Cubic bipolar fuzzy graphs with applications

Abstract

Recently, Jun et al. have introduced the concept of cubic set as a generalization of the concept of fuzzy set and that of the interval valued fuzzy set. This concept has been widely applied in many circumstances like pattern recognition, decision making etc. So far, no attention has been paid towards graph of cubic set therefore leads us in this manuscript to study the concepts of interval valued bipolar fuzzy graph (IVBFG) and cubic bipolar fuzzy graph (CBFG). Some graph theoretic terms for CBFGs are defined along with the several operations. Illustrative examples are provided to explain the defined terms and several results are discussed. As application, a cubic bipolar fuzzy influence graph in a social group is elaborated.

Keywords

Cubic fuzzy graphs bipolar fuzzy graphs cubic bipolar fuzzy graph strong cubic bipolar fuzzy graphs

1 Introduction

The study of cubic set (CS) has been started in 2012 when Jun et al. [1] proposed this concept. Basically, CS is a combination of a fuzzy set (FS) and an interval-valued fuzzy set (IVFS). After its development, much work is being done on cubic sets. In [2], the neutrosophic CS is developed and in [3] some applications of neutrosophic CS in pattern recognition are given. Khan et al. [4] introduced CS in semi-group theory and Yaqoob et al. [5] developed KU-ideals and KU-algebras for CSs. Some cosine similarity measures for CSs were developed in [6] by Lu and Ye and they applied these similarity measures in decision making. The notion of cubic soft set, introduced by Mohiuddin and Al-roqi [7], has gained much importance and some aspects of this notion are discussed in [8]. The notions of G-algebras and G-subalgebras for CSs along with ideals for CSs in semigroup theory are discussed in [9, 10]. For more recent advancements in CSs and their applications, one may be referred to see [11 –14].

After Zadeh introduced the notion of FS [15], the concept of fuzzy graph (FG) was proposed in [16] by Rosenfield which has now become a rich research field [17 –20]. Bhutani [21] defined automorphisms for FGs and Gomez et al. [22] used FGs in image classifications while Mathew and Sunitha [23] discussed the connectivity of nodes of FGs. For some other work on FGs, one may refer to [24 –34].

The theory of interval valued fuzzy set (IVFS) [35] provided a base for interval valued fuzzy graph (IVFG) [36]. IVFG has equal importance in handling uncertain situations as compared to the other notions. Rashmanlou and Jun [37] introduced the notion of complete IVFG and Pal and Rashmanlou [38] initiated the concept of irregular IVFG. Akram [39] introduced IVF line graphs and Rashmanlou and Pal [40, 41] developed the notion of balanced IVFG and isometry on IVFG. Talebi and Rashmanlou [42] worked on isomorphism of IVFG and Pramanik et al. [43] studied the notion of IVF planner graphs. Jan et al. [44] recently challenged some basic concepts and results of FG theory and redefined those concepts demonstrated by examples. Davvaz et al. [45] introduced a novel concept of intuitionistic FGs of nth type and studied their applications in social networks. The concept of picture FG is also introduced by Jan et al. [46] and an improved algorithm has been proposed for clustering and further utilized in human decision making. Jan et al. [47] also introduced the idea of interval valued q-rung ortho pair FGs and analyse some social, communication and transportation networks based on proposed work. For some other notable work, one is referred as [48 –53].

The recent developments in FSs have shown some good improvements among which the concept of Bipolar FS (BFS) [54, 55] is a prominent one. A BFS improved the concept of FS by enlarging its range. The concept of BFS leads to the development of bipolar fuzzy graph (BFG) [56, 57]. For some other work in this direction, one may refer to [58 –63]. Motivated by the developments of FGs, IVFGs and BFGs, this article aims to provide the concept of CBFGs and demonstrates various graph theoretic concepts. Further, the proposed concept of CBFGs shall be utilized in a social network.

This manuscript is about CBFGs which involves the theory of BFGs as well as IVBFGs. The manuscript is based on seven sections. In section one, some history is recalled and some remarkable work on FG, IVFG, BFG and CFG have been discussed. In section Two, some basic definitions are presented. In section Three, the idea of CBFG is introduced as a generalization of FG, IVFG and BFG and related terms are discussed. Section Four is based on operations on CBFG and their results. Section Five contains the concept of strong CBFG and their results. Application of CBFG is discussed in section Six and some concluding remarks are added in section Seven at the end of the article.

2 Preliminaries

This section is about a short introduction of FS, CS, FG and IVFG which are helpful for our graphical work on CBFGs.

Definition 1. [15] A FS in $\dot{X}$ is defined as $A = {〈 x, s_{A} (x) 〉 / x \in \dot{X}}$ where $\dot{X}$ is a non-empty set and $s_{A} : \dot{X} \to [0, 1]$ is the membership function of the FS on A.

Definition 2. [1] Let X be a non-empty set. A CS is a structure $C = {x, A (x), \underline{B} (x) / x \in X}$ where A (x) an interval valued FS and $\underline{B} (x)$ is a FS of x in C and it is denoted by $(A, \underline{B})$ .

Definition 3. [16, 17] A pair Ǧ = $\underset{˙}{V}, \overset{?}{E}$ is known as FG if

$\underset{˙}{V} = {v_{i} : i \in}$ and $s_{1} : \underset{˙}{V} \to [0, 1]$ is the association degree of $v_{i} \in \underset{˙}{V}$ ;

$\overset{?}{E} = {(v_{i}, v_{j}) : (v_{i}, v_{j}) \in \underset{˙}{V} \times \underset{˙}{V}}$ and $s_{2} : \underset{˙}{V} \times \underset{˙}{V} \to [0, 1]$ .

Where s₂ (v_i, v_j) ≤ min [s₁ (v_i) , s₁ (v_j)] for all $(v_{i}, v_{j}) \in \overset{?}{E}$ .

Definition 4. [36] A pair Ǥ = $(A, \underline{B})$ of a graph Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ is said to be IVFG where $A = {< [S_{A}^{L}, S_{A}^{U}] >}$ is an IVFS and $\underline{B} = {< [S_{\underline{B}}^{L}, S_{\underline{B}}^{U}] >}$ be the IVF relation on $\dot{E}$ satisfying the following condition:

$\underset{˙}{V} {v_{1}, v_{2}, . . ., v_{n}}$ such that $S_{A}^{L} : \underset{˙}{V} \to [0, 1]$ , $S_{A}^{U} : \underset{˙}{V} \to [0, 1]$ represents the degree of membership of the element $v \in \underset{˙}{V}$ .

