, v) = μ(v, u) ∀ u and it is said to satisfy transitivity property when μ2μ.

Definition 10 [19]. Suppose G: (σ, μ) as a fuzzy graph. The TFG is a pair T(G) = (σT, μT ) of G and;

• σT is defined over VE such that σT (u) = σ(u), if uV, μT (e) = μ(e), if eE.

• μT is described as;

μT (u, v) = μ(u, v), if u, vV,

μT (u, e) = σ(u)∧ μ(e), if uV, eE, and the vertex

u lies on the edge e,

μT (u, e) = 0, otherwise,

μT (ei, ej) = μ(ei) ∧ μ(ej), if the edges ei and ej have a node in common between them, μT (ei, ej) = 0, otherwise.

This section introduces the homomorphism and isomorphism of TFG. Furthermore, different types of isomorphism such as weak isomorphism and co-weak isomorphism of TFG are defined and elaborated by examples.

Definition 11. A homomorphism of total fuzzy graph h: T(G) → T(G’) is a map h: S(T(G)) → S(T(G’)) satisfying the following two conditions, where G and G’ is fuzzy graphs from which the TFG is defined:

• σT(G)(a) ≤ σT(G’)(h(a)), ∀ aS(T(G)),

• μT(G)(a, b) ≤ μT(G’)(h(a), h(b)), ∀ a, bS(T(G)).

Example 1. Consider the following two TFGs given by T(G) = (σT(G), μT(G)) and T(G’) = (σT(G’), μT(G’)) with the given fuzzy vertices S(T(G)) and S(T(G’)). Let S(T(G)) = {a, b, c, ab, bc, ac}.

Suppose σT(G) be defined over S(T(G)) and it is as given here under:

σT(G)(a)=12,σT(G)(b)=14,σT(G)(c)=13,σT(G)(ab)=15,σT(G)(bc)=15,σT(G)(ac)=14.

Consider μT(G) which is defined over S(T(G)) × S(T(G)) as follows:

μT(G)(a,b)=15,μT(G)(b,c)=15,μT(G)(a,c)=14,μT(G)(a,ab)=15,μT(G)(a,ac)=14,μT(G)(b,ab)=15,μT(G)(b,bc)=15,μT(G)(c,bc)=15,μT(G)(c,ac)=14,μT(G)(ab,bc)=15,μT(G)(ab,ac)=15,μT(G)(bc,ac)=15.

Similarly, Let S(T(G’)) = {a’, b’, c’, a’b’, b’c’, a’c’}.

Again consider σT(G’) which is defined over the set, S(T(G’)) as follows:

σT(G)(a)=12,σT(G)(b)=14,σT(G)(c)=13,σT(G)(ab)=14,σT(G)(bc)=14,σT(G)(ac)=13.

For μT(G) we have it over S(T(G’)) × S(T(G’)) as;

μT(G)(a,b)=14,μT(G)(b,c)=14,μT(G)(a,c)=13,μT(G)(a,ab)=14,μT(G)(a,ac)=13,μT(G)(b,ab)=14,μT(G)(b,bc)=14,μT(G)(c,bc)=14,μT(G)(c,ac)=13,μT(G)(ab,bc)=14,μT(G)(ab,ac)=14,μT(G)(bc,ac)=14.

Our objective in this example is to define a homomorphism mapping h: T(G) → T(G’) which is a map h: S(T(G)) → S(T(G’)). Therefore, let us define h: S(T(G)) → S(T(G’)) by;

h(a)=a,h(b)=b,h(c)=c,h(ab)=ab,h(bc)=bc,h(ac)=ac,

Clearly, h is both one-to-one and onto. Hence, h is a bijective mapping. To show that it is a homomorphism, we need to check the two conditions of a homomorphism of TFG. These are;

• σT(G)(a) ≤ σT(G’)(h(a)), ∀ aS(T(G)) and

• μT(G)(a, b) ≤ μT(G’)(h(a), h(b)), ∀ a, bS(T(G)).

Thus, we have the following results.

12=σT(G)(a)σT(G)(h(a))=σT(G)(a)=12,14=σT(G)(b)σT(G)(h(b))=σT(G)(b)=14,13=σT(G)(c)σT(G)(h(c))=σT(G)(c)=13,15=σT(G)(ab)σT(G)(h(ab))=σT(G)(ab)=14,15=σT(G)(bc)σT(G)(h(bc))=σT(G)(bc)=14,14=σT(G)(ac)σT(G)(h(ac))=σT(G)(ac)=13.

Hence, σT(G)(a) ≤ σT(G’)(h(a)), ∀ aS(T(G)), and the first condition of homomorphism holds. The verification of the second condition will be as follows:

15=μT(G)(a,b)μT(G)(h(a),h(b))=μT(G)(a,b)=14,15=μT(G)(b,c)μT(G)(h(b),h(c))=μT(G)(b,c)=14,14=μT(G)(a,c)μT(G)(h(a),h(c))=μT(G)(a,c)=13,15=μT(G)(a,ab)μT(G)(h(a),h(ab))=μT(G)(a,ab)=14,15=μT(G)(a,ac)μT(G)(h(a),h(ac))=μT(G)(a,a,c)=13,15=μT(G)(b,ab)μT(G)(h(b),h(ab))=μT(G)(b,ab)=14,15=μT(G)(b,bc)μT(G)(h(b),h(bc))=μT(G)(b,bc)=14,15=μT(G)(c,bc)μT(G)(h(c),h(bc))=μT(G)(c,bc)=14,14=μT(G)(c,ac)μT(G)(h(c),h(ac))=μT(G)(c,ac)=13,15=μT(G)(ab,bc)μT(G)(h(ab),h(bc))=μT(G)(ab,bc)=14,15=μT(G)(ab,bc)μT(G)(h(ab),h(ac))=μT(G)(ab,ac)=14,15=μT(G)(bc,ac)μT(G)(h(bc),h(ac))=μT(G)(bc,ac)=14.

