Intuitionistic fuzzy incidence graphs

1 Introduction

In 1736, Euler first proposed the idea of graph theory. In the history of mathematics, the solution provided by Euler of the famous Konigsberg bridge problem is considered to be the first theorem of graph theory. The graph concept is one of the most excellent and widely employed tools for multiple real-world problem representation, modeling, and analysis. To indicate the objects and their relations between themselves, the graph vertices or nodes and edges or arcs are applied, respectively. The idea of the graph is a handy tool for solving combinatorial problems in different areas such as geometry, algebra, number theory, topology, operations research, optimization, and computer science. Gurevich [10] discussed classes of infinite loaded graphs with randomly deleted edges.

There is a drawback of graphs because they are silent when there are unpredictability and vagueness among relationships of different objects. This disadvantage of graphs makes way for Rosenfeld [30] to present an excellent idea of fuzzy graphs (FGs). The study of FGs opens a new door for many researchers to take part in this field, such as Bhutani and Rosenfeld [6] discussed strong arcs in FGs. Later, Mathew and Sunitha [15] examined three types of arcs depending on the strength of the arcs, such as α - strong, β - strong, and δ - arc. Poulik and Ghorai [28] introduced the Wiener index for a bipolar fuzzy graph (BFG) along with its certain properties. They also created the term Wiener absolute index based on the total accurate connectivity between all the pair of vertices and in the whole BFG. They talked about the behavior of Wiener absolute index in various BFGs such as bipolar fuzzy forest, bridge, and tree. Characterization of bipolar fuzzy detour g-eccentric node, bipolar fuzzy detour g-boundary nodes, and bipolar fuzzy detour g-interior nodes in a BFG was examined by Poulik and Ghorai [24]. They also explored some properties of bipolar fuzzy detour g-boundary nodes, bipolar fuzzy detour g-interior nodes, and bipolar fuzzy complete nodes. They also provided an application of detour g-distance, detour g-boundary node, detour g-interior node. Poulik and Ghorai [25] presented different types of indices like connectivity index and average connectivity index of BFG. They also initiated an idea of different types of nodes like bipolar fuzzy connectivity-enhancing node, bipolar fuzzy connectivity-reducing node, and bipolar fuzzy connectivity-neutral node with their properties. Poulik et al. [27] applied an interval-valued fuzzy graph (IVFG) and degree of vertices for performance evaluation in an educational system. They also provided a real-life application using IVFG in the education system in Taiwan. Poulik and Ghorai [26] presented a note on BFGs with an application.

Fuzzy sets (FSs) have some limitations; they give the degree of membership of an element in a given set but do not give any clue about the degree of non-membership of an element in a set. This deficiency in FSs, encourages Atanassov [5] to initiate the notion of intuitionistic fuzzy sets (IFSs). IFSs give both a degree of membership and a degree of non-membership, which are more or less independent of each other; the only requirement is that the sum of these two degrees is not greater than 1. Shannon and Atanassov [32] introduced intuitionistic fuzzy graphs (IFGs) as a generalization of FGs. Later, many researchers worked on IFGs such as Parvathi, and Karunambigai [23] gave certain properties of IFGs. Akram and Davvaz [2] proposed an innovative idea of strong IFGs, self-complementary, and self weak complementary strong IFGs. Akram and Alshehri [1] introduced various types of intuitionistic fuzzy bridges, intuitionistic fuzzy cut-vertices, intuitionistic fuzzy cycles, intuitionistic fuzzy trees in IFGs and investigate some of their interesting properties. Akram and Dudek [3] determined different properties of the sequence of crisp structures that characterize a given property of the intuitionistic fuzzy structure. Alshehri and Akram [4] applied IFSs to multi-graphs, planar graphs, and dual graphs along with their magnetic properties. They also presented an idea of isomorphism between intuitionistic fuzzy planar graphs. For a more deep and comprehensive study on IFGs, we may suggest to the reader [11, 31].

FGs and IFGs cannot give any hint of the impact of a vertex on edge. This deficiency in FGs and IFGs caused to the introduction of FIGs. Dinesh [8] was the first who introduced the fundamental idea of FIGs. Mordeson and Mathew [16] developed a relationship between weak incidence pairs and fuzzy incidence end nodes. Fang et al. [9] presented the idea of connectivity index and Wiener index of FIGs. Malik et al. [12] applied FIGs to the problems involving human trafficking. Mathew et al. [13] proposed an idea of node connectivity and arc connectivity in FIGs. Their relationships with fuzzy connectivity parameters are obtained. Mordeson et al. [17] initiated the notion of a vague incidence graph and its eccentricity. They applied the results to problems including illegal immigration and illegal transfer of people. Mordeson et al. [18] introduced the notion of the degree of incidence of a vertex and an edge in a FG theory. They concentrated on incidence, where the edge is adjacent to the vertex. They examined results related to bridges, cut vertices, cut pairs, fuzzy incidence paths, fuzzy incidence trees for FIGs. For a more detailed study on FIGs, we may refer to the reader [7]. The idea of order, size, domination, strong fuzzy incidence domination, weak fuzzy incidence domination, and a relationship between strong fuzzy incidence domination and weak fuzzy incidence domination for FIGs were presented by Nazeer et al. [20]. Nazeer et al. [21] proposed the idea of domination in the join of FIGs along with an application in the trading system of different countries. Nazeer and Rashid [22] introduced Picture FIGs. A variety of operations in IFIGs, including Cartesian product (CP), composition, tensor product, and normal product, were proposed by Nazeer et al. [19]. They also discussed the application of CP and the composition of two IFIGs in the textile industry.

Mathew and Mordeson [14] presented connectivity concepts in FIGs such as α-SP, β-SP, δ pair, fuzzy incidence tree, and fuzzy incidence cycle. This motivates us to introduce these ideas for IFIGs because these concepts are unknown for IFIGs. In the future, these ideas will help us to study the different properties of IFIGs in detail. The IFIG is a more realistic approach than FIGs because of the degree of non-membership.

The remainder of this article is formulated as follows. Section 2 provides some preliminary results which are required to understand the remaining part of the article. The elementary concepts and notions of IFIGs are discussed in Section 3. In Section 4, some results of connectivity in IFIG with numerical examples are proved. Section 5 contains a real-life application of the network of different computers. Conclusions and prospects are elaborated in Section 6.

