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, v ∈ V, 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:

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

Definition 3 [18]. Let G: (σ, μ) be fuzzy graph. Let u ∈ V be any vertex of G, then the degree of u is defined as; d_G(u) = ∑_{v ≠ u, v ∈ V} μ(u, v).

Definition 4 [4]. Let g: G → G’ be a map g: S → S’ which satisfies the following two conditions:

• σ(u) ≤ σ’ (g(u)), ∀ u ∈ S and

• μ(u, v) ≤ μ’ (g(u), g(v)), ∀ u, v ∈ S 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: G → G’ is a mapping. If g: S → S’ of fuzzy graphs is a bijective homomorphism satisfying σ(u) = σ’ (g(u)), ∀ u ∈ S, then g is called a weak-isomorphism.

Definition 6 [4]. Let g: G → G’ be a map, g: S → S’ which is a bijective homomorphism that satisfies μ(x, y) = μ’ (g(x), g(y)) ∀ x, y ∈ S, 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 u ∈ V.

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 μ² ⊆ μ.

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

• σ_T is defined over V ∪ E such that σ_T (u) = σ(u), if u ∈ V, μ_T (e) = μ(e), if e ∈ E.

• μ_T is described as;

  μ_T (u, v) = μ(u, v), if u, v ∈ V,

  μ_T (u, e) = σ(u)∧ μ(e), if u ∈ V, e ∈ E, and the vertex

  u lies on the edge e,

  μ_T (u, e) = 0, otherwise,

μ_T (e_i, e_j) = μ(e_i) ∧ μ(e_j), if the edges e_i and e_j have a node in common between them, μ_T (e_i, e_j) = 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:

• σ_T(G)(a) ≤ σ_T(G’)(h(a)), ∀ a ∈ S(T(G)),

• μ_T(G)(a, b) ≤ μ_T(G’)(h(a), h(b)), ∀ a, b ∈ S(T(G)).

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

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

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

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

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

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

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;

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;

• σ_T(G)(a) ≤ σ_T(G’)(h(a)), ∀ a ∈ S(T(G)) and

• μ_T(G)(a, b) ≤ μ_T(G’)(h(a), h(b)), ∀ a, b ∈ S(T(G)).

Thus, we have the following results.

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

Hence, μ_T(G)(a, b) ≤ μ_T(G’)(h(a), h(b)), ∀ a, b ∈ S(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:

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

It can be easily observed that σ_T(G)(a) ≤ σ_T(G’)(h(a)), ∀ a ∈ S(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:

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 σ_T(G): S(T(G)) → [0, 1], μ_T(G): S(T(G)) × S(T(G)), and σ_T(G): S(T(G’)) → [0, 1], μ_T(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;

Clearly, h is a bijective mapping. For the homomorphism property of h, we need to check the conditions σ_T(G)(a) ≤ σ_T(G’)(h(a)), ∀ a ∈ S(T(G)) and μ_T(G)(a, b) ≤ μ_T(G’)(h(a), h(b)), ∀ a, b ∈ S(T(G)).

Hence, we have the following:

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

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 μ_T(G)(a, b) = μ_T(G’)(h(a), h(b)), ∀ a, b ∈ S(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:

• σ_T(G)(a) = σ_T(G’)(h(a)), ∀ a ∈ S(T(G)).

• μ_T(G)(a, b) = μ_T(G’)(h(a), h(b)), ∀ a, b ∈ S(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’).

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 σ_T(G)(a) = σ_T(G’)(h(a)), ∀ a ∈ S(T(G)).

Thus;

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

Thus;

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 μ_T(G)(a, b) = μ_T(G’)(h(a), h(b)), ∀ a, b ∈ S(T(G)).

From the definition of vertices of the TFG;

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

Proof. Let T(G): (σ_T(G), μ_T(G)), T(G’): (σ_T(G’), μ_T(G’)) and T(G’): (σ_T(G’), μ_T(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 a ∈ S(T(G)). This I is a bijective mapping which satisfies the conditions σ_T(G)(a) = σ_T(G)(I(a)) = a, ∀ a ∈ S(T(G)) and μ_T(G)(a, b) = μ_T(G)(I(a), I(b)) = (a, b), ∀ a, b ∈ S(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

  (1)h(a)=a′, a∈S(T(G)).

  Satisfying σ_T(G)(a) = σ_T(G)(h(a)) = a, ∀ a ∈ S(T(G)) and

  (2)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;

  (3)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, (g ○ h) is a one-to-one and onto map from S(T(G) onto S(T(G’)), where (g ○ h)(a) = g(h(a)), ∀ a ∈ S(T(G)).

h: S(T(G)) → S(T(G’)) is an isomorphism such that h(a) = a’, a∈S(T(G)) with σ_T(G)(a)=σ_T(G’)(h(a)), ∀ a ∈ S(T(G))), and μ_T(G)(a, b) = μ_T(G’)(h(a), h(b)), ∀ a, b ∈ S(T(G)). That is;

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

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

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

This shows that g ○ h 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.