Hence, μT(G)(a, b) ≤ μT(G’)(h(a), h(b)), ∀ a, bS(T(G)). Therefore, h: T(G) → T(G’) defined on h: S(T(G)) → S(T(G’)) is a homomorphism of a TFG from T(G) onto T(G’). The graphs G and G’ and the graphs of their TFGs, T(G) and T(G’) are shown in Figures 1(a)–1(d), respectively.

Definition 12. A weak isomorphism h: T(G) → T(G’) is a mapping defined on h: S(T(G)) → S(T(G’)), such that h is a bijective homomorphism which satisfies the following condition:

σT(G)(a)=σT(G)(h(a)),   aS(T(G)).

Example 2. Consider the TFG T(G) and T(G’) as given in Example 1.

It can be easily observed that σT(G)(a) ≤ σT(G’)(h(a)), ∀ aS(T(G)). From this condition, the equality holds only for the vertex sets and the inequality holds for the vertices in the edge set of the fuzzy graph G. Hence, h: T(G) → T(G’) is not a weak isomorphism.

Definition 13. The co-weak isomorphism h: T(G) → T(G’) is a map which is bijective homomorphism that fulfills the following condition:

μT(G)(a,b)=μT(G)(h(a),h(b)),   a,bS(T(G)).

Example 3. Consider the following two TFGs T(G) and T(G’). Let the node sets be S(T(G)) = {a, b, c, ab, bc, ca} and S(T(G’)) = {a’, b’, c’, a’b’, b’c’, c’a’}.

Define σT(G): S(T(G)) → [0, 1], μT(G): S(T(G)) × S(T(G)), and σT(G): S(T(G’)) → [0, 1], μT(G): S(T(G’))× S(T(G’)) as follows:

• The fuzzy subsets of T(G):

σT(G)(a)=13,σT(G)(b)=12,σT(G)(c)=14,σT(G)(ab)=13,σT(G)(bc)=15,σT(G)(ca)=14.

• The fuzzy relation of T(G):

μT(G)(a,b)=13,μT(G)(a,c)=14,μT(G)(b,c)=15,μT(G)(a,ab)=13,μT(G)(a,ca)=14,μT(G)(b,ab)=13,μT(G)(b,bc)=15,μT(G)(c,bc)=15,μT(G)(c,ac)=14,μT(G)(ab,bc)=15,μT(G)(ab,ca)=14,μT(G)(bv,ca)=15.

• The fuzzy subsets of T(G’):

σT(G)(a)=12,σT(G)(b)=1,σT(G)(c)=14,σT(G)(ab)=13,σT(G)(bc)=15,σT(G)(ca)=14.

• The fuzzy relation of T(G’):

μT(G)(a,b)=13,μT(G)(a,c)=14,μT(G)(b,c)=15,μT(G)(a,ab)=13,μT(G)(a,ca)=14,μT(G)(b,ab)=13,μT(G)(b,bc)=15,μT(G)(c,ca)=14,μT(G)(c,bc)=15,μT(G)(ab,bc)=15,μT(G)(ab,ca)=14,μT(G)(bc,ca)=15.

Define h: S(T(G)) → S(T(G’)) by;

h(a)=a,h(b)=b,h(c)=c,h(ab)=ab,h(bc)=bc,h(ac)=ac.

Clearly, h is a bijective mapping. For the homomorphism property of h, we need to check the conditions σT(G)(a) ≤ σT(G’)(h(a)), ∀ aS(T(G)) and μT(G)(a, b) ≤ μT(G’)(h(a), h(b)), ∀ a, bS(T(G)).

Hence, we have the following:

13=σT(G)(a)σT(G)(h(a))=σT(G)(a)=12,12=σT(G)(b)σT(G)(h(b))=σT(G)(b)=1,14=σT(G)(c)σT(G)(h(c))=σT(G)(c)=14,13=σT(G)(ab)σT(G)(h(ab))=σT(G)(ab)=13,15=σT(G)(bc)σT(G)(h(bc))=σT(G)(bc)=15,14=σT(G)(ca)σT(G)(h(ca))=σT(G)(ca)=14.

These show that the first condition of homomorphism is illustrated and we need to show the second condition. Thus;

13=μT(G)(a,b)μT(G)(h(a),h(b))=μT(G)(a,b)=13,15=μT(G)(b,c)μT(G)(h(b),h(c))=μT(G)(b,c)=15,14=μT(G)(a,c)μT(G)(h(a),h(c))=μT(G)(a,c)=14,13=μT(G)(a,ab)μT(G)(h(a),h(ab))=μT(G)(a,ab)=13,14=μT(G)(a,ac)μT(G)(h(a),h(ac))=μT(G)(a,ac)=14,13=μT(G)(b,ab)μT(G)(h(b),h(ab))=μT(G)(b,ab)=13,15=μT(G)(b,bc)μT(G)(h(b),h(bc))=μT(G)(b,bc)=15,15=μT(G)(c,bc)μT(G)(h(c),h(bc))=μT(G)(c,bc)=15,14=μT(G)(c,ac)μT(G)(h(c),h(ac))=μT(G)(c,ac)=14,15=μT(G)(ab,bc)μT(G)(h(ab),h(bc))=μT(G)(ab,bc)=15,14=μT(G)(ab,ac)μT(G)(h(ab),h(ac))=μT(G)(ab,ac)=14,15=μT(G)(bc,ac)μT(G)(h(bc),h(ac))=μT(G)(bc,ac)=15.

Hence, the second condition of homomorphism also satisfied and h: T(G) → T(G’) defined in Example 3 is a homomorphism.

It can be easily observed that from the second condition of homomorphism, we have μT(G)(a, b) = μT(G’)(h(a), h(b)), ∀ a, bS(T(G)). This implies that h is a co-weak isomorphism.

Definition 14. An isomorphism h: T(G) → T(G’) is a bijective map h: S(T(G)) → S(T(G’)) which satisfies the following two conditions:

• σT(G)(a) = σT(G’)(h(a)), ∀ aS(T(G)).

• μT(G)(a, b) = μT(G’)(h(a), h(b)), ∀ a, bS(T(G)).

If an isomorphism from T(G) to T(G’) exists, then we say that T(G) is isomorphic to T(G’) and it is denoted by T(G) ≅= T(G’).

