Definition 1 (see [43].)An IFG is a pair G = (N, M) such that

• (1)

  V = {u₁, u₂, u₃, …, u_n} with T_N : V⟶[0,1] and F_N : V⟶[0,1] representing the truth-membership degree and falsity-membership degree of the vertex u_i ∈ V and 0 ≤ T_N(u_i) + F_N(u_i) ≤ 1 for each u_i ∈ V(0 ≤ i ≤ n).

• (2)

  E⊆V × V with T_M : E⟶[0,1] and F_M : E⟶[0,1] being as follows:

  $$ (1)
  and 0 ≤ T_M(u_i, u_j) + F_M(u_i, u_j) ≤ 1 for every edge (u_i, u_j) ∈ E.

The next definition is related to complete IFG given by Parvathi et al. [29].

Definition 2 (see [29].)An IFGG is complete if T_M(u_i, u_j) = min{T_N(u_i), T_N(u_j)} and F_M(u_i, u_j) = max{F_N(u_i), F_N(u_j)} for each (u_i, u_j) ∈ E.

The role of path is extremely famous and important in IFGs. The following definition gives us the concept of path in IFGs.

Definition 3. [29] A sequence u₁, u₂, u₃, …, u_n of distinct vertices is a path P in an IFG, provided it has one of the conditions given below for some i and j.

• (1)

  T_M(u_i, u_j) > 0 and F_M(u_i, u_j) = 0

• (2)

  T_M(u_i, u_j) = 0 and F_M(u_i, u_j) > 0

• (3)

  T_M(u_i, u_j) > 0 and F_M(u_i, u_j) > 0

The strength of paths plays a significant role in IFG settings. The following definition gives us component-wise and whole strength of paths in IFGs.

Definition 4 (see [29].)Let P = u₁, u₂, u₃, …, u_n be a path in an IFG. Then,

• (1)

  The T-strength of P is denoted and defined by S_T = min{T_M(u_i, u_j)}∀i, j

• (2)

  The F-strength of P is denoted and defined by S_F = max{F_M(u_i, u_j)}∀i, j

• (3)

  S_P = (S_T, S_F) is called the strength P if both S_T and S_F exist for the same edge

The highly connected nodes have significant role to a network. The next definition is about the strength of connectedness between the nodes.

Definition 5 (see [29].)The T-strength of connectedness between two vertices u_i and u_j is defined by CONN_T(G)(u_i, u_j) = max{S_T} and F-strength of connectedness between u_i and u_j is defined by CONN_F(G)(u_i, u_j) = min{S_F} for all possible paths between u_i and u_j,

where $$, $$ denote the T-strength of connectedness and F-strength of connectedness between u_i and u_j attained by removing the edge (u_i, u_j) from G, respectively.

The following definition gives us the idea of a bridge whose deletion from an IFG increase its number of connected components.

Definition 6 (see [29].)An edge (v_i, v_j) is called a bridge in G if either

In other words, deletion of (u_i, u_j) reduces the strength of connectedness between any pair of vertices.

The concept of strong and weakest edges is of much importance in IFGs as well as in our study. The next definition is related to the notions of strong and weakest edges.

Definition 7 (see [29].)An edge (u_i, u_j) in an IFG is

• (1)

  Strong if T_M(u_i, u_j) ≥ CONN_T(G)(u_i, u_j) and F_M(u_i, u_j) ≤ CONN_F(G)(u_i, u_j) for each u_i, u_j ∈ V

• (2)

  Weakest if T_M(u_i, u_j) < CONN_T(G)(u_i, u_j) and F_M(u_i, u_j) > CONN_F(G)(u_i, u_j) for each u_i, u_j ∈ V

The coming definition gives us the strongest paths between two vertices. This definition is relevant to our work.

Definition 8 (see [29].)A strongest path between two vertices in an IFGG is a path P having its strength equal to CONN_T(G)(u_i, u_j) and CONN_F(G)(u_i, u_j) lying in the same edge.

The following definition tells us the concept of strong path.

Definition 9 (see [29].)Let G = (N, M) be an IFG. A path P : u_i − u_j in G is called strong path if P consists of only strong edges.

Example 1. In Figure 1, T_M(v₁, v₂) = 0.6 = CONN_T(G)(v₁, v₂) and F_M(v₁, v₂) = 0.6 = CONN_F(G)(v₁, v₂), which implies that (v₁, v₂) is a strong arc. Similarly, (v₂, v₃), (v₁, v₅), (v₂, v₄) are strong arcs and (v₃, v₄), (v₄, v₅) are weakest arcs. Here, P = v₁v₂v₃ is a strong path. In fact, it is the strongest path.

