Graphs have many applications in different fields such as computers, systems analysis, networks, transportation, operations research, and economics. Graphs are usually used to model relationships among objects. But there are many issues that are vague and uncertain as a result of the information loss, lack of evidence, incomplete statistical data, etc. In general, uncertainty exists in many real life problems and is an integral part of them. In a classical graph, for each vertex or edge, the probability of uncertainty existence or non-existence is assumed. Therefore, classical graphs cannot model uncertain problems. However, often real-life problems are uncertain, which makes it difficult to model using conventional methods. Zadeh [1] presented an extended version of sets, called fuzzy set (FS), where objects have different degrees of membership between zero and one. This concept quickly found wide applications in computer science, information science, system science, management science, theoretical mathematics and other fields of sciences. A decade after the introduction of FS, Zadeh [2] presented an interval-valued fuzzy set (IVFS) as a branch of FS in which an interval between 0 and 1 was used as the membership value instead of a fuzzy number. These two concepts gave rise to different types of graphs called fuzzy graphs, which were first introduced by Kaufman [3] in 1973. Later, fuzzy graph theory was developed as a generalization of graph theory by Rosenfeld [4] in 1975. He explained some concepts including tree, cut vertex, cycle, bridge, and end vertex in fuzzy graphs. The researchers studied different types of fuzzy graphs. Talebi [5] had a study on Kayley fuzzy graph. Borzooei et al. [6] had many studies on vague graphs. Atanassov [7] introduced the concept of intuitionistic fuzzy set (IFS) as a generalization of FS. Akram and Dudek [8] gave the idea of an interval-valued fuzzy graph (IVFG) in 2011. Talebi et al. [9, 10] introduced some new concepts of interval-valued intuitionistic fuzzy graph (IVIFG). Kosari et al. [11–13] studied new results in vague graph and vague graph structures. Some trend concepts in fuzzy graphs were explained by Pal et al. [14]. Samanta et al. [15, 16] reviewed some results from fuzzy k–competition graphs.

Graph structures were presented by Sampathkumar [17] in 2006 as a generalization of signed graphs and graphs with labeled or colored edges. Fuzzy graph structure (FGS) is more important than graph structure because uncertainty and ambiguity in many real-world phenomena often occur as two or more separate relationships. Dinesh [18] introduced the notion of an FGS and discussed some related properties. Ramakrishnan and Dinesh [19] generalized this concept in studies. Akram [20] presented new results on m-polar FGSs. Akram and Akmal [21–23] investigated the concepts of bipolar FGSs and intuitionistic FGSs. Akram et al. [24–28] defined new concepts of operations in FGSs. Kou et al. [29] studied vague graph structure. Continuing his studies in 2020, Denish [30] presented the concept of fuzzy incidence graph structure. Akram and Sitara [31] introduced decision-making with q-rung orthopair FGSs. Sitara and Zafar [32] studied the application of q-rung picture FGSs in airline services.

Fuzzy graphs were previously limited to one or more degrees of fuzzy membership or interval-valued fuzzy membership. Jun et al. [33] introduced the idea of a cubic fuzzy set (CFS) in the form of a combination of FS and IVFS, serving as a more general tool for modeling uncertainty and ambiguity. By applying this concept, various problems that arise from uncertainties can be solved and the best choice can be made using CFS in decision making. Jun et al. [34] combined the neutrosophic complex with CFS and proposed the idea of neutrosophic CFS. Jun et al., also, studied some CFS-based algebraic features including cubic IVIFSs [35], cubic structures [36], cubic sets in semigroups [37], cubic soft sets [38], and cubic intuitionistic structures [39]. Muhiuddin et al. [40] presented the stable CFSs idea. Kishore Kumar et al. [41] examined the regularity concept in CFG. Rashid et al. [42] introduced the concept of a CFG where they introduced many new types of graphs and their applications. A modified definition of a CFG is given by Muhiuddin et al [43] along with concepts such as the strong edge, path, path strength, bridge, and cut vertex. Rashmanlou et al [44] explained some of the concepts of the CFG.