2 Preliminaries

Definition 2.1. [20] A mapping μ : T → [0, 1] is named as a fuzzy subset (FSS) of T. A FG with M as the underlying set is a pair G = (σ, χ), where σ : M → [0, 1] is FSS, χ : M × M → [0, 1] is a fuzzy relation on the FSS σ such that χ (a, b) ≤ σ (a) ∧ σ (b) for each a, b ∈ M, and M is a finite set.

Definition 2.2. [14] Let G ˆ = ( η , ϑ , κ ) G = ( V G , E G ) be a graph. Then G = ( V G , E G , I G ) is an incidence graph, where I G ⊆ V G × E G .

In Fig 1, we have an incidence graph.

OPEN IN VIEWER

Fig. 1

Incidence graph.

Definition 2.3. [14] Let G = ( V G , E G ) be a graph and η be a FSS of V G , ϑ be a FSS of E G and κ be a FSS of V G × E G . If κ (v, e) ≤ η (v) ∧ ϑ (e) for each v ∈ V G and e ∈ E G , then κ is known as fuzzy incidence of G .

Definition 2.4. [14] Let G = ( V G , E G ) be a graph and (η, ϑ) be a fuzzy subgraph of G . If κ is a fuzzy incidence of G , then G ★ = ( η , ϑ , κ ) is a FIG of G . FIG is given in Fig 2.

OPEN IN VIEWER Fig. 2

Definition 2.5. [23] An IFG is G = ( V G , E G , η , ϑ ) where η = (η₁, η₂) , ϑ = (ϑ₁, ϑ₂) and

V G = { v 0 , v 1 , . . . , v n } such that η 1 : V G → [ 0 , 1 ] where η₁ (v _k ) represents the grade of membership and η 2 : V G → [ 0 , 1 ] where η₂ (v _k ) represents the grade of non-membership of the vertex v k ∈ V G and 0 ≤ η₁ (v _k ) + η₂ (v _k ) ≤1 for every v k ∈ V G ( k = 1 , 2 , . . . , n ) .

Definition 2.6. [19] Let G = ( V G , E G ) be a graph and G ★ = ( η , ϑ , κ ) be a fuzzy incidence graph of G . Then G ˆ = ( V G , E G , V G × E G , η , ϑ , κ ) is known as an IFIG of G where η = (η₁, η₂) , ϑ = (ϑ₁, ϑ₂) and κ = (κ₁, κ₂).

V G = { v 1 , v 2 , . . . , v n } such that η 1 : V G → [ 0 , 1 ] and η 2 : V G → [ 0 , 1 ] , where η₁ and η₂ represent the membership (MS) value and non-membership (NMS) value of the vertex respectively and 0 ≤ η₁ (v _i ) + η₂ (v _i ) ≤1 for every v i ∈ V G ( i = 1 , 2 , . . . , n ) .

Example 2.7. Here, we include a daily life example of three different branches of hospitals. As an explanatory case, assume a network (IFIG) of three vertices expressing the distinct branches of a hospital. The MS value of the vertices denotes the percentage of patients who visit the hospital for their treatment of different diseases and the NMS value of the vertices indicates the percentage of patients who do not want to come to these hospitals due to their bad reputation. The MS value of the edges represents the percentage of those patients which one hospital recommends to another hospital due to the unavailability of doctors of different diseases in that hospital and the NMS value of the edges expresses the percentage of those patients which one hospital does not recommend to another hospital. The MS value of the pairs demonstrates the percentage of the profit of the hospital and the NMS value of the pairs shows the percentage of loss of the hospital.

Let G ˆ = ( η , ϑ , κ ) be an IFIG associated with G expressing three different branches of hospitals, where η (v) = (η₁ (v) , η₂ (v)), ϑ (uv) = (ϑ₁ (uv) , ϑ₂ (uv)), κ (v, uv) = (κ₁ (v, uv) , κ₂ (v, uv)) Let vertex set be V G = { v 1 , v 2 , v 3 } and η (v₁) = (0.2, 0.8), η (v₂) = (0.3, 0.6), η (v₃) = (0.5, 0.4). Then ϑ (v₁v₂) = (0.15, 0.7), ϑ (v₂v₃) = (0.2, 0.5), ϑ (v₃v₁) = (0.15, 0.6) and κ (v₁, v₁v₂) = (0.1, 0.75), κ (v₂, v₁v₂) = (0.1, 0.65), κ (v₂, v₂v₃) = (0.15, 0.55), κ (v₃, v₂v₃) = (0.15, 0.45), κ (v₃, v₃v₁) = (0.1, 0.55), κ (v₁, v₃v₁) = (0.1, 0.7). The corresponding IFIG is shown in Fig 3.

OPEN IN VIEWER

Fig. 3

IFIG indicating three different branches of hospital.

In this section, some notions concerning IFIG with examples are explored. This section contains some of the ideas which are taken from [14]. We have proposed the concept of connected vertices. Also, we defined intuitionistic fuzzy incidence path (IFIP), intuitionistic fuzzy incidence strength (IFIS), intuitionistic fuzzy cycle (IFC), and intuitionistic fuzzy incidence cycle (IFIC) along with their examples. The idea of an intuitionistic fuzzy incidence subgraph is also provided. Three different kinds of pairs, along with examples, are also elaborated. Complete IFIG is also defined in this section.

Definition 3.1. For a vertex v ∈ V G , if η₁ (v) >0 and η₂ (v) <1, then v is in support of η. For an edge uv ∈ E G , if ϑ₁ (uv) >0 and ϑ₂ (uv) <1, then uv is in support of ϑ, and for pair ( u , uv ) ∈ V G × E G , if κ₁ (u, uv) >0 and κ₂ (u, uv) <1 then (u, uv) is in the support of κ. The support of η, ϑ and κ are represented as η^∗, ϑ^∗ and κ^∗ respectively.

Definition 3.2. Let uv ∈ ϑ^∗. Then uv is an edge of the IFIG G ˆ = ( η , ϑ , κ ) and if (u, uv) , (v, uv) ∈ κ^∗ then (u, uv) and (v, uv) are known as pairs of the IFIG.