The functions $S_{\underline{B}}^{L} : \underset{˙}{V} \times \underset{˙}{V} \to [0, 1]$ , $S_{\underline{B}}^{U} : \underset{˙}{V} \times \underset{˙}{V} \to [0, 1]$ such that $S_{\underline{B}}^{L} \leq min (S_{A}^{L} (x), S_{A}^{L} ($ )), $S_{\underline{B}}^{U} \leq min (S_{A}^{U} (x), S_{A}^{U} ($ )) for all $xy \in \overset{?}{E}$ .

Definition 5. [44, 45] Let X be a non-empty set. Then a mapping $A = (S_{A}^{P}, S_{A}^{N}) : X \times X \to [0, 1] \times [- 1, 0]$ a bipolar fuzzy relation on X such that $S_{A}^{P} (x,$ ) ∈ [0, 1] and $S_{A}^{N} (x,$ ) ∈ [-1, 0].

Definition 6. [46, 47] A pair Ǥ = $(A, \underline{B})$ with underlying graph Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ is said to be BFG where $A = {< [S_{A}^{P}, S_{A}^{N}] >}$ is an BFS and $\underline{B} = = {< [S_{\underline{B}}^{P}, S_{\underline{B}}^{N}] >}$ is an BF relation on $\dot{E}$ such that $S_{\underline{B}}^{P} (x$ $) \leq min (S_{A}^{P} (x), S_{A}^{P} ($ )), $S_{\underline{B}}^{N} (x$ $) \geq max (S_{A}^{N} (x), S_{A}^{N} ($ )),for all x $\in \overset{?}{E}$

Definition 7. [52] A pair Ǥ = $(A, \underline{B})$ of a graph Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ is said to CFG where $A = {< [S_{A}^{L}, S_{A}^{U}], S_{A} >}$ is a cubic FS and $\underline{B} = {< [S_{\underline{B}}^{L}, S_{\underline{B}}^{U}], S_{\underline{B}} >}$ be the cubic fuzzy relation on $\overset{?}{E}$ satisfying the following condition:

$\underset{˙}{V} {v_{1}, v_{2}, . . ., v_{n}}$ such that $S_{A}^{L} : \underset{˙}{V} \to [0, 1]$ , $S_{A}^{U} : \underset{˙}{V} \to [0, 1]$ ,and $S_{A} : \underset{˙}{V} \to [0, 1]$ represents the degree of membership of the element $v \in \underset{˙}{V}$ .

The functions $S_{\underline{B}}^{L}$ : $\underset{˙}{V}$ × $\underset{˙}{V}$ → [0, 1], $S_{A}^{U} :$ $\underset{˙}{V}$ × $\underset{˙}{V}$ → [0, 1] and $S_{\underline{B}} :$ $\underset{˙}{V}$ × $\underset{˙}{V}$ → [0, 1] are such that $S_{\underline{B}}^{L}$ ≤ $min (S_{A}^{L} (x)$ , $S_{A}^{L} (y))$ , $S_{\underline{B}}^{U}$ ≤ $min (S_{A}^{U} (x)$ , $S_{A}^{U}$ ()) and $S_{\underline{B}}$ ≤ (S_A (x), S_A ()) for all x $\in \overset{?}{E}$

Example 1. The following graph is an example of CFG.

Fig.1

Cubic fuzzy graph.

3 Cubic bipolar fuzzy graph

Definition 8. A pair Ǥ = $(A, \underline{B})$ with underlying graph Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ is said to be an IVBFG where $A = {< [S_{A}^{PL}, S_{A}^{PU}], [S_{A}^{NL}, S_{A}^{NU}] >}$ is an IVBFS and $\underline{B} = {< [S_{\underline{B}}^{PL}, S_{\underline{B}}^{PU}], [S_{\underline{B}}^{NL}, S_{\underline{B}}^{NU}] >}$ is an interval valued bipolar fuzzy relation on $\dot{E}$ such that $S_{\underline{B}}^{PL} (x$ $) \leq min (S_{A}^{PL} (x), S_{A}^{PL} ($ )), $S_{\underline{B}}^{PU} (x$ $) \leq min (S_{A}^{PU} (x), S_{A}^{PL} ($ )) with a condition $S_{\underline{B}}^{PU}$ (x) ≥ $min (S_{A}^{PL} (x)$ , $S_{A}^{PL}$ ()) and $S_{\underline{B}}^{NL}$ (x) ≥ $max (S_{A}^{NL} (x)$ , $S_{A}^{NL} ($ )), $S_{\underline{B}}^{NL}$ (x) ≥ $max (S_{A}^{NU} (x)$ , $S_{A}^{NU} ($ )) with a condition $S_{\underline{B}}^{NU} (xy)$ ≥ $max (S_{A}^{NL} (x)$ , $S_{A}^{NL}$ ())forallx $\in \overset{?}{E}$

Example 2. Let Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ be a IVBFG where $\underset{˙}{V}$ be the set of vertices and $\overset{?}{E}$ be the set of edges. Then

Fig.2

Interval valued bipolar fuzzy graph.