The concept of node order and degree plays an important role in graph theory and its applications, including the analysis of social networks, road transmission networks, wireless networks, etc. Vertex degree is an accepted concept to represent the total number of relations of a vertex in a graph that can be used in graph analysis. Gani and Radha [45] offered the notation of the regular FG. Samanta and Pal [46] introduced the concept of the irregular bipolar fuzzy graphs. Borzooei et al. [47] investigated the Regularity of vague graphs. Gani and Lathi [48] defined the concept of irregularity, total irregularity, and total degree in an FG. Huang et al. [49] studied regular and irregular Neutrozophic graphs with real applications. Samanta et al. [50] investigated the completeness and regularity of generalized fuzzy graphs. These concepts have been gradually developed by researchers into different types of FGs.

Cubic fuzzy graph structure (CFGS), as a combination of FGS and CFG, has better flexibility in modeling and solving problems in ambiguous and uncertain fields. The study of regularity in the CFGS that supports multiple relationships is important and decisive in its own way. In fact, checking regularity is essential from the point of view that most of the issues around us are composed of several different relationships. The quality and method of determining the degree of the vertices in the cubic fuzzy graph structure, has fuzzy membership and interval-valued fuzzy membership at the same time besides the multiplicity of existing relations, made us carry out a study on the regularity of cubic fuzzy graph structures. In this paper, we introduce regularity in a CFGS. We were able to investigate the corresponding properties by defining the degree of a vertex and the total degree of a vertex. In the following, by introducing the order and size in the CFGS, some relevant results were studied. Finally, an application of the CFGS in the detection of bank criminals is presented.

In this section, we have an overview of the basic concepts in fuzzy graphs in order to enter the main discussion.

A graph is a pair of G = (V, E), where V is a non-empty set of vertices and E is the set of edges of G. A graph structure (GS) of X = (V, E ₁, E ₂, …, E _k ) consists of a set V with relations of E ₁, E ₂, …, E _k on V, all of which are mutually disjoint and each E _i is irreflexive and symmetric, for i = 1, 2, …, k. If (x, y) ∈ E _i for some i = 1, 2, …, k, then, it is called an E _i −edge and is simply written xy. A GS is complete whenever each E _i −edge appears at least once and between each pair of vertices of x, y ∈ V, xy ∈ E _i for some i = 1, 2, …, k. A path between two vertices of x and y consisting of only E _i −edges is named E _i −path. Reciprocally, E _i −cycle is a cycle consisting of only E _i −edges. A GS is a tree, if it is connected and contains no cycle. If the subgraph structure induced by E _i −edges is a tree, then, it is an E _i −tree. A GS is an E _i −forest, if the subgraph structure induced by E _i −edges is a forest [17].

A fuzzy graph (FG) on V is a pair of G = (τ, μ), where τ is a fuzzy subset (FS) of V and μ is a fuzzy relation on τ so that μ(x, y) ≤ τ(x) ∧ τ(y), ∀x, y ∈ V. The underlying crisp graph of G is the graph G* = (τ*, μ*), where τ* = {x ∈ ∣τ(x) > 0} and μ* = {xy ∈ V × V∣μ(xy) > 0}. An FG S = (λ, η) on V is a partial fuzzy subgraph of G if λ ≤ τ and η ≤ μ. A fuzzy subgraph S is a spanning fuzzy subgraph of G if τ = λ [14].

An interval-valued fuzzy set (IVFS) A on V is described by A = α x , β x ∣ x ∈ V where α and β are FSs of V so that α(x) ≤ β(x) for all x ∈ V. [14]

A cubic fuzzy set (CFS) [33] A on V is described as A = ⟨ α z , β z , γ z ⟩ ∣ z ∈ V , where [α(z), β(z)] is named the interval-valued fuzzy membership degree and γ(z) is named the fuzzy membership degree of z, so that α, β, γ: V → [0, 1].