### 4. Properties of Isomorphism of TFG

Under this section, we introduce some theorems related to properties of isomorphism of total fuzzy graphs with their proofs.

Theorem 1. The order of any two isomorphic TFG is the same.

Proof. Let T(G) and T(G’) be two isomorphic TFGs with an isomorphism h: T(G) → T(G’) between them. Let the underlying set be S(T(G)) and S(T(G’)) respectively. Since h is an isomorphism, we have σT(G)(a) = σT(G’)(h(a)), ∀ aS(T(G)).

Thus;

Order(T(G))=aS(T(G))σT(G)(a)=aS(T(G))σT(G)(h(a))=Order(T(G)).

Theorem 2. The size of any two isomorphic TFG is the same.

Proof. Let h: T(G) → T(G’) be an isomorphism between the TFGs T(G) and T(G’) with the underlying sets S(T(G)) and S(T(G’)) respectively.

Since h is an isomorphism, we have

μT(G)(a,b)=μT(G)(h(a),h(b)),   a,bS(T(G)).

Thus;

Size(T(G))=a,bS(T(G))μT(G)(a,b)=a,bS(T(G))μT(G)(h(a),h(b))=Size(T(G)).

Theorem 3. An isomorphism of TFGs T(G) and T(G’) preserves the degree of the nodes.

Proof. Suppose h: T(G) → T(G’) be an isomorphism T(G) onto T(G’) such that μT(G)(a, b) = μT(G’)(h(a), h(b)), ∀ a, bS(T(G)).

From the definition of vertices of the TFG;

d(a)=abbS(T(G))μT(G)(a,b)=abbS(T(G))μT(G)(h(a),h(b))=d(h(a)).

Theorem 4. An isomorphism between TFG forms an equivalence relation.

Proof. Let T(G): (σT(G), μT(G)), T(G’): (σT(G’), μT(G’)) and T(G’): (σT(G’), μT(G’)) be TFGs with S(T(G)), S(T(G’)) and S(T(G’)) in their order.

• Reflexivity property

Let I: S(T(G)) → S(T(G)) be identity mapping such that I(a) = a for all aS(T(G)). This I is a bijective mapping which satisfies the conditions σT(G)(a) = σT(G)(I(a)) = a, ∀ aS(T(G)) and μT(G)(a, b) = μT(G)(I(a), I(b)) = (a, b), ∀ a, bS(T(G)).

Hence I is an isomorphism of TFG, T(G) onto itself. Therefore, I: S(T(G)) → S(T(G)) has reflexive property.

• Symmetric property

Let h: S(T(G)) → S(T(G’)) be an isomorphism T(G) onto T(G’). Clearly h is a bijective mapping with

h(a)=a,aS(T(G)).

Satisfying σT(G)(a) = σT(G)(h(a)) = a, ∀ aS(T(G)) and

μT(G)(a,b)=μT(G)(h(a),h(b))=(a,b),   a,bS(T(G)).

By using Eq. (1)h is bijective and h−1(a’) = a for all a’S(T(G’)). Using Eq. (2) we get;

σT(G)(h-1(a))=σT(G)((a)),   aS(T(G)),and μT(G)(h(a),h(b))=μT(G)(a,b),a,bS(T(G)).

These shows that the mapping Let h−1: S(T(G’)) → S(T(G)) is a one-to-one and onto mapping and it is an isomorphism from S(T(G’)) onto S(T(G)). Hence, T(G) ≅= T(G’) and it implies that T(G’) ≅= T(G). Therefore, the isomorphism of TFG has the symmetric property.

• Transitive property

Let h: S(T(G)) → S(T(G’)) and g: S(T(G’)) → S(T(G’)) be an isomorphism of T(G) onto T(G’) and T(G’) onto T(G’) respectively. Hence, (gh) is a one-to-one and onto map from S(T(G) onto S(T(G’)), where (gh)(a) = g(h(a)), ∀ aS(T(G)).

h: S(T(G)) → S(T(G’)) is an isomorphism such that h(a) = a’, aS(T(G)) with σT(G)(a)=σT(G’)(h(a)), ∀ aS(T(G))), and μT(G)(a, b) = μT(G’)(h(a), h(b)), ∀ a, bS(T(G)). That is;

σT(G)(a)=σT(G)(a),   aS(T(G)),μT(G)(a,b)=μT(G)(a,b),a,bS(T(G)).

Since g is an isomorphism from S(T(G’)) onto S(T(G’)), we have

g(a)=a,aS(T(G)),andσT(G)(a)=σT(G)(g(a)),aS(T(G)),μT(G)(a,b))=μT(G)(g(a),g(b)),a,bS(T(G)),a,bS(T(G)).

Thus, from Eqs. (4) and (6) and using h(a) = a’, aS(T(G)), we get

σT(G)(a)=σT(G)(a)=σT(G)(g(a)),   aS(T(G))=σT(G)(g(h(a))),   aS(T(G)).

Again, from Eqs. (5) and (7), we have

μT(G)(a,b)=μT(G)(a,b)),   a,bS(T(G))=μT(G)(g(a),g(b)),   a,bS(T(G))=μT(G)(g(h(a)),g(h(b))),   a,bS(T(G)).

This shows that gh is an isomorphism between T(G) and T(G’). Hence, an isomorphism of TFG satisfies the transitive property. Therefore, an isomorphism between TFG is an equivalence relation.

The article aimed to deal with the isomorphism of TFG. To reach to the goal the concepts of morphisms of fuzzy graphs and bipolar fuzzy graphs are recalled to support the need of the study. Having being studied about TFG, we need to extend its study to isomorphism of FG. Hence, this article defined the homomorphism of TFG and equipped it with justification by an example. This article introduces the concept of isomorphism of TFG. Further, the notion of weak homomorphism and co-weak homomorphism of TFG is presented with evidence and some relationship between homomorphism and week homomorphism is also shown in the examples. To the end the formal definition of an isomorphism of TFG is given and new results which arise from this definition are stated and proved.

### Limitations

The manuscript is limited to:

• Homomorphism of total fuzzy graphs.