Definition 3.3. [14] Two vertices v _i and v _j joined by an IFIP v _i , (v _i , v _i v_i+1) , v _i v_i+1, (v_i+1, v _i v_i+1) , v_i+1, . . . , v _j in an IFIG are known as connected.

Example 3.4. Let G ˆ = ( η , ϑ , κ ) be an IFIG associated with G . Let vertex set be V G = { v 1 , v 2 , v 3 } and η (v₁) = (0.5, 0.4), η (v₂) = (0.3, 0.6), η (v₃) = (0.2, 0.8), ϑ (v₁v₂) = (0.2, 0.5), ϑ (v₂v₃) = (0.1, 0.7), and κ (v₁, v₁v₂) = (0.1, 0.45), κ (v₂, v₁v₂) = (0.15, 0.55), κ (v₂, v₂v₃) = (0.05, 0.65), κ (v₃, v₂v₃) = (0.05, 0.75).

It is clear from Fig. 4, v₁, (v₁, v₁v₂) , v₁v₂, (v₂, v₁v₂) , v₂ is an IFIP. Thus v₁ and v₂ are connected. Similarly, there are IFIPs between v₂ to v₃, i.e. v₂, (v₂, v₂v₃) , v₂v₃, (v₃, v₂v₃) , v₃ and v₃ to v₁, i.e.v₁, (v₁, v₁v₂) , v₁v₂, (v₂, v₁v₂) , v₂, (v₂, v₂v₃) , v₂v₃, (v₃, v₂v₃) , v₃. Thus v₂, v₃ and v₃, v₁ are also connected.

OPEN IN VIEWER

Fig. 4

Paths in IFIG.

Definition 3.5. An IFIS of an IFIG G ˆ = ( η , ϑ , κ ) is defined as (⊖ κ₁ (v _i , v _i v _j ) , ⊕ κ₂ (v _i , v _i v _j )); ∀ (v _i , v _i v _j ) ∈ κ^∗.

Definition 3.6. [14] The IFIG G ˆ = ( η , ϑ , κ ) is a cycle, if (η^∗, ϑ^∗, κ^∗) is a cycle and it is an IFC if (η^∗, ϑ^∗, κ^∗) is a cycle and there exists no unique edge uv ∈ ϑ^∗ s.t. ϑ (uv) = (⊖ ϑ₁ (rs) , ⊕ ϑ₂ (rs)); ∀rs ∈ ϑ^∗.

Definition 3.7. [14] The IFIG G ˆ = ( η , ϑ , κ ) is an IFIC if it is a fuzzy cycle and not exists a unique (u, uv) ∈ κ^∗ s.t. κ (u, uv) = (⊖ κ₁ (v _i , v _i v _j ) , ⊕ κ₂ (v _i , v _i v _j )); ∀ (v _i , v _i v _j ) ∈ κ^∗.

Example 3.8. Let G ˆ = ( η , ϑ , κ ) be an IFIG shown in Fig 5 with vertex set V G = { v 1 , v 2 , v 3 , v 4 } and η ( v 1 ) = ( 0.2 , 0.7 ) , η ( v 2 ) = ( 0.3 , 0.5) , η (v₃) = (0.3, 0.6) , η (v₄) = (0.4, 0.5) , ϑ (v₁v₂)= (0.15, 0.65) , ϑ (v₁v₄) = (0.15, 0.65) , ϑ (v₂v₃) =(0.25, 0.55) , ϑ (v₃v₄) = (0.25, 0.55) and κ (v₁, v₁v₂)= (0.1, 0.7) , κ (v₂, v₁v₂) = (0.1, 0.6) , κ (v₁, v₁v₄) = (0.1, 0.7) , κ (v₄, v₁v₄) = (0.1, 0.6) , κ (v₂, v₂v₃) = (0.2, 0.55) , κ (v₃, v₂v₃) = (0.2, 0.6) , κ (v₃, v₃v₄) = (0.2, 0.6) , κ (v₄, v₃v₄) = (0.2, 0.55).

OPEN IN VIEWER

Fig. 5

IFIG which is a cycle.

It is an IFC as there is no unique edge equal to the strength of the cycle. Since ϑ (v₁v₂) = ϑ (v₁v₄) = (0.15, 0.65). It is also an IFIC as there is no unique pair equal to the IFIS of G ˆ . Since κ (v₁, v₁v₂) = κ (v₁, v₁v₄) = (0.1, 0.7).

Example 3.9. Let G ˆ = ( η , ϑ , κ ) be an IFIG as shown in Fig 6 with vertex set V G = { v 1 , v 2 , v 3 } and η ( v 1 ) = ( 0.3 , 0.6 ) , η ( v 2 ) = ( 0.8 , 0.2 ) , η ( v 3 ) = ( 0.2 , 0.8 ) , ϑ ( v 1 v 2 ) = ( 0.25 , 0.45 ) , ϑ (v₁v₃) = (0.18, 0.7) , sϑ (v₂v₃) = (0.18, 0.75)and κ (v₁, v₁v₂) = (0.2, 0.55) , κ (v₂, v₁v₂) = (0.2, 0.4) , κ (v₁, v₁v₃) = (0.16, 0.65) , κ (v₃, v₁v₃) = (0.17, 0.75) , κ (v₂, v₂v₃) = (0.16, 0.7) , κ (v₃, v₂v₃)= (0.15, 0.78). It is not an IFIC because it has a unique pair (v₃, v₂v₃) as κ (v₃, v₂v₃) = (0.15, 0.78) = the IFIS of G ˆ .

OPEN IN VIEWER

Fig. 6

IFIG which is not a cycle.

Definition 3.10 [14] Let G ˆ = ( η , ϑ , κ ) be an IFIG. Then H ˆ = ( υ , φ , ω ) is an intuitionistic fuzzy incidence subgraph of G ˆ if υ ⊆ η, φ ⊆ ϑ and ω ⊆ κ such that υ₁ (r) ≤ η₁ (r), υ₂ (r) ≥ η₂ (r); ∀ r ∈ H ˆ , φ₁ (rs) ≤ ϑ₁ (rs), φ₂ (rs) ≥ ϑ₂ (rs); ∀ rs ∈ H ˆ and ω₁ (r, rs) ≤ κ₁ (r, rs), ω₂ (r, rs) ≥ κ₂ (r, rs); ∀ ( r , rs ) ∈ H ˆ .