The CFS A is called an internal CFS if γ(z) ∈ [α(z), β(z)], and external CFS whenever γ(z)∉[α(z), β(z)], for all z ∈ V.

Definition 2.1 [19]. Let Z = (V, E ₁, E ₂, …, E _k ) be a GS. Then, Z = ( τ , φ 1 , φ 2 , … , φ k ) is named the fuzzy graph structure (FGS) of Z whenever τ, φ ₁, φ ₂, …, φ _k are fuzzy subset on V, E ₁, E ₂, …, E _k , respectively, so that φ i a b ≤ τ a ∧ τ b , ∀ a , b ∈ V , i = 1,2 , … , k .

If ab ∈ supp(φ _i ), then, ab is called a φ _i −edge of Z .

Definition 2.2 [43]. A cubic fuzzy graph (CFG) on a non-empty set V is a pair of G = ( A , B ) where A = { ⟨ [ α ( z ) , β ( z ) ] , γ ( z ) ⟩ ∣ z ∈ V } is a CFS on V and B = { ⟨ [ α ( w z ) , β ( w z ) ] , γ ( w z ) ⟩ ∣ w z ∈ E } is a CFS on V × V, so that for all z, w ∈ V , α B z w ≤ α A z ∧ α A w , β B z w ≤ β A z ∧ β A w , γ B z w ≤ γ A z ∧ γ A w .

Definition 2.3. Let V be a non-empty set and G* = (V, E ₁, E ₂, …, E _k ) be a GS. Then, G = ( A , B 1 , B 2 , … , B k ) is named a cubic fuzzy graph structure (CFGS) on G* if A = { ⟨ [ α ( z ) , β ( z ) ] , γ ( z ) ⟩ ∣ z ∈ V } is a CFS on V and B i = { ⟨ [ α B i ( w z ) , β B i ( w z ) ] , γ B i ( w z ) ⟩ ∣ w z ∈ E i } is CFS on E _i , respectively, so that α B i z w ≤ α A z ∧ α A w , β B i z w ≤ β A z ∧ β A w , γ B i z w ≤ γ A z ∧ γ A w , f o r a l l z w ∈ E i a n d i = 1,2 , … , k . If z w ∈ s u p p ( B i ) , then, zw is named as B i -edge of CFGS G . Obviously, [α _i , β _i ] and γ _i are named the membership function of B i − edges. Furthermore, B 1 , B 2 , … , B k are mutually disjoint so that each α _i , β _i and γ _i is symmetric and irreflexive, for 1 ≤ i ≤ k .

Example 2.4. Consider the GS G* = (V, E ₁, E ₂, E ₃) where V = {x, y, z, t, u, w}, E ₁ = {xy, tu}, E ₂ = {yz, tw}, and E ₃ = {wy, tz}. We define the CFSs A , B 1 , B 2 , and B 3 on V , E ₁ , E ₂, and E ₃, respectively, as follows: A = x , 〈 0.2 , 0.5 , 0.6 〉 , y , 〈 0.4 , 0.6 , 0.7 〉 , z , 〈 0.3 , 0.5 , 0.4 〉 , t , 〈 0.3 , 0.4 , 0.5 〉 , × u , 〈 0.6 , 0.7 , 0.2 〉 , w , 〈 0.5 , 0.7 , 0.8 〉 , B 1 = { x y , 〈 0.2 , 0.4 , 0.5 〉 , t u , 〈 0.2 , 0.4 , 0.2 〉 } , B 2 = { y z , 〈 0.2 , 0.3 , 0.4 〉 , t w , 〈 0.3 , 0.4 , 0.4 〉 } , B 3 = { w y , 〈 0.4 , 0.6 , 0.7 〉 , t z , 〈 0.3 , 0.4 , 0.4 〉 } . Then, the CFGS G = ( A , B 1 , B 2 , B 3 ) on G* is shown in Figure 1 .

Definition 2.5. A CFGS G = ( A , B 1 , B 2 , … , B k ) is B i -strong if α B i z w = α A z ∧ α A w , β B i z w = β A z ∧ β A w , γ B i z w = γ A z ∧ γ A w , f o r a l l z w ∈ E i . If G is B i -strong for all i = 1, 2, …, k, then, G is named strong CFGS.