• Weak and co-weak isomorphism of total fuzzy graphs.

• Isomorphism of total fuzzy graphs and their properties.

Theories and properties related with TFGs is well presented and discussed. The future intention of the researchers is to deal with isomorphic properties of 1-quasi and 2-quasi TFGs which are the new concepts of fuzzy graphs they studied recently. Currently the applications of isomorphic properties of fuzzy graphs are emerging and are associated with image visualization, social networking and communication networks as well as transportation networks. Hence, the researchers other future work is to investigate these real-life applications on isomorphism of TFGs and quasi TFGs to get more appropriate and better results.

The data used to in these findings are included within the manuscript.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Both authors contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.

Fig. 1.

(a) Fuzzy graph G. (b) T(G), total fuzzy graph of G. (c) Fuzzy graph G’. (d) T(G’), total fuzzy graph of G’.

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Fekadu Tesgera Agama is one of the staff of Wollega University who has received his PhD in Graph Theory and Combinatoric at Wollega University, Ethiopia. He published three articles at international peer reviewed journals, two of them are Scopus and Web of Science indexed. He has been graduated with his first degree B.Sc. in Mathematics from Addis Ababa University Ethiopia in 2001 and 2nd-degree M.Sc. in Mathematics from Bahir Dar University Ethiopia in 2011. In addition he got his master of business administration (MBA) from Wollega University in 2019. He had 20 years of teaching experience as a lecturer in Mathematics.

V.N.SrinivasaRao Reaplle has been working as an Associate Professor and Ph.D. advisor in the Department of Mathematics at Wollega University, Ethiopia since 2017. He completed his M.Sc. and Ph.D. in Mathematics from Acharya Nagarjuna University, Andhra Pradesh, India. He has 24 years of teaching and research experience. He published 24 research articles in internationally reputed journals. Further, he presented and published 19 articles at international national and national conferences. He was a life member in various professional bodies. Also, he was serving as a reviewer for various international journals. Two Students are graduated Ph.D. under his guidance and 4 students are working under his major advisory. His research interests are in the areas of Applied Mathematics, particularly in Graph Theory & Combinatory, Fuzzy Graph Theory, Discrete Mathematics and Lattice Theory.

### Article

#### Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(4): 382-390

Published online December 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.4.382

## The Isomorphism of Total Fuzzy Graphs

Fekadu Tesgera Agama and V. N. Srinivasa Rao Repalle

Department of Mathematics, College of Natural and Computational Sciences, Wollega University, Nekemte, Ethiopia

Received: February 3, 2022; Revised: June 6, 2022; Accepted: December 12, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This manuscript is aimed to deal with the isomorphism properties of total fuzzy graph. In order to achieve the desired objective, we consider two fuzzy graphs and their total fuzzy graphs. For these total fuzzy graphs, the notion of homomorphism of total fuzzy graphs is defined and the existence of a homomorphism between these two total fuzzy graphs is shown by examples. This is achieved by both sketching their graph and verifying the condition of homomorphism. Having a mapping being a homomorphism of the total fuzzy graphs the concept of weak isomorphism and co-weak isomorphism are presented with separate and distinct definition. These definitions are accompanied with supportive examples and the case where a homomorphism between two total fuzzy graphs is not weak-isomorphism is observed and illustrated by supportive example. In addition to these, the notion of co-weak isomorphism is also defined and illustrated by using a typical example. A very great attention is given to the definition of an isomorphism of total fuzzy graph and results arising from this definition. Accordingly, certain theorems related to isomorphic properties of total fuzzy graphs are stated and each of these theorems are thoroughly proved to show the results. Finally the manuscript states the future research works, limitations of the manuscript.

Keywords: Fuzzy graph, Total fuzzy graph, Homomorphism, Isomorphism, Week isomorphism

### 1. Introduction

Fuzzy sets and fuzzy relations are introduced and discussed well by Zadeh [1] in 1965. The emerging of these concepts attracted the attention of many scholars and forced them to associate with many research fields. Hence, notion goes beyond graph theory and took attention in the fields medicine, engineering, statistics, science of management and the like. The distinctions of this fuzzy set is that each element is associated with a membership value chosen from the interval [0, 1]. As a result of these, Rosenfeld [2] well-thought-out fuzzy relations on fuzzy sets and advanced the theory of fuzzy graphs in 1975. The notion of isomorphism of fuzzy graphs also followed the foot step and Perchant [3] familiarized a generic meaning of fuzzy morphism amongst graphs that comprises standard graph related definitions as graph and sub-graph isomorphism. In addition to these, Bhutani in [4] studied different types of isomorphism of fuzzy graphs and some graceful theorems on weak and co-weak isomorphism of m-polar fuzzy graph was studied by S. Satyanarayana & et al. [5].

Currently, the study on fuzzy graph is more inclined to the sub-part of a fuzzy graph called bipolar fuzzy graphs. Accordingly Poulik S. [6] have determined connectivity index and Wiener index in bipolar fuzzy graphs and in [7] the study disclosed that the totally accurate communication between all connected nodes is explained by introducing completely open neighborhood degree and completely closed neighborhood degree of nodes and edges in a bipolar fuzzy graph. Moreover, Soumitra Poulik and Ganesh Ghorai [8] described the empirical results on operations of bipolar fuzzy graphs with their degree. Other studies focus on the isomorphism of fuzzy graphs. Thus, Vijaya M. and Mekala B. [9] studied about weak isomorphism on bipolar total fuzzy graphs and put results related to bipolar TFG. In the book of “Modern Trends in Fuzzy Graphs” [10], the homomorphism products of two intuitionistic fuzzy graphs are presented and results related to these are also elaborated. Isomorphism in generalized fuzzy graphs has been introduced by Samanta [11] to capture the similarity of uncertainties in different networks. Homomorphism, weak isomorphism, co-weak isomorphism and nearly isomorphism are defined and an application of image visualization was described. A new method to explain the homeomorphic between some fuzzy topological graphs which will be applied in smart cities has been presented by Atefa et al. [12]. The idea of an isomorphic picture fuzzy graph as well as its application on a social network have been described by Zuo [13]. The notations of μ-complement, homomorphism, isomorphism, weak isomorphism and co weak isomorphism of regular picture fuzzy graph and mathematical model of communication network and transportation network by using picture fuzzy multigraph is also studies and its application on transportation network/communication network is also presented by Xiao [14]. The concept total fuzzy graph (TFG) was discussed by Nagoor Gani [15]. 2-qusi TFG definition, properties, and its coloring are well studied by Fekadu Tesgera Agama and R.V.N. Srinivasa Rao Repalle [16]. Some related studies on line graph of fuzzy graphs, homeomorphisms and others are also used for this study from [15]. The inverse fuzzy mixed graphs and its application is clearly introduce by Poulik and Ghorai [17]. In their study, they have discussed the isomorphic properties of inverse of mixed fuzzy graphs.

