Definition 1 (see [66].)A neutrosophic cubic graph $$ for a crisp graph G = (A, B) is a pair with

Definition 2. An open neighborhood ℕ_ncg(g) for any vertex g in $$ is given by

where

It consists of membership functions for all vertices adjacent to g, excluding s.

Definition 3. The degree of open neighborhood θ(ℕ_ncg(g)) for any vertex g in $$ is defined by

Example 4. Let G = (A, B) with vertices A = {g₁, g₂, g₃} and edges B = {g₁g₂, g₂g₃, g₁g₃}. Also, $$ such that

Definition 5. A closed neighborhood degree for a vertex g in $$ is given by

It consists of the sum of membership functions of all vertices adjacent to x and membership function of x.

Example 6. Consider Example 4, closed neighborhood degree θ(ℕ_ncg[g]) for each element g ∈ A in $$ is given by

Definition 7. If every vertex in $$ has the same open neighborhood degree n, i.e., if θ(ℕ_ncg(g)) = n, for all g ∈ A, then $$ is called an n regular neutrosophic cubic graph.

Definition 8. If closed neighborhood degree is the same for all vertices in $$, i.e., if θ(ℕ_ncg[g]) = m, for all g ∈ A, then $$ is called an m-totally regular neutrosophic cubic graph.

Example 9. Consider Example 4; here, $$ is a totally regular but not a regular neutrosophic cubic graph.

Example 10. Consider $$ for any graph G = (A, B) with A = {v₁, v₂, v₃}, B = {v₁v₂, v₂v₃, v₁v₃} and let

Example 11. Let $$ be a neutrosophic cubic graph of G = (A, B) with A = {v₁, v₂, v₃}, B = {v₁v₂, v₂v₃, v₁v₃} such that Γ and Λ are given by Tables 2 and 3.

Theorem 12. Let $$ be a neutrosophic cubic graph of G, with Γ showing $$ for vertex set A and Λ is $$ for edge set B. Then,

• (I)

  $$ is regular

• (II)

  $$ is totally regular

Proof. Suppose

(I)⇒(II) Let $$ be a regular, then θ(ℕ_ncg(g)) = n for all g ∈ A. So,

Hence, for all g ∈ A, we have

Thus,

(II)⇒(I) Suppose that $$ is totally regular. Then, for all g ∈ A,

Let

Then,

a constant no for every g ∈ A. Hence,

for all g ∈ A is a constant function.

Theorem 13. Consider $$ as a neutrosophic cubic graph with crisp graph G of an odd cycle. Then, $$ is regular if and only if Λ is a constant function.

Proof. Let

be a constant function for all xy ∈ B. Then,

Now,

Since G is an odd cycle, $$ is regular. Conversely, suppose that $$ is an n- regular, where

Let e₁, e₂, e₃, ⋯, e_2n+1 be edges of $$ in that order. Let

This implies

So, if e₁ and e_2n+1 are incident at a vertex v₁, then

Hence,

Thus, $$ So, if e₂ and e_2n are incident at a vertex v₂, then

Hence,

and so, k₂ = n_2t/2, which shows that $$ is a constant function.

Similar results hold for membership functions $$. This shows that

is a constant function.

Definition 14. Consider $$ be a neutrosophic cubic graph for any arbitrary graph G = (A, B). Then, $$ is complete if

Example 15. Consider $$ for a graph G = (A, B) with A = {v₁, v₂, v₃}, B = {v₁v₂, v₂v₃, v₁v₃} and let

Also,

Definition 16. Let $$ be a neutrosophic cubic graph for some graph G = (V, E). The density of $$ is defined as

Example 17. Consider $$ for a graph G = (A, B), with vertex set A = {v₁, v₂, v₃} and edge set B = {v₁v₂, v₂v₃, v₁v₃}. Also, let Γ and Λ be neutrosophic membership functions for vertices and edges, respectively, shown in Tables 4 and 5.

Definition 18. $$ is balanced if $$ for all subgraphs H of $$.

Definition 19. $$ is strictly balanced if $$ for all nonempty subgraphs H of $$.

Example 20. Consider $$ as given in Example 17. Let H₁ = {a, b}, H₂ = {a, c}, H₃ = {b, c}. Then,

Hence,

Remark 21. All regular neutrosophic cubic graphs are not necessarily balanced.

Definition 22. If there is at least one vertex in $$ adjacent to vertices having different open neighborhood degrees, then $$ is called irregular, i.e., if θ(ℕ_ncg(v)) ≠ n for all v ∈ A.