Definition 2.6. A CFGS G = ( A , B 1 , B 2 , … , B k ) is named complete CFGS if α B i z w = α A z ∧ α A w , β B i z w = β A z ∧ β A w , γ B i z w = γ A z ∧ γ A w , f o r a l l z , w ∈ V .

Definition 2.7. A CFGS G = ( A , B 1 , B 2 , … , B k ) is named B i -connected if all vertices are connected by B i -edges.

Some abbreviations in the article are listed in Table 1.

In this section, vertex regularity in cubic fuzzy graph structures is discussed and some of its properties are examined.

Definition 3.1. Let G = ( A , B 1 , B 2 , … , B k ) be a CFGS. B i -degree of a vertex z is determined as D B i ( z ) = ⟨ [ D α i ( z ) , D β i ( z ) ] , D γ i ( z ) ⟩ , where D α i z = ∑ w z ∈ E i , z ≠ w α B i w z , D β i z = ∑ w z ∈ E i , z ≠ w β B i w z , D γ i z = ∑ w z ∈ E i , z ≠ w γ B i w z . Also, B i 1 i 2 ⋯ i r -degree of a vertex z is determined as D B i 1 i 2 ⋯ i r ( z ) = ⟨ [ D α i 1 i 2 ⋯ i r ( z ) , D β i 1 i 2 ⋯ i r ( z ) ] , D γ i 1 i 2 ⋯ i r ( z ) ⟩ , where D α i 1 i 2 ⋯ i r z = ∑ j = 1 r ∑ w z ∈ E i j , z ≠ w α B i j w z , D β i 1 i 2 ⋯ i r z = ∑ j = 1 r ∑ w z ∈ E i j , z ≠ w β B i j w z , D γ i 1 i 2 ⋯ i r z = ∑ j = 1 r ∑ w z ∈ E i j , z ≠ w γ B i j w z . The full-degree z is defined as D G ( z ) = ⟨ [ D α ( z ) , D β ( z ) ] , D γ ( z ) ⟩ , where D α z = ∑ i = 1 k ∑ w z ∈ E i , z ≠ w α B i w z , D β z = ∑ i = 1 k ∑ w z ∈ E i , z ≠ w β B i w z , D γ z = ∑ i = 1 k ∑ w z ∈ E i , z ≠ w γ B i w z .

Definition 3.2. Let G = ( A , B 1 , B 2 , … , B k ) be a CFGS. If all vertices have the same B i -degree ⟨[a, b], c⟩, then, G is named a ⟨[a, b], c⟩- B i -vertex regular CFGS. Also, G is named a ⟨[a, b], c⟩- B i 1 i 2 ⋯ i r -vertex regular CFGS whenever all vertices have the same B i 1 i 2 ⋯ i r -degree ⟨[a, b], c⟩. It is clear that every connected CFGS with two vertices is regular.

Considering the membership degree of the vertex, we define the total degree of the vertex as follows:

Example 3.3. Consider CFGS G = ( A , B 1 , B 2 ) is shown in Figure 2, where A = 〈 z 1 , 0.6 , 0.7 , 0.8 〉 , 〈 z 2 , 0.5 , 0.6 , 0.7 〉 , 〈 z 3 , 0.7 , 0.8 , 0.9 〉 , 〈 z 4 , 0.8 , 0.9 , 1 〉 , B 1 = 〈 z 1 z 2 , 0.4 , 0.5 , 0.6 〉 , 〈 z 3 z 4 , 0.4 , 0.5 , 0.6 〉 , B 2 = 〈 z 1 z 4 , 0.3 , 0.4 , 0.5 〉 , 〈 z 2 z 3 , 0.3 , 0.4 , 0.5 〉 . The B 1 -degree of vertices is equal to ⟨[0.4, 0.5], 0.6⟩. Also, B 2 -degree of vertices equals ⟨[0.3, 0.4], 0.5⟩. Therefore, B 1,2 -degree of vertices is equal to ⟨[0.7, 0.9], 1.1⟩. Hence, G is a ⟨[0.4, 0.5], 0.6⟩- B 1 -vertex regular, ⟨[0.3, 0.4], 0.5⟩- B 2 -vertex regular, and ⟨[0.7, 0.9], 1.1⟩- B 1,2 -vertex regular CFGS.