Definition 9. A pair Ǥ = $(A, \underline{B})$ with underlying graph Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ is said to be CBFG where $A = {< [S_{A}^{PL}, S_{A}^{PU}], [S_{A}^{NL}, S_{A}^{NU}], S_{A}^{P}, S_{A}^{N} >}$ is a CBFS and $\underline{B} = {< [S_{\underline{B}}^{PL}, S_{\underline{B}}^{PU}], [S_{\underline{B}}^{NL}, S_{\underline{B}}^{NU}], S_{\underline{B}}^{P}, S_{\underline{B}}^{N} >}$ is a cubic bipolar fuzzy relation on $\dot{E}$ such that $S_{\underline{B}}^{PL}$ (x) ≤ $min (S_{A}^{PL} (x)$ , $S_{A}^{PL}$ ()), $S_{\underline{B}}^{PU}$ (x) ≤ $min (S_{A}^{PU} (x)$ , $S_{A}^{PU} ($ )) with a condition $S_{\underline{B}}^{PU}$ (x) ≥ $min (S_{A}^{PL} (x)$ , $S_{A}^{PL} ($ )), $S_{\underline{B}}^{NL}$ (x) ≥ $max (S_{A}^{NL} (x)$ , $S_{A}^{NL} ($ )), $S_{\underline{B}}^{NL}$ (x) ≥ $max (S_{A}^{NU} (x)$ , $S_{A}^{NU} ($ )) with a condition $S_{\underline{B}}^{NU} (x$ ) ≥ $max (S_{A}^{NL} (x)$ , $S_{A}^{NL} ($ )) and $S_{\underline{B}}^{P} (x$ ) ≤ $min (S_{A}^{P} (x)$ , $S_{A}^{P} ($ )) $S_{\underline{B}}^{N} (x$ ) ≥ $max (S_{A}^{N} (x)$ , $S_{A}^{N} ($ )) for all $x y \in \overset{?}{E}$

Example 3. The following graph is an example of CBFG.

Fig.3

Cubic bipolar fuzzy graph.

The values of all the vertices shown separately in Table 1 below.

Table 1

(Values of vertices Fig. 3)

	v ₁	v ₂	v ₃	v ₄
$S_{A}^{PL}$	0.3	0.2	0.4	0.2
$S_{A}^{NL}$	–0.5	–0.6	–0.3	–0.5
$S_{A}^{PU}$	0.5	0.6	0.5	0.3
$S_{A}^{NU}$	–0.3	–0.2	–0.1	–0.3
$S_{A}^{P}$	0.3	0.5	0.2	0.4
$S_{A}^{N}$	–0.6	–0.3	–0.3	–0.5

Definition 10. The order of a CBFG Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ is denoted and defined by $\begin{matrix} O (Ǥ) = ((\sum_{x \in y} S_{A}^{PL} (x), \sum_{x \in Y} S_{A}^{PU} (x)), \\ (\sum_{x \in y} S_{A}^{nL} (x), \sum_{x \in y} S_{A}^{nU} (x)), \\ \sum_{x \in Y} S_{A}^{P} (x), \sum_{x \in Y} S_{A}^{n} (x)) \end{matrix}$

Here $(\sum_{x \in Y} S_{A}^{PL} (x), \sum_{x \in Y} S_{A}^{PU} (x))$ represents the sum of all the positive lower and upper values and $(\sum_{x \in Y} S_{A}^{nL} (x), \sum_{x \in Y} S_{A}^{nU} (x))$ represents the sum of all the negative lower and upper values in an interval bipolar fuzzy numbers and $\sum_{x \in Y} S_{A}^{P} (x), \sum_{x \in Y} S_{A}^{n} (x)$ represents the sum of all the values of positive and negative bipolar fuzzy numbers respectively in $\underset{˙}{V}$ .

And the size of cubic graph Ǧ is:

Definition 11. The degree of a vertex in a CBFG Ǧ = ( $\underset{˙}{V}, \overset{?}{E}$ ) is denoted and defined by $\begin{matrix} deg (x) = ([d_{⌢}^{PL} (x), d_{⌢}^{PU} (x)], [d_{⌢}^{nL} (x), d_{⌢}^{nU} (x)], \\ d_{⌢}^{P} (x), d_{⌢}^{n} (x)) \end{matrix}$

Where $d_{⌢}^{PL} (x)$ = ∑_x≠_x∈Y $S_{B}^{PL} (x$ ), $d_{⌢}^{PU} (x)$ = ∑_x≠_x∈Y $S_{B}^{PU} (x$ ), $d_{⌢}^{nU} (x)$ = ∑_x≠_x∈Y $S_{B}^{nU} (x$ ), $d_{⌢}^{PL} (x)$ = ∑_x≠_x∈Y $S_{B}^{PL} (x$ ) and $d_{⌢}^{P} (x)$ = ∑_x≠_x∈Y $S_{B}^{P} (x$ ), $d_{⌢}^{n} (x)$ = ∑_x≠_x∈Y $S_{B}^{n} (x$ ) here ∑_x≠_x∈Y $S_{B}^{PL} (x$ ), ∑_x≠_x∈Y $S_{B}^{PU} (x$ ), represents the sum of all the positive lower and positive upper values in an interval bipolar fuzzy numbers, ∑_x≠_x∈Y $S_{B}^{nL} (x$ ), ∑_x≠_x∈Y $S_{B}^{nU} (x$ ) denote the sum of all the negative lower and negative upper values of bipolar fuzzy numbers and ∑_x≠_x∈Y $S_{B}^{P} (x$ ) , ∑_x≠_x∈Y $S_{B}^{n} (x$ ) represents the sum of the values of positive and negative bipolar fuzzy numbers respectively in $\overset{?}{E}$

Example 4. Let Ǧ = ( $\underset{˙}{V}, \overset{?}{E}$ ) be a CBFG where $\underset{˙}{V}$ be the set of vertices and be the set of edges. Then

Fig.4

Cubic bipolar fuzzy graph.

The degree of cubic bipolar fuzzy vertices are: $\underline{d} (v_{1})$ = ([0.3, 0.7], [-0.8, -0.3], (0.3, -0.7)), $\underline{d} (v_{2})$ = ([0.2, 0.8], [-0.6, -0.2], (0.3, - 0.5)), $\underline{d} (v_{3})$ = ([0.2, 0.6], [-0.4, -0.2], (0.3, -0.4)), and $\underline{d} (v_{4})$ = ([0.3, 0.5], [-0.6, -0.3], (0.3, -0.6)).

Theorem 1. CBFG generalizes BFG and IVBFG.

Proof: We prove this result as follows:

If we take $[S_{A}^{PL}, S_{A}^{PU}], [S_{A}^{NL}, S_{A}^{NU}]$ as zero then CBFG reduces to BFG.

If we take $S_{A}^{P}, S_{A}^{N}$ as zero then CBFG reduces to IVBFG.

This result shows the significance of the new concept as it is the generalization of all the existing structures and can deal with the situations where the existing structures fail due to their limitations.

Definition 12. In a CBFG, a set of distinct nodes v_i (i = 1, 2, 3 . . . m) is considered as a path if there exist an edge between every two vertices v_i and V_j for i, j = 1, 2, 3 . . . m.