The next definition provides us types of strong arcs in IFG.

Definition 10 (see [29].)An arc (u_i, u_j) in an IFGG = (N, M) is

• (1)

  Called α -strong if $$ and $$

• (2)

  Called β -strong if $$ and $$

• (3)

  Called δ -weak if $$ and $$

Example 2. In Figure 2, the arcs (v₁, v₄), (v₂, v₃), (v₄, v₅), (v₁, v₃) are α-strong, (v₃, v₄) is β-strong, and (v₁, v₂), (v₃, v₅) are δ-weak.

The following definition gives us different types of strong paths in IFGs.

Definition 11 (see [29].)A path in an IFG containing only α-strong arcs is called α-strong and a path having only β-strong arcs is called β-strong.

The concept of a cycle has a vital role in IFGs. The following definition gives us the concept of a cycle in IFGs environment.

Definition 12 (see [35].)

• (1)

  G = (N, M) is called a cycle if G^∗ = (N^∗, M^∗) is cycle

• (2)

  G is called an IF cycle if G^∗ = (N^∗, M^∗) is cycle and ∄ unique (x, y) ∈ M^∗ such that

  $$ (3)

Example 3. In this example, we take (T_N(u), F_N(u)) = (0.3, 0.2) for all u ∈ N^∗. Here, min{T_M(u, v)} = 0.2 and max{F_M(u, v)} = 0.3. Clearly, G = (N, M) is an IF cycle. The graph is shown in Figure 3.

The main goal of our study is to bring more accuracy and precision to the study of topological indices, especially in the context of connectivity indices. FGs have less information in comparison with IFGs. In particular situations like vagueness and uncertainty, FGs are described by only membership grades, but IFGs are characterized by the two grades known by membership and nonmembership. Due to the description of opinions using two membership grades, IFGs have less information loss as compared to FGs. So, that is why we aim to propose the concepts of several $$ for IFGs and study their applications.

Definition 13. The $$ of an IFGG = (N, M) is defined as

Example 4. Refer to Figure 1,

So, $$.

It may be observed that TCI(G) > FCI(G), which shows that the level of FCI(G) is lower than the level of TCI(G) in this problem. This comparison is interesting and useful in applications of connectivity index.

Theorem 1. Let G = (N, M) be a complete IFG with N^∗ = {v₁, v₂, v₃, …, v_n} such that t₁ ≤ t₂ ≤ ⋯≤t_n and s₁ ≥ s₂ ≥ s₃ ≥ ⋯≥s_n, where t_i = T_N(v_i) and s_i = F_N(v_i). Then,

Proof. Let v₁ be the vertex having least truth-membership value t₁. For a complete IFG, CONN_T(G)(u, v) = T_M(u, v) for each u, v ∈ N^∗. So, T_M(v₁, v_i) = t₁ ; 2 ≤ i ≤ n and hence, $$ ; 2 ≤ i ≤ n. Taking summation over i, we have

Similarly, for vertex v₂, we obtain

By adding all the above equations, we get

Now, let v₁ be the vertex with the largest falsity-membership value s₁. For a complete IFG, CONN_F(G)(u, v) = F_M(u, v) for each u, v ∈ N^∗. Thus, F_M(v₁, v_i) = s₁ ; 2 ≤ i ≤ n and hence, $$ ; 2 ≤ i ≤ n. Summing over i, we have

Similarly, for vertex v₂, we have

By adding all the above equations, we get

Hence, by the definition of connectivity, we see

Example 5. In Figure 4, it can be easily seen that k₃ is a complete IFG. So,

Therefore, $$.

Now, we use above theorem

Adding these two summations, we get

Hence, it is verified that

Example 6. Taking the $$G = (N, M) shown in Figure 5. Here, (v₁, v₄), (v₂, v₃), (v₄, v₅) are α-strong arcs, (v₁, v₃), (v₃, v₄) are β-strong arcs, and (v₁, v₂), (v₃, v₅) are δ-arcs. Then,

So, we have $$.

So, we have $$. Thus, $$, which means that $$ of G has been reduced by deleting α -strong edge (v₁, v₄). The IFG, G − (v₁, v₂) is shown in Figure 6(a). If we delete the β -strong edge (v₁, v₃), then the strength of connectedness between every pair of vertices is invariant, i.e., $$ and $$, so $$. The graph of G − (v₁, v₃) is shown in Figure 6(b). Similarly, when we delete the δ -arc (v₁, v₂), then the strength of connectedness between every pair of vertices does not change, and so is the $$. The graph of G − (v₁, v₂) is shown in Figure 6(c).