Definition 3.4. Let G = ( A , B 1 , B 2 , … , B k ) be a CFGS. The total B i -degree of a vertex z is shown as T D B i ( z ) = ⟨ [ T D α i ( z ) , T D β i ( z ) ] , T D γ i ( z ) ⟩ , where T D α i z = ∑ w z ∈ E i , z ≠ w α B i w z + α A z , T D β i z = ∑ w z ∈ E i , z ≠ w β B i w z + β A z , T D γ i z = ∑ w z ∈ E i , z ≠ w γ B i w z + γ A z .

Also, total B i 1 i 2 ⋯ i r -degree of a vertex z is determined as T D B i 1 i 2 ⋯ i r z = ⟨ T D α i 1 i 2 ⋯ i r z , T D β i 1 i 2 ⋯ i r z , T D γ i 1 i 2 ⋯ i r z ⟩ , where T D α i 1 i 2 ⋯ i r z = ∑ j = 1 r ∑ w z ∈ E i j , z ≠ w α B i j w z + α A z , T D β i 1 i 2 ⋯ i r z = ∑ j = 1 r ∑ w z ∈ E i j , z ≠ w β B i j w z + β A z , T D γ i 1 i 2 ⋯ i r z = ∑ j = 1 r ∑ w z ∈ E i j , z ≠ w γ B i j w z + γ A z . The totally full-degree z is defined as T D G ( z ) = ⟨ [ T D α ( z ) , T D β ( z ) ] , T D γ ( z ) ⟩ , where T D α z = ∑ i = 1 k ∑ w z ∈ E i , z ≠ w α B i w z + α A z , T D β z = ∑ i = 1 k ∑ w z ∈ E i , z ≠ w β B i w z + β A z , T D γ z = ∑ i = 1 k ∑ w z ∈ E i , z ≠ w γ B i w z + γ A z .

Definition 3.5. Let G = ( A , B 1 , B 2 , … , B k ) be a CFGS. If all vertices have the same total B i -degree ⟨[a, b], c⟩, then, G is named a ⟨[a, b], c⟩- B i -total vertex regular CFGS. Also, G is named a ⟨[a, b], c⟩- B i 1 i 2 ⋯ i r -total vertex regular CFGS whenever all vertices have the same total B i 1 i 2 ⋯ i r -degree ⟨[a, b], c⟩.

The following definitions determine the maximum or minimum degree of a vertex in CFGS.

Definition 3.6. Let G = ( A , B 1 , B 2 , … , B k ) be a CFGS. The minimum vertex B i -degree of G is defined as δ B i ( G ) = ⟨ [ δ α i ( G ) , δ β i ( G ) ] , δ γ i ( G ) ⟩ , where δ α i G = ⋀ D α i z , z ∈ V , δ β i G = ⋀ D β i z , z ∈ V , δ γ i G = ⋀ D γ i z , z ∈ V . Also, the minimum vertex B i 1 i 2 ⋯ i r -degree of G is determined as δ B i 1 i 2 ⋯ i r G = ⟨ δ α i 1 i 2 ⋯ i r G , δ β i 1 i 2 ⋯ i r G , δ γ i 1 i 2 ⋯ i r G ⟩ , where δ α i 1 i 2 ⋯ i r G = ⋀ D α i 1 i 2 ⋯ i r z , z ∈ V , δ β i 1 i 2 ⋯ i r G = ⋀ D β i 1 i 2 ⋯ i r z , z ∈ V , δ γ i 1 i 2 ⋯ i r G = ⋀ D γ i 1 i 2 ⋯ i r z , z ∈ V .