An intuitionistic fuzzy incidence subgraph H ˆ of G ˆ is an intuitionistic fuzzy incidence spanning subgraph of G ˆ if υ^∗ = η^∗.

Definition 3.11. Let G ˆ = ( η , ϑ , κ ) be an IFIG. An IFIS of ρ is described as: κ (r, rs) = (⊖ κ₁ (t, tu) , ⊕ κ₂ (t, tu)); ∀ (t, tu) ∈ ρ .

where ρ is a IFIP between any two nodes, arcs and pairs belong to η^∗, ϑ^∗ and κ^∗. The IFIS of the ρ from r to rs is denoted by IICONN G ( r , rs ) .

Definition 3.12. Let G ˆ = ( η , ϑ , κ ) be an IFIG. A pair (r, rs) is said to be a SP if κ₁ (r, rs) ≥ ⊖ κ₁ (t, tu) and κ₂ (r, rs) ≤ ⊕ κ₂ (t, tu); ∀ ( t , tu ) ∈ G ˆ - ( r , rs ) where IICONN G ˆ - ( r , rs ) ( r , rs ) = ( ⊖ κ 1 ( t , tu ) , ⊕ κ 2 ( t , tu ) ) represents the greatest IFIS of path r - rs.

Definition 3.13. A pair (r, rs) is called an α-SP if κ₁ (r, rs) > ⊖ κ₁ (t, tu) and κ₂ (r, rs) < ⊕ κ₂ (t, tu); ∀ ( t , tu ) ∈ G ˆ - ( r , rs ) .

Definition 3.14. A pair (r, rs) is called a β-SP if κ ( r , rs ) = IICONN G ˆ - ( r , rs ) ( r , rs ) = ( ⊖ κ 1 ( t , tu ) , ⊕ κ 2 ( t , tu ) ) ; ∀ ( t , tu ) ∈ G ˆ - ( r , rs ) .

Definition 3.15. Let G ˆ = ( η , ϑ , κ ) be an IFIG. A pair (r, rs) is called δ-pair if κ₁ (r, rs) < ⊖ κ₁ (t, tu) and κ₂ (r, rs) > ⊕ κ₂ (t, tu); ∀ ( t , tu ) ∈ G ˆ - ( r , rs ) .

Definition 3.16. [14] Let G ˆ = ( η , ϑ , κ ) be an IFIG. An IFIP ρ in G ˆ is called a strong if all of its pairs are strong.

Example 3.17. Consider the IFIG G ˆ as shown in Fig 7.

OPEN IN VIEWER

Fig. 7

α-strong IFIP between (v₂, v₂v₃) and v₁.

All the vertices are α-strong except v₂ as η₁ (v₂) =0.3 but ⊖η₁ (v _i ) =0.4 > 0.3 and η₂ (v₂) =0.6 but ⊕η₂ (v _i ) =0.5 < 0.6 ∀ v i ∈ G ˆ - v 2 . All the edges are α-strong except v₁v₂ as ϑ (v₁v₂) =0.15 but ⊖ϑ₁ (v _i v _j ) =0.2 > 0.15 and ⊕ϑ₂ (v _i v _j ) =0.45 < 0.5 ∀ v i v j ∈ G ˆ - v 1 v 2 . All the pairs are α-strong except (v₁, v₁v₂) and (v₂, v₁v₂) as ⊖κ₁ (v _i , v _i v _j ) =0.1 = κ₁ (v₁, v₁v₂) and ⊕κ₂ (v _i , v _i v _j ) =0.55 > 0.45 = κ₂ (v₁, v₁v₂); ∀ ( v i , v i v j ) ∈ G ˆ - ( v 1 , v 1 v 2 ) . Similarly we can see (v₂, v₁v₂). Thus (v₂, v₂v₃) , v₂v₃, (v₃, v₂v₃) , v₃, (v₃, v₃v₄) , v₃v₄, (v₄, v₃v₄) , v₄, (v₄, v₁v₄) , v₁v₄, (v₁, v₁v₄) , v₁ is an α-strong IFIP.

Definition 3.18. Let G ˆ = ( η , ϑ , κ ) be an IFIG. An intuitionistic fuzzy incidence subgraph H ˆ = ( υ , φ , ω ) is called a subgraph of G ˆ if υ₁ (r) = η₁ (r); ∀r ∈ υ^∗, φ₁ (rs) = ϑ₁ (rs) , φ₂ (rs) = ϑ₂ (rs); ∀rs ∈ φ^∗ and ω₁ (r, rs) = κ₁ (r, rs) , ω₂ (r, rs) = κ₂ (r, rs); ∀ (r, rs) ∈ ω^∗.

Definition 3.19. Let G ˆ = ( η , ϑ , κ ) be an IFIG. Then it is called an IFIT if there is a spanning subgraph F ˆ = ( υ , φ , ω ) of G ˆ , that is a tree as for all pair (r, rs) ∈ κ^∗ - ω^∗, κ₁ (r, rs) < ⊖ {κ₁ (t, tu) | (t, tu) ∈ ω^∗} and κ₂ (r, rs) > ⊕ {κ₂ (t, tu) | (t, tu) ∈ ω^∗}. It can be stated as there is an IFIP in spanning subgraph F ˆ from r to rs s.t. all of its pair has maximum κ₁ MS value than κ₁ (r, rs) and has minimum κ₂ NMS value than κ₂ (r, rs).

Example 3.20. Let G ˆ = ( η , ϑ , κ ) be an IFIG given in Fig 8 with vertex set V G = { v 1 , v 2 , v 3 , v 4 } and η (v₁) = (0.4, 0) , η (v₂) = (0.3, 0.05) , η (v₃) = (0.5, 0.03) , η (v₄) = (0.4, 0.1) , ϑ (v₁v₂) = (0.2, 0.04) , ϑ (v₁v₄) = (0.3, 0.05) , ϑ (v₂v₃) = (0.2, 0.04) , ϑ (v₃v₄) = (0.3, 0.05) and κ (v₁, v₁v₂) = (0.1, 0.03) , κ (v₂, v₁v₂) = (0.2, 0.045) , κ (v₁, v₁v₄) = (0.2, 0.04) , κ (v₄, v₁v₄) = (0.1, 0.1) , κ (v₂, v₂v₃) = (0.2, 0.045) , κ (v₃, v₂v₃) = (0.2, 0.035) , κ (v₃, v₃v₄) = (0.2, 0.04) , κ (v₄, v₃v₄) = (0.2, 0.07).