### 1.1 Objective of the Study

In this paper, we give the definition of homomorphism on TFG which is supported by an elaborative example. With this, we define and give examples of weak isomorphism, co-weak isomorphism, and an isomorphism of TFG. In addition, we state some properties regarding the isomorphism of total fuzzy graph and give the proof of these properties.

### 1.2 Organization of the Paper

This manuscript is organized in to four different sections. The first section deals with introduction and the second section discusses some basic terms or definitions required to discuss about the main concept of the manuscript in section three. In section four we present some properties of the isomorphism of total fuzzy graphs and lastly we provide the conclusion to the study.

### 2. Preliminaries

Under this sub-topic we present definitions of fuzzy graphs and TFG which are necessary for this article.

Definition 1 [18]. A fuzzy graph G: (σ, μ) is a triple consisting of a non-empty set V along with the functions σ: V → [0, 1] and μ: V × V → [0, 1] such that for all u, vV, μ(u, v) ≤ σ(u) ∧ σ(v), where ∧ denotes the minimum.

Definition 2 [18]. Let G: (σ, μ) be a fuzzy graph. The order (Order(G)) of G is defined as:

$Order(G)=∑u∈Vσ(u),$

and the size (Size(u)) of G denoted by

$Size(u)=∑u,v∈Vμ(u,v).$

Definition 3 [18]. Let G: (σ, μ) be fuzzy graph. Let uV be any vertex of G, then the degree of u is defined as; dG(u) = ∑vu, vV μ(u, v).

Definition 4 [4]. Let g: GG’ be a map g: SS’ which satisfies the following two conditions:

• σ(u) ≤ σ’ (g(u)), ∀ uS and

• μ(u, v) ≤ μ’ (g(u), g(v)), ∀ u, vS where σ and σ’ are membership functions of vertices, μ and μ’ are membership functions of edges, then g is called a homomorphism of fuzzy graphs.

Definition 5 [4]. Suppose g: GG’ is a mapping. If g: SS’ of fuzzy graphs is a bijective homomorphism satisfying σ(u) = σ’ (g(u)), ∀ uS, then g is called a weak-isomorphism.

Definition 6 [4]. Let g: GG’ be a map, g: SS’ which is a bijective homomorphism that satisfies μ(x, y) = μ’ (g(x), g(y)) ∀ x, yS, and then g is called co-weak isomorphism.

Definition 7 [18]. A fuzzy relation μ on a fuzzy subset σ of a set S is said to be a fuzzy equivalence relation if it is reflexive, symmetric, and transitive.

Definition 8 [18]. Let G = (μ, σ) be a fuzzy graph, then μ is called reflexive if μ(u, u) = σ(u) for all uV.

Definition 9 [18]. For a fuzzy graph, G: (σ, μ), μ is said to be symmetric when μ(u, v) = μ(v, u) ∀ u and it is said to satisfy transitivity property when μ2μ.

Definition 10 [19]. Suppose G: (σ, μ) as a fuzzy graph. The TFG is a pair T(G) = (σT, μT ) of G and;

• σT is defined over VE such that σT (u) = σ(u), if uV, μT (e) = μ(e), if eE.

• μT is described as;

μT (u, v) = μ(u, v), if u, vV,

μT (u, e) = σ(u)∧ μ(e), if uV, eE, and the vertex

u lies on the edge e,

μT (u, e) = 0, otherwise,

μT (ei, ej) = μ(ei) ∧ μ(ej), if the edges ei and ej have a node in common between them, μT (ei, ej) = 0, otherwise.

### 3. Isomorphism of TFG

This section introduces the homomorphism and isomorphism of TFG. Furthermore, different types of isomorphism such as weak isomorphism and co-weak isomorphism of TFG are defined and elaborated by examples.

Definition 11. A homomorphism of total fuzzy graph h: T(G) → T(G’) is a map h: S(T(G)) → S(T(G’)) satisfying the following two conditions, where G and G’ is fuzzy graphs from which the TFG is defined:

• σT(G)(a) ≤ σT(G’)(h(a)), ∀ aS(T(G)),

• μT(G)(a, b) ≤ μT(G’)(h(a), h(b)), ∀ a, bS(T(G)).

Example 1. Consider the following two TFGs given by T(G) = (σT(G), μT(G)) and T(G’) = (σT(G’), μT(G’)) with the given fuzzy vertices S(T(G)) and S(T(G’)). Let S(T(G)) = {a, b, c, ab, bc, ac}.

Suppose σT(G) be defined over S(T(G)) and it is as given here under:

$σT(G)(a)=12, σT(G)(b)=14, σT(G)(c)=13,σT(G)(ab)=15, σT(G)(bc)=15, σT(G)(ac)=14.$

Consider μT(G) which is defined over S(T(G)) × S(T(G)) as follows:

$μT(G)(a,b)=15, μT(G)(b,c)=15, μT(G)(a,c)=14,μT(G)(a,ab)=15, μT(G)(a,ac)=14, μT(G)(b,ab)=15,μT(G)(b,bc)=15, μT(G)(c,bc)=15, μT(G)(c,ac)=14,μT(G)(ab,bc)=15, μT(G)(ab,ac)=15,μT(G)(bc,ac)=15.$

Similarly, Let S(T(G’)) = {a’, b’, c’, a’b’, b’c’, a’c’}.