Definition 3.7. Let G = ( A , B 1 , B 2 , … , B k ) be a CFGS. The maximum vertex B i -degree of G is defined as Δ B i ( G ) = ⟨ [ Δ α i ( G ) , Δ β i ( G ) ] , Δ γ i ( G ) ⟩ , where Δ α i G = ⋁ D α i z , z ∈ V , Δ β i G = ⋁ D β i z , z ∈ V , Δ γ i G = ⋁ D γ i z , z ∈ V . Also, the maximum vertex B i 1 i 2 ⋯ i r -degree of G is defined as Δ B i 1 i 2 ⋯ i r ( G ) = ⟨ [ Δ α i 1 i 2 ⋯ i r ( G ) , Δ β i 1 i 2 ⋯ i r ( G ) ] , Δ γ i 1 i 2 ⋯ i r ( G ) ⟩ , where Δ α i 1 i 2 ⋯ i r G = ⋁ D α i 1 i 2 ⋯ i r z , z ∈ V , Δ β i 1 i 2 ⋯ i r G = ⋁ D β i 1 i 2 ⋯ i r z , z ∈ V , Δ γ i 1 i 2 ⋯ i r G = ⋁ D γ i 1 i 2 ⋯ i r z , z ∈ V .

Definition 3.8. Let G = ( A , B 1 , B 2 , … , B k ) be a CFGS. The minimum total vertex B i -degree of G is defined as δ B i t ( G ) = ⟨ [ δ α i t ( G ) , δ β i t ( G ) ] , δ γ i t ( G ) ⟩ , where δ α i t G = ⋀ T D α i z , z ∈ V , δ β i t G = ⋀ T D β i z , z ∈ V , δ γ i t G = ⋀ T D γ i z , z ∈ V . Also, the minimum total vertex B i 1 i 2 ⋯ i r -degree of G is determined as δ B i 1 i 2 ⋯ i r t G = ⟨ δ α i 1 i 2 ⋯ i r t G , δ β i 1 i 2 ⋯ i r t G , δ γ i 1 i 2 ⋯ i r t G ⟩ , where δ α i 1 i 2 ⋯ i r t G = ⋀ T D α i 1 i 2 ⋯ i r z , z ∈ V , δ β i 1 i 2 ⋯ i r t G = ⋀ T D β i 1 i 2 ⋯ i r z , z ∈ V , δ γ i 1 i 2 ⋯ i r t G = ⋀ T D γ i 1 i 2 ⋯ i r z , z ∈ V .

Definition 3.9. Let G = ( A , B 1 , B 2 , … , B k ) be a CFGS. The maximum total vertex B i -degree of G is defined as Δ B i t ( G ) = ⟨ [ Δ α i t ( G ) , Δ β i t ( G ) ] , Δ γ i t ( G ) ⟩ , where Δ α i t G = ⋁ T D α i z , z ∈ V , Δ β i t G = ⋁ T D β i z , z ∈ V , Δ γ i t G = ⋁ T D γ i z , z ∈ V . Also, the maximum total vertex B i 1 i 2 ⋯ i r -degree of G is determined as Δ B i 1 i 2 ⋯ i r t G = ⟨ Δ α i 1 i 2 ⋯ i r t G , Δ β i 1 i 2 ⋯ i r t G , Δ γ i 1 i 2 ⋯ i r t G ⟩ , where Δ α i 1 i 2 ⋯ i r t G = ⋁ T D α i 1 i 2 ⋯ i r z , z ∈ V , Δ β i 1 i 2 ⋯ i r t G = ⋁ T D β i 1 i 2 ⋯ i r z , z ∈ V , Δ γ i 1 i 2 ⋯ i r t G = ⋁ T D γ i 1 i 2 ⋯ i r z , z ∈ V .