OPEN IN VIEWER

Fig. 8

IFIT.

The spanning subgraph F ˆ = ( υ , φ , ω ) of G ˆ is the IFIG. It has the same MS and NMS value for all nodes, arcs and pairs, except for ω (d, da) = (0.18, 0.08) and ω (a, da) =0. Then (υ^∗, φ^∗, ω^∗) is a tree. (a, da) ∈ κ^∗ - ω^∗. There is an IFIP: ϱ : da, (d, da) , d, (d, dc) , cd, (c, cd) , c, (c, cb) , bc, (b, bc) , b, (b, ba) , ab, (a, ab) , a with IICONN F ˆ ( a , da ) = IIstrength ( ϱ ) = ( 0.18 , 0.08 ) . But κ (a, da) = (0.1, 0.1) where 0.18 > 0.1 and 0.08 < 0.1. Hence, G ˆ is an IFIT.

Definition 3.21. An IFIG G ˆ = ( η , ϑ , κ ) is called complete if ϑ₁ (v _i v _j ) = η₁ (v _i ) ⊖ η₁ (v _j ); ϑ₂ (v _i v _j ) = η₂ (v _i ) ⊕ η₂ (v _j ) and κ₁ (v _i , v _i v _j ) = η₁ (v _i ) ⊖ ϑ₁ (v _i v _j ); κ₂ (v _i , v _i v _j ) = η₂ (v _i ) ⊕ ϑ₂ (v _i v _j ) for all (v _i , v _i v _j ) ∈ κ^∗.

Example 3.22. Let G ˆ = ( η , ϑ , κ ) be an IFIG shown in Fig 9 with vertex set V G = { v 1 , v 2 , v 3 } and η (v₁) = (0.3, 0.6) , η (v₂) = (0.4, 0.5) , η (v₃) = (0.7, 0.3) , ϑ (v₁v₂) = (0.3, 0.6) , ϑ (v₁v₃) = (0.3, 0.6)  ϑ (v₂v₃) = (0.4, 0.5) and κ (v₁, v₁v₂) = (0.3, 0.6)  κ (v₂, v₁v₂) = (0.3, 0.6) , κ (v₁, v₁v₃) = (0.3, 0.6)  κ (v₃, v₁v₃) = (0.3, 0.6) , κ (v₂, v₂v₃) = (0.4, 0.5)  κ (v₃, v₂v₃) = (0.4, 0.5). It is a complete IFIG.

OPEN IN VIEWER

Fig. 9

Complete IFIG.

In this section, we proved some important results on IFIGs. It is proved that every IFIC is always a strong cycle. Some novel results related to SPs are discussed. A very interesting result that every IFICP is a SP is also given in this section. The proof of an IFIT is always connected iff it does not contain any β-SP is also presented in this section. At the end of this section, a very fascinating result that a complete IFIG does not contain any δ-pair is provided.

Proposition 4.1. An IFIC is a strong cycle.

Proof. Consider G ˆ = ( η , ϑ , κ ) an IFIC. Then by Definition 3.7, G ∗ = ( η ∗ , ϑ ∗ , κ ∗ ) is a cycle so there does not exist the only rs ∈ ϑ^∗ and pair (r, rs) ∈ κ^∗ such that ϑ (rs) = (⊖ ϑ₁ (tu) , ⊕ ϑ₂ (tu)); ∀tu ∈ ϑ^∗ and κ (r, rs) = (⊖ κ₁ (t, tu) , ⊕ κ₂ (t, tu)); ∀ (t, tu) ∈ κ^∗.

Now we have to prove that each arc and pair in G ˆ are strong. It is quite obvious that all arcs in a fuzzy cycle are strong. Let (r, rs) belongs to G ˆ and assume that it is not strong. This implies (r, rs) is a δ-pair. Then by Definition 3.15, κ₁ (r, rs) < ⊖ κ₁ (t, tu) and κ₂ (r, rs) > ⊕ κ₂ (t, tu); ∀ ( t , tu ) ∈ G ˆ - ( r , rs ) where IICONN G ˆ - ( r , rs ) = ( ⊖ κ 1 ( t , tu ) , ⊕ κ 2 ( t , tu ) ) ; ∀ (t, tu) ∈ κ^∗. As G ∗ is a cycle so every pair in the incidence subgraph G ˆ - ( r , rs ) will have value κ₁ (t′, t′u′) > ⊖ κ₁ (t, tu) and κ₂ (t′, t′u′) < ⊕ κ₂ (t, tu); ∀ ( t ′ , t ′ u ′ ) ∈ G ˆ - ( r , rs ) . Which is a contradiction to our assumption that G ˆ is an IFIC. Hence, (r, rs) is strong. ■

Proposition 4.2. Let G ˆ = ( η , ϑ , κ ) be an IFIG and H ˆ = ( υ , φ , ω ) an intuitionistic fuzzy incidence subgraph of G ˆ .Then IICONN H ˆ ( r , rs ) ≤ IICONN G ˆ ( r , rs ) ; ∀ (r, rs) ∈ ω^∗

Proof. It is obvious because different r - rs IFIPs in G ˆ may not be in H ˆ or IFIPs have less IFIS in H ˆ . ■

Proposition 4.3. Consider G ˆ = ( η , ϑ , κ ) an IFIG. A pair (r, rs) ∈ κ^∗ is a SP if κ (r, rs) = (⊖ κ₁ (t, tu) , ⊕ κ₂ (t, tu)); ∀ (t, tu) ∈ κ^∗.

Proof. Since IFIS of r - rs IFIP in G ˆ will has κ₁ (r, rs) ≥ ⊖ κ₁ (t, tu) and κ₂ (r, rs) ≤ ⊕ κ₂ (t, tu) where (t, tu) ∈ r - rs an IFIP.