Example 5. Let Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ be a CBFG where $\underset{˙}{V}$ be the set of vertices and $\overset{?}{E}$ be the set of edges. Then

Fig.5

Cubic bipolar fuzzy graph for a path.

Here v₁v₄v₅ is a path.

Definition 13. An edge in a CBFG Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ is said to be bridge, if deleting that edge reduces the strength of connectedness between some pair of vertices.

Example 6. Let Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ be a graph such that $\underset{˙}{V} = {v_{1}, v_{2}, v_{3}}$ and $\dot{E} = {v_{1}, v_{2}, v_{3}, v_{3} v_{1}}$ . Then the following graph is an example of CBFG for an edge.

Fig.6

Cubic bipolar fuzzy graph for a bridge.

The strength of v₁v₃ in Ǧ is ([0.2,0.4], [-0.3, -0.1], (0.1, -0.3)) by definition of (13), (v₁, v₃) is a bridge.

Definition 14. The power of edge relation in a CBFG is defined as $e_{ij}^{1} = (e_{ij}$ , $([S_{\underline{B} ij}^{-}$ , $S_{\underline{B} ij}^{+}]$ , $S_{\underline{B} ij}))$ , $e_{ij}^{2}$ = (e_ij * e_ij = (e_ij, $[S_{\underline{B} ij}^{-}$ , $S_{\underline{B} ij}^{+}]^{2}$ , $(S_{\underline{B} ij})^{2})$ , $e_{ij}^{3}$ = (e_ij * e_ij * e_ij = (e_ij, $[S_{\underline{B} ij}^{-}$ , $S_{\underline{B} ij}^{+}]^{3}$ , $(S_{\underline{B} ij}^{3}))$ and $e_{ij}^{\infty} = (e_{ij}$ , $([S_{\underline{B} ij}^{-}$ , $S_{\underline{B} ij}^{+}]^{\infty}$ , $S_{\underline{B} ij}^{\infty}))$ . Here $[S_{\underline{B} ij}^{PL}$ , $S_{\underline{B} ij}^{PU}]^{\infty}$ = $min ({[S_{\underline{B} ij}^{PL})^{k}$ , $(S_{\underline{B} ij}^{PU})^{k}]})$ , $[S_{\underline{B} ij}^{nL}$ , $S_{\underline{B} ij}^{nU}]^{\infty}$ = $max ({[(S_{\underline{B} ij}^{nL})^{k}$ , $(S_{\underline{B} ij}^{nU})^{k}]})$ and $(S_{\underline{B} ij}^{p})^{\infty}$ = $min {(S_{\underline{B} ij}^{p})^{k}}$ , $(S_{\underline{B} ij}^{n})^{\infty}$ = $max {(S_{\underline{B} ij}^{n})^{k}}$ are the s-strength and of the connectedness between two vertices (v_i, v_j).

Theorem 2. Consider Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ be a CBFG. Then for any v_i, v_j in Ǧ, the following are equivalent.

(v_i, v_j) is a bridge.

$s_{\underline{B} ij}^{{PL}^{\circ} \infty}$ < $s_{\underline{B} ij}^{PL}$ , $s_{\underline{B} ij}^{{PU}^{\circ} \infty}$ < $s_{\underline{B} ij}^{PU}$ , $s_{\underline{B} ij}^{{nL}^{\circ} \infty}$ > $s_{\underline{B} ij}^{nL}$ , $s_{\underline{B} ij}^{{nU}^{\circ} \infty}$ > $s_{\underline{B} ij}^{nU}$ and $s_{\underline{B} ij}^{P^{\circ} \infty}$ < $s_{\underline{B} ij}^{P}$ , $s_{\underline{B} ij}^{n^{\circ} \infty}$ > $s_{\underline{B} ij}^{n}$

(v_i, v_j) is not an edge of any cycle.