Again consider σT(G’) which is defined over the set, S(T(G’)) as follows:

$σT(G′)(a′)=12, σT(G′)(b′)=14, σT(G′)(c′)=13,σT(G′)(a′b′)=14, σT(G′)(b′c′)=14, σT(G′)(a′c′)=13.$

For μT(G) we have it over S(T(G’)) × S(T(G’)) as;

$μT(G′)(a′,b′)=14, μT(G′)(b′,c′)=14,μT(G′)(a′,c′)=13, μT(G′)(a′,a′b′)=14,μT(G′)(a′,a′c′)=13, μT(G′)(b′,a′b′)=14,μT(G′)(b′,b′c′)=14, μT(G′)(c′,b′c′)=14,μT(G′)(c′,a′c′)=13, μT(G′)(a′b′,b′c′)=14,μT(G′)(a′b′,a′c′)=14, μT(G′)(b′c′,a′c′)=14.$

Our objective in this example is to define a homomorphism mapping h: T(G) → T(G’) which is a map h: S(T(G)) → S(T(G’)). Therefore, let us define h: S(T(G)) → S(T(G’)) by;

$h(a)=a′, h(b)=b′, h(c′)=c′, h(ab)=a′b′,h(bc)=b′c′, h(ac)=a′c′,$

Clearly, h is both one-to-one and onto. Hence, h is a bijective mapping. To show that it is a homomorphism, we need to check the two conditions of a homomorphism of TFG. These are;

• σT(G)(a) ≤ σT(G’)(h(a)), ∀ aS(T(G)) and

• μT(G)(a, b) ≤ μT(G’)(h(a), h(b)), ∀ a, bS(T(G)).

Thus, we have the following results.

$12= σT(G)(a)≤σT(G′)(h(a))= σT(G′)(a′)=12,14= σT(G)(b)≤σT(G′)(h(b))= σT(G′)(b′)=14,13= σT(G)(c)≤σT(G′)(h(c))= σT(G′)(c′)=13,15= σT(G)(ab)≤σT(G′)(h(ab))= σT(G′)(a′b′)=14,15= σT(G)(bc)≤σT(G′)(h(bc))= σT(G′)(b′c′)=14,14= σT(G)(ac)≤σT(G′)(h(ac))= σT(G′)(a′c′)=13.$

Hence, σT(G)(a) ≤ σT(G’)(h(a)), ∀ aS(T(G)), and the first condition of homomorphism holds. The verification of the second condition will be as follows:

$15 =μT(G)(a,b)≤μT(G′)(h(a),h(b)) =μT(G′)(a′,b′)=14,15 =μT(G)(b,c)≤μT(G′)(h(b),h(c)) =μT(G′)(b′,c′)=14,14 =μT(G)(a,c)≤μT(G′)(h(a),h(c)) =μT(G′)(a′,c′)=13,15 =μT(G)(a,ab)≤μT(G′)(h(a),h(ab)) =μT(G′)(a′,a′b′)=14,15 =μT(G)(a,ac)≤μT(G′)(h(a),h(ac)) =μT(G′)(a′,a′,c′)=13,15 =μT(G)(b,ab)≤μT(G′)(h(b),h(ab)) =μT(G′)(b′,a′b′)=14,15 =μT(G)(b,bc)≤μT(G′)(h(b),h(bc)) =μT(G′)(b′,b′c′)=14,15 =μT(G)(c,bc)≤μT(G′)(h(c),h(bc)) =μT(G′)(c′,b′c′)=14,14 =μT(G)(c,ac)≤μT(G′)(h(c),h(ac)) =μT(G′)(c′,a′c′)=13,15 =μT(G)(ab,bc)≤μT(G′)(h(ab),h(bc)) =μT(G′)(a′b′,b′c′)=14,15 =μT(G)(ab,bc)≤μT(G′)(h(ab),h(ac)) =μT(G′)(a′b′,a′c′)=14,15 =μT(G)(bc,ac)≤μT(G′)(h(bc),h(ac)) =μT(G′)(b′c′,a′c′)=14.$

Hence, μT(G)(a, b) ≤ μT(G’)(h(a), h(b)), ∀ a, bS(T(G)). Therefore, h: T(G) → T(G’) defined on h: S(T(G)) → S(T(G’)) is a homomorphism of a TFG from T(G) onto T(G’). The graphs G and G’ and the graphs of their TFGs, T(G) and T(G’) are shown in Figures 1(a)–1(d), respectively.

Definition 12. A weak isomorphism h: T(G) → T(G’) is a mapping defined on h: S(T(G)) → S(T(G’)), such that h is a bijective homomorphism which satisfies the following condition:

$σT(G)(a)=σT(G′)(h(a)), ∀a∈S(T(G)).$

Example 2. Consider the TFG T(G) and T(G’) as given in Example 1.

It can be easily observed that σT(G)(a) ≤ σT(G’)(h(a)), ∀ aS(T(G)). From this condition, the equality holds only for the vertex sets and the inequality holds for the vertices in the edge set of the fuzzy graph G. Hence, h: T(G) → T(G’) is not a weak isomorphism.

Definition 13. The co-weak isomorphism h: T(G) → T(G’) is a map which is bijective homomorphism that fulfills the following condition:

$μT(G)(a,b)=μT(G′)(h(a),h(b)), ∀a, b∈S(T(G)).$

Example 3. Consider the following two TFGs T(G) and T(G’). Let the node sets be S(T(G)) = {a, b, c, ab, bc, ca} and S(T(G’)) = {a’, b’, c’, a’b’, b’c’, c’a’}.