Theorem 2. Let H be the IF subgraph of an IFG G = (N, M) formed by removing an edge uv ∈ M_G from G. Then, $$ or $$ iff uv is a bridge.

Proof. Take uv as a bridge. According to the definition, there exit u and v such that their strength of connectedness will be decreased. So, we conclude that $$ or $$.

Conversely, suppose that $$ or $$ and consider the possibilities given below.

•  

  Case 1: suppose that uv is a δ-arc. Then, CONN_T(G)−uv(u, v) = CONN_T(G)(u, v) and CONN_F(G)−uv(u, v) = CONN_F(G)(u, v). So, we have $$ and $$ and therefore, $$.

•  

  Case 2: take uv as β-strong edge. Then, T_M(u, v) = CONN_T(G)−uv(u, v) , F_M(u, v) = CONN_F(G)−uv(u, v). This implies that there is another u − v path different from uv edge. Therefore, the removal of the arc uv will have no effect on the strength of connectedness between u and v. So, $$.

•  

  Case 3: now, take uv as α-strong edge. Then, T_M(u, v) > CONN_T(G)−uv(u, v), F_M(u, v) < CONN_F(G)−uv(u, v). So, the only strongest path is uv edge having strength equal to (T_M(u, v), F_M(u, v)). Then, clearly $$, or $$$$$$ or $$$$ since α-strong edges are IF bridges. This implies that uv is a bridge.

Corollary 1. Let H be the IF subgraph of an IFG G = (N, M) formed by removing an edge uv ∈ M_G from G. Then, $$ iff uv is either δ edge or β-strong.

Corollary 2. Suppose that uv an edge of a complete IFG G = (N, M). Then, $$ iff uv is a unique IF bridge of G.

Proof. Let G = (N, M) be a complete IFG. Suppose that $$. Then, uv is the only IF bridge of G.

Conversely, let uv be a unique IF bridge of G. Hence, by Theorem 2, it follows that $$.

Theorem 3. Let G₁ = (N₁, M₁) and G₂ = (N₁, M₂) be the two isomorphic IFGs. Then, $$.

Proof. Suppose that G₁ = (N₁, M₁) and G₂ = (N₂, M₂) are isomorphic IFGs. Then, ∃ is a mapping h : N₁⟶N₂ such that h is bijective and $$ and $$ for all u_i ∈ N^∗ as well as $$ and $$ for (u_i, u_j) ∈ M^∗. As G₁ and G₂ are isomorphic, then the strength of any strongest path between u_i and u_j is equal to that between h(u_i) and h(u_j) in G₂. Thus, for u, v ∈ N^∗

So, we have

Thus, $$. This implies that $$.

Definition 14. The truth and falsity values of the weakest edge in a cycle C are defined to be the strength of C in an IFGG.

Definition 15. Let C denote a cycle in an IFGG. Then, C is called IF strongest strong cycle (IFSSC) if it is the union of two strongest strong u − v paths for each of u and v in C with the exception when uv in C is an IF bridge of G.

Remark 1. We observe that when uv is an IF bridge of G and it lies in C, the condition for C to be the union of two strongest strong u to v paths can be omitted for u and v. Also, CONN_T(G)(a, b) = CONN_T(C)(a, b) and CONN_F(G)(a, b) = CONN_F(C)(a, b)∀a, b ∈ G.

Definition 16. If C is a cycle in an IFGG, then C is said to be strong, provided each of its edges is strong.

Example 7. In Figure 7, we take (T_N(u), F_N(u)) = (0.6, 0.3) for all u ∈ N^∗. The edges (v₁, v₂), (v₃, v₅) are bridges in G. C₁ = v₁v₂v₃v₅v₁ and C₂ = v₅v₃v₄v₅ are strongest strong cycles as C₃ = v₁v₂v₃v₄v₅v₁ is not the union of two strongest strong v₃ − v₅ paths. So, it is not a strong cycle. Also, CONN_T(G)(v₃, v₅) = 0.6 and CONN_F(G)(v₃, v₅) = 0.3. But C₃ has no strongest v₃ − v₅ path. Moreover, we see that (v₃, v₅) is a fuzzy bridge of G that is outside of C₃.