Example 23. Consider $$ for some graph G = (A, B), with A = {v₁, v₂, v₃, v₄} and

Definition 24. A connected $$ is totally irregular, if at least one vertex is adjacent to the vertices having different closed neighborhood degrees.

Example 25. Consider $$ for G = (A, B), with

and let

Then,

Definition 26. The complement of $$ is a neutrosophic cubic graph $$ where

similarly, for indeterminate membership functions, we have

Proposition 27. For self-complementary $$, we have

Proof. Given $$ is self-complementary, so $$; also, by definition of a self-complementary neutrosophic cubic graph, we have

Dividing both sides of equation (77) by min{Φ(x), Φ(y)},

Hence,

Proposition 28. Let $$ be strictly balanced, let $$ be its complement, then $$

Proof. Let $$ be strictly balanced; let $$ be its complement. Let H be a nonempty subgraph of $$ Since $$ is strictly balanced, $$ for every subset $$ and for any x, y ∈ A. In $$, we have $$, and so, for truth membership functions, we have

Dividing equation (82) by $$, we get

Hence,

Multiplying both sides by 2,we get

Hence, $$. Similarly for right end point of interval in truth valued membership functions, we have $$. Similar results hold for the rest of membership functions. Hence,

This completes the proof.

Definition 29. A connected $$ is neighborly irregular if every two adjacent vertices in $$ have different closed neighborhood degrees.

Example 30. Consider $$ for any G = (A, B) with A = {a, b, c, d} and B = {ab, bc, cd, da} and let neutrosophic cubic membership functions for vertices and edges be represented in Tables 6 and 7, respectively.

Definition 31. $$ is said to be neighborly totally irregular if every two adjacent vertices of $$ have different closed neighborhood degrees.

Example 32. Consider $$ for any G = (A, B) with A = {v₁, v₂, v₃, v₄} and B = {v₁v₂, v₂v₃, v₃v₄, v₁v₄}. Also, let

Definition 33. Consider a connected $$; then, $$ is highly irregular if every vertex in $$ is adjacent to vertices with different neighborhood degrees.

Example 34. Consider $$ for a graph G = (A, B) with A = {v₁, v₂, v₃, v₄, v₅, v₆} and B = {v₁v₂, v₂v₃, v₂v₆, v₃v₄, v₃v₅, v₄v₅, v₁v₅} and let

By routine computations, we have

Theorem 35. $$ is highly irregular and neighborly irregular if and only if open neighborhood degrees for all vertices of $$ are different.

Proof. Suppose $$ has n vertices v₁, v₂, ⋯, v_n. Also, let $$ be highly irregular and neighborly irregular.

Claim 1. The open neighborhood degrees for all vertices in $$ are different. Let

for all i = 1, 2, 3, ⋯, n. Let the adjacent vertices of v₁ be v₂, v₃, ⋯, v_n with open neighborhood degrees:

Claim 2. $$ is highly irregular and neighborly irregular. Let

Remark 36. A complete $$ may not be neighborly irregular.

Example 37. Let $$ for any G = (A, B), with A = {v₁, v₂, v₃} and B = {v₁v₂, v₂v₃, v₁v₃} such that

By simple computation, we get

Theorem 38. If $$ is neighborly irregular and

Proof. Assume that $$ is a neighborly irregular. Then, open neighborhood degrees of every two adjacent vertices are different. Let v_i, v_j ∈ A be adjacent vertices with different open neighborhood degrees. Then, θ(ℕ_ncg(v_i)) ≠ θ(ℕ_ncg(v_j)) for all i ≠ j; let θ(ℕ_ncg(v_i)) = d₁&θ(ℕ_ncg(v_j)) = d₂ then d₁ ≠ d₂. Also, as

Hence,

Theorem 39. If $$ is neighborly totally irregular and

Proof. Assume that $$ is a neighborly totally irregular. Then, closed neighborhood degrees of every two adjacent vertices are distinct. Let v_i, v_j ∈ A be adjacent vertices with distinct closed neighborhood degrees. Then, for all i ≠ j,

Proposition 40. If $$ is neighborly irregular as well as neighborly totally irregular, then

Remark 41. If $$ is neighborly irregular, then a neutrosophic cubic subgraph H of $$ may not be neighborly irregular.

Remark 42. If $$ is neighborly totally irregular, then a neutrosophic cubic subgraph H of $$ may not be neighborly totally irregular.