Define σT(G): S(T(G)) → [0, 1], μT(G): S(T(G)) × S(T(G)), and σT(G): S(T(G’)) → [0, 1], μT(G): S(T(G’))× S(T(G’)) as follows:

• The fuzzy subsets of T(G):

$σT(G)(a)=13, σT(G)(b)=12,σT(G)(c)=14, σT(G)(ab)=13,σT(G)(bc)=15, σT(G)(ca)=14.$

• The fuzzy relation of T(G):

$μT(G)(a,b)=13, μT(G)(a,c)=14,μT(G)(b,c)=15, μT(G)(a,ab)=13,μT(G)(a,ca)=14, μT(G)(b,ab)=13,μT(G)(b,bc)=15, μT(G)(c,bc)=15,μT(G)(c,ac)=14, μT(G)(ab,bc)=15,μT(G)(ab,ca)=14, μT(G)(bv,ca)=15.$

• The fuzzy subsets of T(G’):

$σT(G′)(a′)=12, σT(G′)(b′)=1,σT(G′)(c′)=14, σT(G′)(a′b′)=13,σT(G′)(b′c′)=15, σT(G′)(c′a′)=14.$

• The fuzzy relation of T(G’):

$μT(G′)(a′,b′)=13, μT(G′)(a′,c′)=14,μT(G′)(b′,c′)=15, μT(G′)(a′,a′b′)=13,μT(G′)(a′,c′a′)=14, μT(G′)(b′,a′b′)=13,μT(G′)(b′,b′c′)=15, μT(G′)(c′,c′a′)=14,μT(G′)(c′,b′c′)=15, μT(G′)(a′b′,b′c′)=15,μT(G′)(a′b′,c′a′)=14, μT(G′)(b′c′,c′a′)=15.$

Define h: S(T(G)) → S(T(G’)) by;

$h(a)=a′, h(b)=b′, h(c′)=c′, h(ab)=a′b′,h(bc)=b′c′, h(ac)=a′c′.$

Clearly, h is a bijective mapping. For the homomorphism property of h, we need to check the conditions σT(G)(a) ≤ σT(G’)(h(a)), ∀ aS(T(G)) and μT(G)(a, b) ≤ μT(G’)(h(a), h(b)), ∀ a, bS(T(G)).

Hence, we have the following:

$13= σT(G)(a)≤σT(G′)(h(a))= σT(G′)(a′)=12,12= σT(G)(b)≤σT(G′)(h(b))= σT(G′)(b′)=1,14= σT(G)(c)≤σT(G′)(h(c))= σT(G′)(c′)=14,13= σT(G)(ab)≤σT(G′)(h(ab))= σT(G′)(a′b′)=13,15= σT(G)(bc)≤σT(G′)(h(bc))= σT(G′)(b′c′)=15,14= σT(G)(ca)≤σT(G′)(h(ca))= σT(G′)(c′a′)=14.$

These show that the first condition of homomorphism is illustrated and we need to show the second condition. Thus;

$13 =μT(G)(a,b)≤μT(G′)(h(a),h(b)) =μT(G′)(a′,b′)=13,15 =μT(G)(b,c)≤μT(G′)(h(b),h(c)) =μT(G′)(b′,c′)=15,14 =μT(G)(a,c)≤μT(G′)(h(a),h(c)) =μT(G′)(a′,c′)=14,13 =μT(G)(a,ab)≤μT(G′)(h(a),h(ab)) =μT(G′)(a′,a′b′)=13,14 =μT(G)(a,ac)≤μT(G′)(h(a),h(ac)) =μT(G′)(a′,a′c′)=14,13 =μT(G)(b,ab)≤μT(G′)(h(b),h(ab)) =μT(G′)(b′,a′b′)=13,15 =μT(G)(b,bc)≤μT(G′)(h(b),h(bc)) =μT(G′)(b′,b′c′)=15,15 =μT(G)(c,bc)≤μT(G′)(h(c),h(bc)) =μT(G′)(c′,b′c′)=15,14 =μT(G)(c,ac)≤μT(G′)(h(c),h(ac)) =μT(G′)(c′,a′c′)=14,15 =μT(G)(ab,bc)≤μT(G′)(h(ab),h(bc)) =μT(G′)(a′b′,b′c′)=15,14 =μT(G)(ab,ac)≤μT(G′)(h(ab),h(ac)) =μT(G′)(a′b′,a′c′)=14,15 =μT(G)(bc,ac)≤μT(G′)(h(bc),h(ac)) =μT(G′)(b′c′,a′c′)=15.$

Hence, the second condition of homomorphism also satisfied and h: T(G) → T(G’) defined in Example 3 is a homomorphism.

It can be easily observed that from the second condition of homomorphism, we have μT(G)(a, b) = μT(G’)(h(a), h(b)), ∀ a, bS(T(G)). This implies that h is a co-weak isomorphism.

Definition 14. An isomorphism h: T(G) → T(G’) is a bijective map h: S(T(G)) → S(T(G’)) which satisfies the following two conditions:

• σT(G)(a) = σT(G’)(h(a)), ∀ aS(T(G)).

• μT(G)(a, b) = μT(G’)(h(a), h(b)), ∀ a, bS(T(G)).

If an isomorphism from T(G) to T(G’) exists, then we say that T(G) is isomorphic to T(G’) and it is denoted by T(G) ≅= T(G’).

### 4. Properties of Isomorphism of TFG

Under this section, we introduce some theorems related to properties of isomorphism of total fuzzy graphs with their proofs.

Theorem 1. The order of any two isomorphic TFG is the same.

Proof. Let T(G) and T(G’) be two isomorphic TFGs with an isomorphism h: T(G) → T(G’) between them. Let the underlying set be S(T(G)) and S(T(G’)) respectively. Since h is an isomorphism, we have σT(G)(a) = σT(G’)(h(a)), ∀ aS(T(G)).

Thus;

$Order(T(G))=∑a∈S(T(G))σT(G)(a)=∑a∈S(T(G))σT(G′)(h(a))=Order(T(G′)).$

Theorem 2. The size of any two isomorphic TFG is the same.

Proof. Let h: T(G) → T(G’) be an isomorphism between the TFGs T(G) and T(G’) with the underlying sets S(T(G)) and S(T(G’)) respectively.

Since h is an isomorphism, we have

$μT(G)(a,b)=μT(G′)(h(a),h(b)), ∀a, b∈S(T(G)).$

Thus;

$Size(T(G))=∑a,b∈S(T(G))μT(G)(a,b)=∑a,b∈S(T(G))μT(G′)(h(a),h(b))=Size(T(G′)).$

Theorem 3. An isomorphism of TFGs T(G) and T(G’) preserves the degree of the nodes.