Definition 17. Let G be an IFG. Then, Tθ -evaluation of two vertices u_i and u_j in G is the set θ_T(u_i, u_j) defined by

Note 1.  If cycles through u and v do not exist, then θ_T(u_i, u_j) = ϕ and θ_F(u_i, v_j) = ϕ. With this θ-evaluation, we define another connectivity measure in IFGs called cyclic connectivity (CC).

Definition 18. Let G be an IFG. Then, cycle T-connectivity between u_i and u_j in G is denoted and defined by

Similarly, cycle F-connectivity between u_i and u_j in G is denoted and defined by

Note 2. If θ_T(u_i, u_j) = ϕ and θ_F(u_i, u_j) = ϕ for any two vertices u_i and u_j, then we define cycle T-connectivity and cycle F-connectivity to be zero, i.e., $$ and $$.

Example 8. From Figure 8, we have θ_T(v₁, v₃) = {0.7, 0.6} and hence, $$. Similarly, θ_F(v₁, v₃) = {0.3, 0.4} and therefore, $$.

Theorem 4. Let G be an IFG and for any u_i, u_j ∈ N^∗, both u_i and u_j lie on a common SC. Then,

Proof. Suppose that u_i, u_j ∈ N^∗ such that both of them lie on a common IFSSC. Then, $$, where θ_T(u_i, u_j) = {α ∈ (0,1] : α is T − strength of a strong cycle through u and v}. Therefore, $$ and hence,

Similarly, $$, where θ_F(u_i, u_j) = {β ∈ (0,1] : β is F − strength of a strong cycle through u and v}. Thus, we obtain $$ and hence,

So, we obtain

Example 9. Let G = (N, M) be the IFG in Figure 9, with (T_N(v), F_N(v)) = (0.9, 0.1) for all v ∈ N^∗. Then, by routine calculations, we have $$ and $$.

From Example 10, $$ and the number of pairs in G is $$. By averaging the $$ and $$, we get $$, $$ and $$. Now, consider G − V₄, and we have $$ and $$. On averaging them, we have $$ and $$. The overall connectivity of G is increased by deleting vertex v₄ from G. The following definitions and results are led by this example.

Definition 19. Let G = (N, M) be an IFG. Then, the average T-connectivity index of G is denoted and defined as

Definition 20. Let G = (N, M) be an IFG. The average connectivity index of G is defined to be the sum of average T- connectivity index and average F- connectivity index of G, i.e.,

Note 3.  Obviously, $$ will not be enhanced by the removal of an edge, and therefore, $$ also. For an IFG, $$.

Definition 21. Let G = (N, M) be an IFG and u ∈ N^∗. Then, u is said to be an IF connectivity reducing node (IFCRN) of G if $$. u is said to be an IF connectivity enhancing node (IFCEN) of G if $$. u is said to be an IF neutral node of G if $$.

Example 10. Consider the IFGG as shown in Figure 10. We have taken here (T_N(u), F_N(u)) = (0.5, 0.5) for all u ∈ N^∗. $$, $$ and $$.Thus v₁, v₃ are IFCRN s, v₂ is an IF neutral node, and v₄ is a IFCEN. We characterize these nodes using $$ in the following theorem.

Theorem 5. Let G = (N, M) be an IFG and u ∈ N^∗ with |N^∗| ≥ 3. Let $$. u is a IFCEN iff r < n/(n − 2). u is a IFCRN iff r > n/(n − 2). u is an IF neutral node iff r = n/(n − 2).

Proof. Suppose that u is an IF neutral node. Then, by definition, $$. By definition of average connectivity index, we obtain

From here, we get

The converse part can be proved by reversing the arguments. Similarly, we can prove other cases.

Note 4. An isolated vertex w in an IFG, and G = (N, M) is an IFCEN. Further, if $$ denote complement of IFGG = (N, M), then $$ and therefore,

Theorem 6. Let G = (N, M) be an IFG with |N^∗| ≥ 3. If w ∈ N^∗ is an end vertex of G, let $$. Then,

• (1)

  w is an IFCEN if $$

• (2)

  w is an IFCRN if $$

• (3)

  w is an IF neutral node if $$

Proof. Let w be an IF neutral node. Then, by definition, $$. We see that

The converse part can be proved by reversing these steps. Similarly, we can prove other parts.

At this stage, we can classify an IFG depending on the nature of vertices in it.

Definition 22. An IFGG containing at least one IFCEN is called an IF connectivity enhancing graph. If there is no IFCEN node in G and at least one IFCRN, then G is said to be an IF connectivity reducing graph. If G has all vertices as IF neutral nodes, then it is said to be an IF neutral graph.