Proof. (ii) ⇒ (i) suppose $s_{\underline{B} ij}^{{PL}^{\circ} \infty} < s_{\underline{B} ij}^{PL}, s_{\underline{B} ij}^{{PU}^{\circ} \infty} < s_{\underline{B} ij}^{PU}, s_{\underline{B} ij}^{{nL}^{\circ} \infty} > s_{\underline{B} ij}^{nL}, s_{\underline{B} ij}^{{nU}^{\circ} \infty} > s_{\underline{B} ij}^{nU}$ and $s_{\underline{B} ij}^{P^{\circ} \infty} < s_{\underline{B} ij}^{P}, s_{\underline{B} ij}^{n^{\circ} \infty} > s_{\underline{B} ij}^{n}$ . To prove (v_i, v_j) is a bridge, then $s_{\underline{B} ij}^{{PL}^{\circ} \infty} = s_{\underline{B} ij}^{{PL}^{\infty}} \geq s_{\underline{B} ij}^{PL}, s_{\underline{B} ij}^{{PU}^{\circ} \infty} = s_{\underline{B} ij}^{{PU}^{\infty}} \geq s_{\underline{B} ij}^{PU}, s_{\underline{B} ij}^{P^{\circ} \infty} = s_{\underline{B} ij}^{P^{\infty}} \geq s_{\underline{B} ij}^{P}$ and $s_{\underline{B} ij}^{{nL}^{\circ} \infty} = s_{\underline{B} ij}^{{nL}^{\infty}} \leq s_{\underline{B} ij}^{nL}, s_{\underline{B} ij}^{{nU}^{\circ} \infty} = s_{\underline{B} ij}^{{nU}^{\infty}} \leq s_{\underline{B} ij}^{nU}, s_{\underline{B} ij}^{n^{\circ} \infty} = s_{\underline{B} ij}^{n^{\infty}} \leq s_{\underline{B} ij}^{n} \Rightarrow s_{\underline{B} ij}^{{PL}^{\infty}} \geq s_{\underline{B} ij}^{PL}, s_{\underline{B} ij}^{{PU}^{\infty}} \geq s_{\underline{B} ij}^{PU}, s_{\underline{B} ij}^{P^{\infty}} \geq s_{\underline{B} ij}^{P}$ and $s_{\underline{B} ij}^{{nL}^{\infty}} \leq s_{\underline{B} ij}^{nL}, s_{\underline{B} ij}^{{nU}^{\infty}} \leq s_{\underline{B} ij}^{nU}, s_{\underline{B} ij}^{n^{\infty}} \leq s_{\underline{B} ij}^{n}$ . Which is contradiction. Therefore, (v_i, v_j) is a bridge. Now (i) ⇒(iii) Consider (v_i, v_j) is a bridge. To show that (v_i, v_j) is not an edge of any cycle. Then any path involving the edge (v_i, v_j) can be converted into a path involving (v_i, v_j) by using the rest of the cycle as a path from v_i and v_j. This implies (v_i, v_j) can not be bridge which is contradiction. Therefore, (v_i, v_j) is not an edge of any cycle. Now (iii)⇒ (i) suppose (v_i, v_j) is not an edge of any cycle. To show that $s_{\underline{B} ij}^{{PL}^{\circ} \infty} < s_{\underline{B} ij}^{PL}, s_{\underline{B} ij}^{{PU}^{\circ} \infty} < s_{\underline{B} ij}^{PU}, s_{\underline{B} ij}^{{nL}^{\circ} \infty} > s_{\underline{B} ij}^{nL}, s_{\underline{B} ij}^{{nU}^{\circ} \infty} > s_{\underline{B} ij}^{nU}$ and $s_{\underline{B} ij}^{P^{\circ} \infty} < s_{\underline{B} ij}^{P}, s_{\underline{B} ij}^{n^{\circ} \infty} > s_{\underline{B} ij}^{n}$ . Suppose that $s_{\underline{B} ij}^{{PL}^{\infty}} \geq s_{\underline{B} ij}^{PL}, s_{\underline{B} ij}^{{PU}^{\infty}} \geq s_{\underline{B} ij}^{PU}, s_{\underline{B} ij}^{P^{\infty}} \geq s_{\underline{B} ij}^{P}$ , and $s_{\underline{B} ij}^{{nL}^{\infty}} \leq s_{\underline{B} ij}^{nL}, s_{\underline{B} ij}^{{nU}^{\infty}} \leq s_{\underline{B} ij}^{nU}, s_{\underline{B} ij}^{n^{\infty}} \leq s_{\underline{B} ij}^{n}$ . Then there is a path from v_i to v_j not involving (v_i, v_j) that has strength greater are equal to $S_{\underline{B} ij}^{PL}, S_{\underline{B} ij}^{PU}, S_{\underline{B} ij}^{P}$ and less are equal to $S_{\underline{B} ij}^{nL}, S_{\underline{B} ij}^{nU}, S_{\underline{B} ij}^{n}$ . Which is contradiction. Hence $s_{\underline{B} ij}^{{PL}^{\circ} \infty} < s_{\underline{B} ij}^{PL}, s_{\underline{B} ij}^{{PU}^{\circ} \infty} < s_{\underline{B} ij}^{PU}, s_{\underline{B} ij}^{{nL}^{\circ} \infty} > s_{\underline{B} ij}^{nL}, s_{\underline{B} ij}^{{nU}^{\circ} \infty} > s_{\underline{B} ij}^{nU}$ and $s_{\underline{B} ij}^{P^{\circ} \infty} < s_{\underline{B} ij}^{P}, s_{\underline{B} i j}^{n^{\circ} \infty} > s_{\underline{B} i j}^{n}$ . Therefore, the statements (i), (ii) and (iii) are equivalent.

Definition 15. A vertex in a CBFG Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ is said to be cut-vertex if deleting that vertex reduces the strength of connectedness between some pair of vertices.

Example 7. Let Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ be a CBFG where $\underset{˙}{V}$ be the set of vertices and $\overset{?}{E}$ be the set of edges. Then

Fig.7

Cubic bipolar fuzzy graph for a cut-vertex.

Here v₁ is a cut-vertex.

4 Operations on cubic bipolar fuzzy graph

In this section, the operations on CBFG are defined and their results are studied.

Definition 16. The Cartesian product of two CBFGs Ǧ₁ = $(A_{1}, \underline{B} 1)$ and Ǧ₂ = $(A_{2}, \underline{B_{2}}),$ Ǧ₁ × Ǧ₂ = $(A_{1} \times A_{2}, \underline{B} 1 \times \underline{B} 2)$ of the CBFGs Ǥ₁ and Ǥ₂ is defined as:

$(i) {\begin{matrix} (S_{A_{1}}^{PL} \times S_{A_{2}}^{PL}) (x_{1}, x_{2}) = min (S_{A_{1}}^{PL} (x_{1}), S_{A_{2}}^{PL} (x_{2})) \\ (S_{A_{1}}^{PU} \times S_{A_{2}}^{PU}) (x_{1}, x_{2}) = min (S_{A_{1}}^{PU} (x_{1}), S_{A_{2}}^{PU} (x_{2})) \\ (S_{A_{1}}^{P} \times S_{A_{2}}^{P}) (x_{1}, x_{2}) = min (S_{A_{1}}^{P} (x_{1}), S_{A_{2}}^{P} (x_{2})) \end{matrix}$

$(ii) {\begin{matrix} (S_{A_{1}}^{nL} \times S_{A_{2}}^{nL}) (x_{1}, x_{2}) = max (S_{A_{1}}^{nL} (x_{1}), S_{A_{2}}^{nL} (x_{2})) \\ (S_{A_{1}}^{nU} \times S_{A_{2}}^{nU}) (x_{1}, x_{2}) = max (S_{A_{1}}^{nU} (x_{1}), S_{A_{2}}^{nU} (x_{2})) \\ (S_{A_{1}}^{n} \times S_{A_{2}}^{n}) (x_{1}, x_{2}) = max (S_{A_{1}}^{n} (x_{1}), S_{A_{2}}^{n} (x_{2})) \end{matrix}$

For all $(x_{1}, x_{2}) \in \underset{˙}{V}$

$(iii) {\begin{matrix} (S_{\underline{B} 1}^{PL} \times S_{\underline{B} 2}^{PL}) (x, x_{2}) (x, y_{2}) = min (S_{A_{1}}^{PL} (x), S_{\underline{B} 2}^{PL} (x_{2} y_{2})) \\ (S_{\underline{B} 1}^{PU} \times S_{\underline{B} 2}^{PU}) (x, x_{2}) (x, y_{2}) = min (S_{A_{1}}^{PU} (x), S_{\underline{B} 2}^{PU} (x_{2} y_{2})) \\ (S_{\underline{B} 1}^{P} \times S_{\underline{B} 2}^{P}) (x, x_{2}) (x, y_{2}) = min (S_{A_{1}}^{P} (x), S_{\underline{B} 2}^{P} (x_{2} y_{2})) \end{matrix}$