If (r, rs) is the unique pair with κ₁ (r, rs) = ⊖ κ₁ (t, tu) and κ₂ (r, rs) = ⊕ κ₂ (t, tu) then all other IFIPs r - rs in G ˆ will have IFIS: κ₁ (r, rs) > ⊖ κ₁ (t, tu) and κ₂ (r, rs) < ⊕ κ₂ (t, tu); ∀ ( t , tu ) ∈ G ˆ - ( r , rs ) . Hence, (r, rs) is an α-SP.

Example 4.4. Consider G ˆ = ( η , ϑ , κ ) is an IFIG as given in the Fig 10 with vertex set V G = { v 1 , v 2 , v 3 } and η (v₁) = (0.4, 0.6) , η (v₂) = (0.4, 0.6) , η (v₃) = (0.4, 0.6) , ϑ (v₁v₂) = (0.3, 0.5) , ϑ (v₁v₃) = (0.3, 0.5) , ϑ (v₂v₃) = (0.3, 0.5) and κ (v₁, v₁v₂) = (0.2, 0.55) , κ (v₂, v₁v₂) = (0.2, 0.55) , κ (v₁, v₁v₃) = (0.2, 0.55) , κ (v₃, v₁v₃) = (0.2, 0.55) , κ (v₂, v₂v₃) = (0.2, 0.55) , κ (v₃, v₂v₃) = (0.2, 0.55).

Hence, G ˆ is an IFIC which is strong because all the pairs in the Fig 10 are strong.

OPEN IN VIEWER

Fig. 10

IFIG having all strong pairs.

Definition 4.5. Let G ˆ = ( η , ϑ , κ ) be an IFIG. If the removal of a pair in an IFIG increases the number of connected components, then the pair is known as an IFICP.

Proposition 4.6. Let G ˆ = ( η , ϑ , κ ) be an IFIG. A pair (r, rs) that is defined by κ (r, rs) = (⊖ {η₁ (r) , ϑ₁ (rs)} , ⊕ {η₂ (r) , ϑ₂ (rs)}) is strong.

Proof. Let G ˆ - ( r , rs ) be the intuitionistic fuzzy incidence subgraph of G ˆ .

Suppose that G ˆ - ( r , rs ) is disconnected, then (r, rs) is an IFICP. But given κ (r, rs) = (κ₁ (r, rs) , κ₂ (r, rs)) = (⊖ {η₁ (r) , ϑ₁ (rs)} , ⊕ {η₂ (r) , ϑ₂ (rs)}). As G ˆ - ( r , rs ) is a subgraph of G ˆ then κ₁ (r, rs) = ⊖ {η₁ (r) , ϑ₁ (rs)} ≥ ⊖ {η₁ (t) , ϑ₁ (tu)} and κ 2 ( r , rs ) = ⊕ { η 2 ( r ) , ϑ 2 ( rs ) } ≤ ⊕ { η 2 ( t ) , ϑ 2 ( tu ) } ; ∀ ( t , tu ) ∈ G ˆ - ( r , rs )

This shows that a pair (r, rs) is strong.

Example 4.7. Let G ˆ = ( η , ϑ , κ ) be an IFIG. In Fig 11, only κ (v₁, v₁v₂) = (⊖ {η₁ (v₁) , ϑ₁ (v₁v₂)}   ⊕ {η₂ (v₁) , ϑ₂ (v₁v₂)})

κ₁ (v₁, v₁v₂) =0.2 > 0.1 = ⊖ {κ₁ (r, rs) | (r, rs) ∈ κ^∗}; κ₂ (v₁, v₁v₂) =0.6 < 0.65 = ⊕ {κ₂ (r, rs) | (r, rs) ∈ κ^∗}. Hence, (v₁, v₁v₂) is a SP.

Proposition 4.8. Every IFICP in an IFIG, is a SP.

Proof. Consider G ˆ = ( η , ϑ , κ ) an IFIG and (r, rs) ∈ κ^∗ be an IFICP. As G ˆ - ( r , rs ) is a subgraph of G ˆ . Then ⊖ {η₁ (r) , ϑ₁ (rs)} ≥ ⊖ {η₁ (r) , ϑ₁ (rs)} and ⊕ {η₂ (r) , ϑ₂ (rs)} ≤ ⊕ {η₂ (r) ϑ₂ (rs)}; ∀ ( t , tu ) ∈ G ˆ - ( r , rs ) . Assume that (r, rs) is not a SP. Then κ₁ (r, rs) < ⊖ {η₁ (t) , ϑ₁ (tu)} and κ₂ (r, rs) > ⊕ {η₂ (t) , ϑ₂ (tu)}; ∀ ( u , uv ) ∈ G ˆ - ( r , rs ) . Let a strongest IFIP ϱ from r - rs in G ˆ - ( r , rs ) . Then ϱ with pair ( r , rs ) ∈ G ˆ form an incidence cycle where (r, rs) is the weakest pair. But it is contradict our assumption, i.e. (r, rs) is an IFICP. Hence, (r, rs) is a SP. ■

Theorem 4.9. Let G ˆ = ( η , ϑ , κ ) be a connected IFIG. Then there is a strong IFIP from any node r ∈ η^∗ to any arc st ∈ ϑ^∗.

Proof. As G ˆ is connected, so ∃ an IFIP ϱ from r ∈ η^∗ to st ∈ ϑ^∗ such that ρ : r = r 0 , ( r 0 , r o r 1 ) , r 0 r 1 , ( r 1 , r 0 r 1 ) . ̇ . . . . , r n = s , ( s , st ) , st , ( t , st ) ;(r _n can be taken t also). Since κ (r _k , r _k r_k+1) >0 ∀ k = 0, 1, 2, 3, . . . , n - 1 and κ (s, st) , κ (t, st) >0. Suppose that (r_j-1, r_j-1r _j ) is not a SP. Then (r_j-1, r_j-1r _j ) is δ-pair and it implies κ ( r j - 1 , r j - 1 r j ) < IICONN G ˆ - ( r j - 1 , r j - 1 r j ) ( r j - 1 , r j - 1 r j ) such that κ₁ (r_j-1, r_j-1r _j ) < ⊖ {κ₁ (r_l-1, r_l-1r _l ) | (r_l-1, r_l-1r _l ) ∈ κ^∗} and κ₂ (r_j-1, r_j-1u _j ) > ⊕ {κ₂ (r_l-1, r_l-1r _l ) | (r_l-1, r_l-1r _l ) ∈ κ^∗} ∀ ( u l - 1 , u l - 1 u l ) ∈ G ˆ - ( r j - 1 , r j - 1 u j ) Let ϱ be an IFIP from r to st in G ˆ - ( r j - 1 , r j - 1 r j ) with IFIS IICONN G ˆ - ( r j - 1 , r j - 1 r j ) ( r j - 1 , r j - 1 r j ) . We observe that every pair in ϱ will have κ₁ value greater than κ₁ (r_j-1, r_j-1r _j ) and κ₂ value smaller than κ₂ (r_j-1, r_j-1r _j ). If ϱ is strong, then proof is completed. In other case, find out δ-pairs in ϱ and supersede these by IFIPs having strength more than κ₁ and less than κ₂ values. We will continue this procedure ultimately we will have an IFIP ς from r to st contain only SPs. ■