Proof. Suppose h: T(G) → T(G’) be an isomorphism T(G) onto T(G’) such that μT(G)(a, b) = μT(G’)(h(a), h(b)), ∀ a, bS(T(G)).

From the definition of vertices of the TFG;

$d(a)=∑a≠bb∈S(T(G))μT(G)(a,b)=∑a≠bb∈S(T(G))μT(G′)(h(a),h(b))=d(h(a)).$

Theorem 4. An isomorphism between TFG forms an equivalence relation.

Proof. Let T(G): (σT(G), μT(G)), T(G’): (σT(G’), μT(G’)) and T(G’): (σT(G’), μT(G’)) be TFGs with S(T(G)), S(T(G’)) and S(T(G’)) in their order.

• Reflexivity property

Let I: S(T(G)) → S(T(G)) be identity mapping such that I(a) = a for all aS(T(G)). This I is a bijective mapping which satisfies the conditions σT(G)(a) = σT(G)(I(a)) = a, ∀ aS(T(G)) and μT(G)(a, b) = μT(G)(I(a), I(b)) = (a, b), ∀ a, bS(T(G)).

Hence I is an isomorphism of TFG, T(G) onto itself. Therefore, I: S(T(G)) → S(T(G)) has reflexive property.

• Symmetric property

Let h: S(T(G)) → S(T(G’)) be an isomorphism T(G) onto T(G’). Clearly h is a bijective mapping with

$h(a)=a′, a∈S(T(G)).$

Satisfying σT(G)(a) = σT(G)(h(a)) = a, ∀ aS(T(G)) and

$μT(G)(a,b)=μT(G)(h(a),h(b))=(a,b), ∀a, b∈S(T(G)).$

By using Eq. (1)h is bijective and h−1(a’) = a for all a’S(T(G’)). Using Eq. (2) we get;

$σT(G)(h-1(a′))=σT(G′)((a′)), ∀a′∈S(T(G′)),and μT(G)(h′(a′),h′(b′))=μT(G′)(a′,b′),∀a, b∈S(T(G′)).$

These shows that the mapping Let h−1: S(T(G’)) → S(T(G)) is a one-to-one and onto mapping and it is an isomorphism from S(T(G’)) onto S(T(G)). Hence, T(G) ≅= T(G’) and it implies that T(G’) ≅= T(G). Therefore, the isomorphism of TFG has the symmetric property.

• Transitive property

Let h: S(T(G)) → S(T(G’)) and g: S(T(G’)) → S(T(G’)) be an isomorphism of T(G) onto T(G’) and T(G’) onto T(G’) respectively. Hence, (gh) is a one-to-one and onto map from S(T(G) onto S(T(G’)), where (gh)(a) = g(h(a)), ∀ aS(T(G)).

h: S(T(G)) → S(T(G’)) is an isomorphism such that h(a) = a’, aS(T(G)) with σT(G)(a)=σT(G’)(h(a)), ∀ aS(T(G))), and μT(G)(a, b) = μT(G’)(h(a), h(b)), ∀ a, bS(T(G)). That is;

$σT(G)(a)=σT(G′)(a′), ∀a∈S(T(G)),$$μT(G)(a,b)=μT(G′)(a′,b′), ∀a, b∈S(T(G)).$

Since g is an isomorphism from S(T(G’)) onto S(T(G’)), we have

$g(a′)=a″, a′∈S(T(G′)), andσT(G′)(a′)=σT(G″)(g(a′)), ∀a′S(T(G′)),$$μT(G′)(a′,b′))=μT(G″)(g(a′),g(b′)),∀a′, b′∈S(T(G′)), ∀a, b∈S(T(G)).$

Thus, from Eqs. (4) and (6) and using h(a) = a’, aS(T(G)), we get

$σT(G)(a)=σT(G′)(a′)=σT(G″)(g(a′)), ∀a′∈S(T(G′))=σT(G″)(g(h(a))), ∀a∈S(T(G)).$

Again, from Eqs. (5) and (7), we have

$μT(G)(a,b)=μT(G′)(a′,b′)), ∀a, b∈S(T(G))=μT(G″)(g(a′),g(b′)), ∀a′, b′∈S(T(G′))=μT(G″)(g(h(a)),g(h(b))), ∀a, b∈S(T(G)).$

This shows that gh is an isomorphism between T(G) and T(G’). Hence, an isomorphism of TFG satisfies the transitive property. Therefore, an isomorphism between TFG is an equivalence relation.

### 5. Conclusion

The article aimed to deal with the isomorphism of TFG. To reach to the goal the concepts of morphisms of fuzzy graphs and bipolar fuzzy graphs are recalled to support the need of the study. Having being studied about TFG, we need to extend its study to isomorphism of FG. Hence, this article defined the homomorphism of TFG and equipped it with justification by an example. This article introduces the concept of isomorphism of TFG. Further, the notion of weak homomorphism and co-weak homomorphism of TFG is presented with evidence and some relationship between homomorphism and week homomorphism is also shown in the examples. To the end the formal definition of an isomorphism of TFG is given and new results which arise from this definition are stated and proved.

### Limitations

The manuscript is limited to:

• Homomorphism of total fuzzy graphs.

• Weak and co-weak isomorphism of total fuzzy graphs.

• Isomorphism of total fuzzy graphs and their properties.

### 6. FutureWork

Theories and properties related with TFGs is well presented and discussed. The future intention of the researchers is to deal with isomorphic properties of 1-quasi and 2-quasi TFGs which are the new concepts of fuzzy graphs they studied recently. Currently the applications of isomorphic properties of fuzzy graphs are emerging and are associated with image visualization, social networking and communication networks as well as transportation networks. Hence, the researchers other future work is to investigate these real-life applications on isomorphism of TFGs and quasi TFGs to get more appropriate and better results.

### Fig 1.

Figure 1.

(a) Fuzzy graph G. (b) T(G), total fuzzy graph of G. (c) Fuzzy graph G’. (d) T(G’), total fuzzy graph of G’.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 382-390https://doi.org/10.5391/IJFIS.2022.22.4.382

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