$(iv) {\begin{matrix} (S_{\underline{B} 1}^{nL} \times S_{\underline{B} 2}^{nL}) (x, x_{2}) (x, y_{2}) = max (S_{A_{1}}^{nL} (x), S_{\underline{B} 2}^{nL} (x_{2} y_{2})) \\ (S_{\underline{B} 1}^{nU} \times S_{\underline{B} 2}^{nU}) (x, x_{2}) (x, y_{2}) = max (S_{A_{1}}^{nU} (x), S_{\underline{B} 2}^{nU} (x_{2} y_{2})) \\ (S_{\underline{B} 1}^{n} \times S_{\underline{B} 2}^{n}) (x, x_{2}) (x, y_{2}) = max (S_{A_{1}}^{n} (x), S_{\underline{B} 2}^{n} (x_{2} y_{2})) \end{matrix}$

For all $x \in {\underset{˙}{V}}_{1}$ and $x_{2} y_{2} \in {\overset{?}{E}}_{2}$ $(v) {\begin{matrix} (S_{\underline{B} 1}^{PL} \times S_{\underline{B} 2}^{PL}) (x_{1}, z) (y_{1}, z) = min (S_{\underline{B} 1}^{PL} (x_{1} y_{1}), S_{A_{2}}^{PL} (z)) \\ (S_{\underline{B} 1}^{PU} \times S_{\underline{B} 2}^{PU}) (x_{1}, z) (y_{1}, z) = min (S_{\underline{B} 1}^{PU} (x_{1} y_{1}), S_{A_{2}}^{PU} (z)) \\ (S_{\underline{B} 1}^{p} \times S_{\underline{B} 2}^{p}) (x_{1}, z) (y_{1}, z) = min (S_{\underline{B} 1}^{p} (x_{1} y_{1}), S_{A_{2}}^{p} (z)) \end{matrix}$ $(vi) {\begin{matrix} (S_{\underline{B} 1}^{nL} \times S_{\underline{B} 2}^{nL}) (x_{1}, z) (y_{1}, z) = max (S_{\underline{B} 1}^{nL} (x_{1} y_{1}), S_{A_{2}}^{nL} (z)) \\ (S_{\underline{B} 1}^{nU} \times S_{\underline{B} 2}^{nU}) (x_{1}, z) (y_{1}, z) = max (S_{\underline{B} 1}^{nU} (x_{1} y_{1}), S_{A_{2}}^{nU} (z)) \\ (S_{\underline{B} 1}^{n} \times S_{\underline{B} 2}^{n}) (x_{1}, z) (y_{1}, z) = max (S_{\underline{B} 1}^{n} (x_{1} y_{1}), S_{A_{2}}^{n} (z)) \end{matrix}$

For all $z \in {\underset{˙}{V}}_{2}$ and $x_{1} y_{1} \in {\overset{?}{E}}_{1}$ .

Proposition 1. The Cartesian product Ǧ₁× Ǧ₂ = $(A_{1} \times A_{2}, {\underline{B}}_{1} \times {\underline{B}}_{2})$ of two CBFGs Ǧ₁ and Ǧ₂ is a CBFG.

Proof. The condition on A₁ × A₂ is obvious, we need only to verify the conditions for ${\underline{B}}_{1} \times {\underline{B}}_{2}$ .

Assume that $x \in {\underset{˙}{V}}_{1}$ and $x_{2} y_{2} \in {\overset{?}{E}}_{2}$

$(s_{\underline{B} 1}^{PL} \times s_{\underline{B} 2}^{PL}) (x, x_{2}) (x, y_{2}) = min (s_{A_{1}}^{PL} (x), s_{\underline{B} 2}^{PL} (x_{2} y_{2})) min (min (s_{A_{1}}^{PL} (x), s_{A_{2}}^{PL} (x_{2})), min (s_{A_{1}}^{PL} (x), s_{A_{2}}^{PL} (y_{2}))) = min (S_{A_{1}}^{PL} \times S_{A_{2}}^{PL}) (x_{,} x_{2}), (S_{A_{1}}^{PL} \times S_{A_{2}}^{PL}) (x_{,} y_{2})$ $(S_{\underline{B} 1}^{PU} \times S_{\underline{B} 2}^{PU}) (x, x_{2}) (x, y_{2}) = min (S_{A_{1}}^{PU} (x) \times S_{\underline{B} 2}^{PU} (x_{2} y_{2})) \leq min (S_{A_{1}}^{PU} (x), min (S_{A_{2}}^{PU} (x_{2}), S_{A_{2}}^{PU} (y_{2}))) min (min (S_{A_{1}}^{PU} (x_{1}), S_{A 2}^{PU} (x_{2})), min (S_{A_{1}}^{PU} (x), S_{A_{2}}^{PU} (y_{2}))) = min (S_{A_{1}}^{PU} \times S_{A_{2}}^{PU}) (x, x_{2}), (S_{A_{1}}^{PU} \times S_{A_{2}}^{PU}) (x, y_{2})$ $(S_{\underline{B} 1}^{P} \times S_{\underline{B} 2}^{P}) (x, x_{2}) (x, y_{2}) = min (S_{A_{1}}^{P} (x), S_{\underline{B} 2}^{P} (x_{2} y_{2})) \leq min (S_{A_{1}}^{P} (x), min (S_{A_{2}}^{P} (x_{2}), S_{A_{2}}^{P} (y_{2}))) = min (min (S_{A_{1}}^{P} (x), S_{A_{2}}^{P} (x_{2})), min (S_{A_{1}}^{P} (x), S_{A_{2}}^{P} (y_{2}))) = min (S_{A_{1}}^{P} \times S_{A_{2}}^{P}) (x, x_{2}), (S_{A_{1}}^{P} \times S_{A_{2}}^{P}) (x, y_{2})$