Theorem 4.10. Let G ˆ = ( η , ϑ , κ ) be an IFIG. (r, rs) ∈ κ^∗ is an IFICP iff it is α-strong.

Proof. As G ˆ = ( η , ϑ , κ ) is an IFIG. Let (r, rs) ∈ κ^∗ be an IFICP of G ˆ . Then by Proposition 4.6, (r, rs) is an α-strong. i.e. κ₁ (r, rs) > ⊖ {κ₁ (t, tu) | (t, tu) ∈ κ^∗} and κ₂ (r, rs) < ⊕ {κ₂ (t, tu) | (t, tu) ∈ κ^∗} ∀ ( t , tu ) ∈ G ˆ - ( r , rs ) . Conversely, let (r, rs) be an α-strong. Then r, (r, rs) , rs is the unique strongest IFIP from r to rs. If a pair (r, rs) is removed from this IFIP then it will be reduced the IFIS between r and rs. Thus (r, rs) is an IFICP. ■

Theorem 4.11. Let G ˆ = ( η , ϑ , κ ) be an IFIT. Any pair in κ^∗ is strong iff it belongs to tree F ˆ = ( υ , φ , ω ) .

Proof. As G ˆ = ( η , ϑ , κ ) is an IFIT. Let F ˆ = ( υ , φ , ω ) be the tree and (r, rs) be a strong pair in κ^∗. Suppose that ( r , rs ) ∉ F ˆ then κ 1 ( r , r s ) < ⊖ { κ 1 ( t , t u ) | ( t , t u ) ∈ ω * } ; κ 2 ( r , r s ) > ⊕ { κ 2 ( t , t u ) | ( t , t u ) ∈ ω * }(4.1)

Since (r, rs) is strong,

κ 1 ( r , r s ) < ⊖ { κ 1 ( t , t u ) | ( t , t u ) ∈ κ * } ; κ 2 ( r , r s ) > ⊕ { κ 2 ( t , t u ) | ( t , t u ) ∈ κ * }(4.2) ∀ ( t , tu ) ∈ G ˆ - ( r , rs ) . Since ( r , rs ) ∉ F ˆ , as F ˆ is a subgraph of G ˆ - ( r , rs ) . So above inequality(4.2) implies κ₁ (r, rs) ≥ ⊖ {κ₁ (t, tu) | (t, tu) ∈ κ^∗}; ∀ ( t , tu ) ∈ G ˆ - ( r , rs ) ≥ ⊖ {κ₁ (t, tu) | (t, tu) ∈ ω^∗}; ∀ ( t , tu ) ∈ F ˆ and κ₂ (r, rs) ≤ ⊕ {κ₂ (t, tu) | (t, tu) ∈ κ^∗}; ∀ ( t , tu ) ∈ G ˆ - ( r , rs ) ≤ ⊕ {κ₂ (t, tu) | (t, tu) ∈ ω^∗}; ∀ ( t , tu ) ∈ F ˆ a contradiction of inequality(4.1). Hence, ( r , rs ) ∈ F ˆ . Conversely, let (r, rs) ∈ ω^∗ then by Theorem 3.4 of [8], (r, rs) is an IFICP. This implies (r, rs) is strong. ■

Note:

The pairs of F ˆ are the SPs of G ˆ .

Proposition 4.12. A connected IFIG G ˆ = ( η , ϑ , κ ) is an IFIT iff it has no β-SPs.

Proof. Suppose G ˆ is an IFIT. Then there is the only F ˆ = ( υ , φ , ω ) such that all pairs belong to F ˆ are IFICPs so these pairs are α-strong. Thus, all pairs of G ˆ , not belong to F ˆ are δ-pairs (By Definition of an IFIT). Hence, there is no β-SP in G ˆ . Conversely, assume that there is no β-SP in G ˆ . Now we have to prove that G ˆ is an IFIT.

If G ˆ has no cycle, then G ˆ is an IFIT.

Theorem 4.13. A connected IFIG G ˆ = ( η , ϑ , κ ) is an IFIT iff there is a strong IFIP between any node arc pair. Particularly, this is an α-strong IFIP.

Proof. Let G ˆ be an IFIT then there is a strong IFIP ξ between a node x ∈ η^∗ and arc yz ∈ ϑ^∗. By Theorem 4.8, ξ belongs to the corresponding maximum spanning tree F ˆ of G ˆ . As F ˆ is a tree, so ξ is the only IFIP in F ˆ . By Theorem 4.9 F ˆ has no β-strong pairs. Thus, ξ is an α-strong IFIP. Conversely, suppose that G ˆ is an IFIG and there is the only one α-strong IFIP between any node arc pair. Now assume that G ˆ is not an IFIT then there is a cycle ϑ ∈ G ˆ so for (r, rs) ∈ ϑ; κ₁ (r, rs) ≥ ⊖ {κ₁ (t, tu) | (t, tu) ∈ κ^∗} and κ₂ (r, rs) ≤ ⊕ {κ₂ (t, tu) | (t, tu) ∈ κ^∗} for all ( t , tu ) ∈ G ˆ - ( r , rs ) which shows that (r, rs) ∈ ϑ is a SP, which contradict the assumption that there is the only one strong IFIP between a node arc pair. Hence G ˆ is an IFIT. ■

Proposition 4.14. Let G ˆ = ( η , ϑ , κ ) be an IFIG and ξ be a strong IFIP between any node r and arc rs. Then ξ is a strongest r - rs path in the following cases.