Example 3.10. Consider CFGS of G = ( A , B 1 , B 2 ) as shown in Figure 3, where A = 〈 z 1 , 0.7 , 0.9 , 1 〉 , 〈 z 2 , 0.5 , 0.6 , 0.6 〉 , 〈 z 3 , 0.6 , 0.8 , 1 〉 , 〈 z 4 , 0.4 , 0.5 , 0.6 〉 , B 1 = 〈 z 1 z 2 , 0.3 , 0.4 , 0.5 〉 , 〈 z 2 z 4 , 0.2 , 0.3 , 0.4 〉 , 〈 z 3 z 4 , 0.4 , 0.5 , 0.5 〉 , B 2 = 〈 z 1 z 4 , 0.3 , 0.4 , 0.5 〉 , 〈 z 1 z 3 , 0.1 , 0.2 , 0.3 〉 , 〈 z 2 z 3 , 0.4 , 0.5 , 0.6 〉 . The total B 1 -degree of vertices is equal to ⟨[1, 1.3], 1.5⟩. Therefore, G is a ⟨[1, 1.3], 1.5⟩- B 1 -total vertex regular CFGS. As it can be seen δ B 1 t G = Δ B 1 t G = ⟨ 1,1.3 , 1.5 ⟩ .

Remark 3.11. A CFGS G = ( A , B 1 , B 2 , … , B k ) is named ⟨[a, b], c⟩- B i -vertex regular if δ B i G = Δ B i G = ⟨ a , b , c ⟩ , and G is named ⟨[a, b], c⟩- B i 1 i 2 ⋯ i r -vertex regular if δ B i 1 i 2 ⋯ i r G = Δ B i 1 i 2 ⋯ i r G = ⟨ a , b , c ⟩ .

Also, G is named ⟨[a, b], c⟩- B i -total vertex regular if δ B i t G = Δ B i t G = ⟨ a , b , c ⟩ , and G is named ⟨[a, b], c⟩- B i 1 i 2 ⋯ i r -total vertex regular if δ B i 1 i 2 ⋯ i r t G = Δ B i 1 i 2 ⋯ i r t G = ⟨ a , b , c ⟩ .

Theorem 3.12. Let G = ( A , B 1 , B 2 , … , B k ) be a CFGS which is both a B i -vertex regular and a B i -total vertex regular, then, α A , β A , and γ A are constant.

Proof. Suppose G = ( A , B 1 , B 2 , … , B k ) is a ⟨[a, b], c⟩- B i -vertex regular and a ⟨[a′, b′], c′⟩- B i -total vertex regular CFGS. Then, for all z ∈ V D B i z = 〈 D α i z , D β i z , D γ i z 〉 = 〈 a , b , c 〉 , T D B i z = 〈 T D α i z , T D β i z , T D γ i z 〉 = 〈 a ′ , b ′ , c ′ 〉 .

Thus, by definition T D α i z = D α i z + α A z , T D β i z = D β i z + β A z , T D γ i z = D γ i z + γ A z .

Therefore, α A z = a ′ − a , β A z = b ′ − b , γ A z = c ′ − c , f o r a l l z ∈ V . Hence, α A , β A , and γ A are constant.

The following example shows that the opposite of the above theorem is not necessarily true.

Example 3.13. Consider CFGS G = ( A , B 1 , B 2 ) is shown in Figure 4, where z i = ⟨ 0.4 , 0.5 , 0.6 ⟩ , i = 1,2 , … , 6 , B 1 = ⟨ z 1 z 2 , 0.3 , 0.4 , 0.5 ⟩ , ⟨ z 3 z 4 , 0.4 , 0.5 , 0.6 ⟩ , ⟨ z 3 z 6 , 0.3 , 0.4 , 0.5 ⟩ , B 2 = ⟨ z 1 z 3 , 0.1 , 0.2 , 0.3 ⟩ , ⟨ z 2 z 4 , 0.5 , 0.6 , 0.7 ⟩ , ⟨ z 3 z 5 , 0.2 , 0.3 , 0.4 ⟩ .

Here, α A , β A , and γ A are a constant functions. But G is neither B i 1 i 2 -vertex regular CFGS nor a B i 1 i 2 -total vertex regular CFGS.