And $(S_{A_{1}}^{nL} \times S_{\underline{B} 2}^{nL}) (x, x_{2}) (x, y_{2}) = max (S_{A_{1}}^{nL} (x), S_{\underline{B} 2}^{nL} (x_{2} y_{2})) \geq max (S_{A_{1}}^{nL} (x), min (S_{A_{2}}^{nL} (x_{2}), S_{A_{2}}^{nL} (y_{2}))) = max (max (S_{A_{1}}^{nL} (x), S_{A_{2}}^{nL} (x_{2})), max (S_{A_{1}}^{nL} (x), S_{A_{2}}^{nL} (y_{2}))) = max (S_{A_{1}}^{nL} \times S_{A_{2}}^{nL}) (x, x_{2}), (S_{A_{1}}^{nL} \times S_{A_{2}}^{nL}) (x, y_{2})$ $(S_{\underline{B} 1}^{nU} \times S_{\underline{B} 2}^{nU}) (x, x_{2}) (x, y_{2}) = max (S_{A_{1}}^{nU} (x), (S_{\underline{B} 2}^{nU} (x_{2} y_{2})) \geq max (S_{A_{1}}^{nU} (x), max (S_{A_{2}}^{nU} (x_{2}), S_{A_{2}}^{nU} (y_{2}))) max (max (S_{A_{1}}^{nU} (x), S_{A_{2}}^{nU} (x_{2})), max (S_{A_{1}}^{nU} (x), S_{A_{2}}^{nU} (y_{2}))) = max (S_{A_{1}}^{nU} \times S_{A_{2}}^{nU}) (x, x_{2}), (S_{A_{1}}^{nU} \times S_{A_{2}}^{nU}) (x, y_{2})$ $(S_{\underline{B} 1}^{n} \times S_{\underline{B} 2}^{n}) (x, x_{2}) (x, y_{2}) = max (S_{A_{1}}^{n} (x), S_{\underline{B} 2}^{n} (x_{2} y_{2})) \geq max (S_{A_{1}}^{n} (x), max (S_{A_{2}}^{n} (y_{2}))) = max (max (S_{A_{1}}^{n} (x), S_{A_{2}}^{n} (x_{2})), max (S_{A_{1}}^{n} (x), S_{A_{2}}^{n} (y_{2}))) = max (S_{A_{1}}^{n} \times S_{A_{2}}^{n}) (x, x_{2}), (S_{A_{1}}^{n} \times S_{A_{2}}^{n}) (x, y_{2})$

For $z \in {\underset{˙}{V}}_{2}$ and $x_{1} y_{1} \in \overset{?}{E}$ we have $(S_{\underline{B} 1}^{PL} \times S_{\underline{B} 2}^{PL}) (x_{1}, z) (y_{1,} z) = min (S_{\underline{B} 1}^{PL} (x_{1} y_{1}), S_{A_{2}}^{PL} (z)) \leq min (min (S_{A_{1}}^{PL} (x_{1}), S_{A_{1}}^{PL} (y_{1}), S_{A_{2}}^{PL} (z)) = min (min (S_{A_{1}}^{PL} (x_{1}), S_{A_{2}}^{PL} (z)), min (S_{A_{1}}^{PL} (y_{2}), S_{A_{2}}^{PL} (z)) = min ((S_{A_{1}}^{PL} \times S_{A_{2}}^{PL}) (x_{1}, z), (S_{A_{1}}^{PL} \times S_{A_{2}}^{PL}) (y_{1}), z))$ $(S_{\underline{B} 1}^{PU} \times S_{\underline{B} 2}^{PU}) (x_{1}, z) (y_{1}, z) = min (S_{\underline{B} 1}^{PU} (x_{1} y_{1}), \times S_{A_{2}}^{PU} (z)) \leq min (min (S_{A_{1}}^{PU} (x_{1}) \times S_{A_{1}}^{PU} (y_{1}), S_{A_{2}}^{PU} (z)) = min (min (S_{A_{1}}^{PU} (x_{1}), S_{A_{2}}^{PU} (z), min (S_{A_{1}}^{PU} (y_{1}), S_{A_{2}}^{PU} (z)) = min ((S_{A_{1}}^{PU} \times S_{A_{2}}^{PU}) (x_{1}, z), (S_{A_{1}}^{PU} \times S_{A_{2}}^{PU}) (y_{1}, z))$ $(S_{\underline{B} 1}^{P} \times S_{\underline{B} 2}^{P}) (x_{1}, z) (y_{1}, z) - min (S_{\underline{B} 1}^{P} (x_{1} y_{1}), S_{A_{2}}^{P} (z)) \leq min (min (S_{A_{1}}^{P} (x_{1}), S_{A_{1}}^{P} (y_{2}), S_{A_{2}}^{P} (z)) = min (min (S_{A_{1}}^{P} \times S_{A_{2}}^{P}) (x_{1}, z), (S_{A_{1}}^{P} \times S_{A_{2}}^{P}) (y_{1}, z))$