ξ has only α-SPs.

Proposition 4.15. A complete IFIG has no δ-pairs.

Proof. Suppose that a complete IFIG G ˆ = ( η , ϑ , κ ) contains (r, rs) as a δ-pair. Then By Definition 3.15, κ₁ (r, rs) < ⊖ κ₁ (t, tu) and κ₂ (r, rs) > ⊕ κ₂ (t, tu); ∀ ( t , tu ) ∈ G ˆ - ( r , rs ) . Thus there is a stronger IFIP ξ in G ˆ from r to rs. Suppose that (m, n) is the IFIS of ξ. If κ (r, rs) = κ (s, rs) = (p, q). So p < m and q > n. Every pair belongs to ξ will have κ₁ MS value more than p and κ₂ NMS value less than q. Let z be the first node in ξ next to r. Then κ₁ (r, rz) > p and κ₂ (r, rz) < q. But this is not possible because κ (r, rz) = (⊖ {η₁ (r) , ϑ₁ (rz)} , ⊕ {η₂ (r) , ϑ₂ (rz)}) = (p, q). Hence, there is no δ-pair in G ˆ . Thus, in a complete IFIG, every pair is strong. ■

Here we include a real-life application of SPs of IFIGs. As an explanatory case, consider the network (IFIG) of five different computers. The vertices in a network are indicating five different computers. The MS value of the vertices represents the percentage of stored data in a computer system and the NMS value of the vertices expresses the empty space in a computer system. The MS values of the edges are representing the percentage of the total capacity of transfer of data among different computers and the NMS value of the edges is expressing the percentage of the total capacity of untransferable data among these computers. The MS value of the pairs is indicating the percentage of sharing data from one computer to another computer and the NMS value of the pairs is expressing the percentage of data that is not shareable from one computer to another computer. To transfer maximum data from one computer to another computer, the pairs should be strong either α-strong or β-strong because through SPs we can easily transfer maximum data from one computer to another computer. But in the case of non-SP (δ-pair) it will be difficult to share the huge amount of data from one computer to another computer.

Let G ˆ = ( η , ϑ , κ ) be an IFIG shown in Fig. 12 expressing a network of different computers. Let vertex set be V G = { C 1 , C 2 , C 3 , C 4 , C 5 } and η (C₁) = (0.3, 0.5), η (C₂) = (0.3, 0.7), η (C₃) = (0.4, 0.3), η (C₄) = (0.5, 0.5), η (C₅) = (0.3, 0.2); ϑ (C₁C₂) = (0.3, 0.7), ϑ (C₁C₃) = (0.2, 0.4), ϑ (C₁C₅) = (0.2, 0.5), ϑ (C₂C₃) = (0.2, 0.5), ϑ (C₃C₄) = (0.3, 0.4), ϑ (C₃C₅) = (0.2, 0.2), ϑ (C₄C₅) = (0.1, 0.4), and κ (C₁, C₁C₂) = (0.3, 0.7), κ (C₂, C₁C₂) = (0.2, 0.6), κ (C₁, C₁C₃) = (0.2, 0.4), κ (C₃, C₁C₃) = (0.1, 0.3), κ (C₁, C₁C₅) = (0.03, 0.4), κ (C₅, C₁C₅) = (0.2, 0.5), κ (C₂, C₂C₃) = (0.2, 0.6), κ (C₃, C₂C₃) = (0.1, 0.4), κ (C₃, C₃C₄) = (0.3, 0.4), κ (C₄, C₃C₄) = (0.3, 0.4), κ (C₃, C₃C₅) = (0.2, 0.3), κ (C₅, C₃C₅) = (0.2, 0.2), κ (C₄, C₄C₅) = (0.07, 0.4), κ (C₅, C₄C₅) = (0.05, 0.3).

OPEN IN VIEWER

Fig. 11

(v₁, v₁v₂) a SP.

OPEN IN VIEWER

Fig. 12

A network of different computers.

Since, κ₁ (C₁, C₁C₂) =0.3 > ⊖ κ₁ (t, tu) =0.03 and κ₂ (C₁, C₁C₂) =0.7 ≰  ⊕ κ₂ (t, tu) =0.6; ∀ ( t , tu ) ∈ G ˆ - ( C 1 , C 1 C 2 ) . Therefore, κ (C₁, C₁C₂) is not a SP. Similarly, κ (C₁, C₁C₅) is not a SP. Now, κ₁ (C₂, C₁C₂) =0.2 > ⊖ κ₁ (t, tu) =0.03 and κ₂ (C₂, C₁C₂) =0.6 < ⊕ κ₂ (t, tu) =0.7; ∀ ( t , tu ) ∈ G ˆ - ( C 2 , C 1 C 2 ) . Therefore, κ (C₂, C₁C₂) is an α-SP. In the same way, κ (C₁, C₁C₃), κ (C₃, C₁C₃), κ (C₅, C₁C₅), κ (C₂, C₂C₃), κ (C₃, C₂C₃), κ (C₃, C₃C₄), κ (C₄, C₃C₄), κ (C₃, C₃C₅), κ (C₅, C₃C₅), κ (C₄, C₄C₅) and κ (C₅, C₄C₅) are all an α-SPs. So, we can share the huge amount of data from one computer to another computer through all α-SPs. In this way, we can save time and energy.

IFIGs are studied and introduced in this paper. Their fundamental notions and properties are being established to comprehend the IFIGs comprehensively. SPs and strong IFIPs are founded in an IFIG. In a complete IFIG, every pair is a strong pair is discussed. The IFIG is a more realistic approach than FIGs because of the non-availability of the NMS degree in FIG, which is present in IFIG. Classification of pairs in IFIGs such as an α-strong, β-strong, and δ-pair is also examined. IFIGs are helpful to form vital networking and software programming. A real-life application of a network of different computers is also provided. In the future, our main ambition is to extend our work for soft IFIGs, Hamiltonian IFIGs, competition IFIGs, interval-valued IFIGs, threshold IFIGs, and bipolar IFIGs. Further work on these ideas will be reported in upcoming papers. The readers can work on the concept of domination, accurate domination, strong and weak domination in IFIGs. They can work on Wiener index and Wiener absolute index to compute boiling point of some molecular structures.