And $(S_{\underline{B} 1}^{nL} \times S_{\underline{B} 2}^{nL}) (x_{1}, z) (y_{1}, z) = max (S_{\underline{B} 1}^{nL} (x_{1} y_{1}), S_{A_{2}}^{nL} (z)) \geq max (max (S_{A_{1}}^{nL} (x_{1}), S_{A_{1}}^{nL} (y_{1}), S_{A_{2}}^{nL} (z)) = max (max (S_{A_{1}}^{nL} (x_{1}), S_{A_{2}}^{nL} (z)), max (S_{A_{1}}^{nL} (y_{1}), S_{A_{2}}^{nL} (z)) = max ((S_{A_{1}}^{nL} \times S_{A_{2}}^{nL}) (x_{1}, z), (S_{A_{1}}^{nL} \times S_{A_{2}}^{nL}) (y_{1}, z))$ $\begin{matrix} \begin{matrix} (S_{\underline{B} 1}^{nU} \times S_{\underline{B} 2}^{nU}) (x_{1}, z) (y_{1}, z) = max (S_{\underline{B} 1}^{nU} (x_{1} y_{1}), S_{A_{2}}^{nU} (z)) \geq max (max (S_{A_{1}}^{nU} (x_{1}), S_{A_{1}}^{nU} (y_{1}), S_{A_{2}}^{nU} (z)) \\ = max (max (S_{A_{2}}^{nU} (x_{1}), S_{A_{2}}^{nU} (z)), max (S_{A_{1}}^{nU} (y_{1}), S_{A_{2}}^{nU} (z)) = max ((S_{A_{1}}^{nU} \times S_{A_{2}}^{nU}) (x_{1}, z), (S_{A_{1}}^{nU} \times S_{A_{2}}^{nU}) (y_{1}, z)) \end{matrix} \end{matrix}$ $\begin{matrix} (S_{\underline{B} 1}^{n} \times S_{\underline{B} 2}^{n}) (x_{1}, z) = max (S_{\underline{B} 1}^{n} (x_{1} y_{1}), S_{A_{2}}^{n} (z)) \geq max (max (S_{A_{1}}^{n} (x_{1}), S_{A_{1}}^{n} (y_{1}), S_{A_{2}}^{n} (z)) \\ = max (max (S_{A_{1}}^{n} (x_{1}), S_{A_{2}}^{n} (z)), max (S_{A_{1}}^{n} (y_{1}), S_{A_{2}}^{n} (z)) = max ((S_{A_{1}}^{n} \times S_{A_{1}}^{n}) (x_{1}, z), (S_{A_{1}}^{n} \times S_{A_{2}}^{n}) (y_{1}, z)) \end{matrix}$

Definition 17. The composition of two CBFGs Ǧ₁ $(A_{1}, {\underline{B}}_{1})$ and Ǧ₂ = $(A_{2}, {\underline{B}}_{2})$ with underlying CBFGs Ǥ₁ and Ǥ₂ is denoted by Ǧ₁ Ǧ₂ = $(A_{1}^{\circ} A_{2}, {\underline{B}}_{1}^{\circ} {\underline{B}}_{2})$ and defined by: $(i) {\begin{matrix} (S_{A_{1}}^{PL \circ} S_{A_{2}}^{PL}) (x_{1}, x_{2}) = min (S_{A_{1}}^{PL} (x_{1}), S_{A_{2}}^{PL} (x_{2})) \\ (S_{A_{1}}^{PU \circ} S_{A_{2}}^{PU}) (x_{1}, x_{2}) = min (S_{A_{1}}^{PU} (x_{1}), S_{A_{2}}^{PU} (x_{2})) \\ (S_{A_{1}}^{P \circ} \times S_{A_{2}}^{P}) (x_{1}, x_{2}) = min (S_{A_{1}}^{P} (x_{1}), S_{A_{2}}^{P} (x_{2})) \end{matrix}$ $(ii) {\begin{matrix} (S_{A_{1}}^{nL \circ} S_{A_{2}}^{nL}) (x_{1}, x_{2}) = max (S_{A_{1}}^{nL} (x_{1}), S_{A_{2}}^{nL} (x_{2})) \\ (S_{A_{1}}^{nU \circ} S_{A_{2}}^{nU}) (x_{1}, x_{2}) = max (S_{A_{1}}^{nU} (x_{1}), S_{A_{2}}^{nU} (x_{2})) \\ (S_{A_{1}}^{n \circ} S_{A_{2}}^{n}) (x_{1}, x_{2}) = max (S_{A_{1}}^{n} (x_{1}), S_{A_{2}}^{n} (x_{2})) \end{matrix}$ For all $(x_{1}, x_{2}) \in \underset{˙}{V}$ $(iii) {\begin{matrix} (S_{\underline{B} 1}^{PL \circ} S_{\underline{B} 2}^{PL}) (x, x_{2}) (x, y_{2}) = min (S_{A_{1}}^{PL} (x), S_{\underline{B} 2}^{PL} (x_{2} y_{2})) \\ (S_{\underline{B} 1}^{PU \circ} S_{\underline{B} 2}^{PU}) (x, x_{2}) (x, y_{2}) = min (S_{A_{1}}^{PU} (x), S_{\underline{B} 2}^{PU} (x_{2} y_{2})) \\ (S_{\underline{B} 1}^{P \circ} S_{\underline{B} 2}^{P}) (x, x_{2}) (x, y_{2}) = min (S_{A_{1}}^{P} (x), S_{\underline{B} 2}^{P} (x_{2} y_{2})) \end{matrix}$ $(iv) {\begin{matrix} (S_{\underline{B} 1}^{nL \circ} S_{\underline{B} 2}^{nL}) (x, x_{2}) (x, y_{2}) = max (S_{A_{1}}^{nL} (x), S_{\underline{B} 2}^{nL} (x_{2} y_{2})) \\ (S_{\underline{B} 1}^{nU \circ} S_{\underline{B} 2}^{nU}) (x_{1}, x_{2}) (x, y_{2}) = max (S_{A_{1}}^{nU} (x), S_{\underline{B} 2}^{nU} (x_{2} y_{2})) \\ (S_{\underline{B} 1}^{n \circ} S_{\underline{B} 2}^{n}) (x, x_{2}) (x, y_{2}) = max (S_{A_{1}}^{n} (x), S_{\underline{B} 2}^{n} (x_{2} y_{2})) \end{matrix}$ For all $x \in {\underset{˙}{V}}_{1}$ and $x_{2} y_{2} \in {\overset{?}{E}}_{2}$ $(v) {\begin{matrix} (S_{\underline{B} 1}^{PL \circ} S_{\underline{B} 2}^{PL}) (x_{1}, z) (y_{1}, z) = min (S_{\underline{B} 1}^{PL} (x_{1} y_{1}), S_{A_{2}}^{PL} (z)) \\ (S_{\underline{B} 1}^{PU \circ} S_{\underline{B} 2}^{PU}) (x_{1}, z) (y_{1}, z) = min (S_{\underline{B} 1}^{PU} (x_{1} y_{1}), S_{A_{2}}^{PU} (z)) \\ (S_{\underline{B} 1}^{P \circ} S_{\underline{B} 2}^{P}) (x_{1}, z) (y_{1}, z) = min (S_{\underline{B} 1}^{P} (x_{1} y_{1}), S_{A_{2}}^{P} (z)) \end{matrix}$ $(vi) {\begin{matrix} (S_{\underline{B} 1}^{nL \circ} S_{\underline{B} 2}^{nL}) (x_{1}, z) (y_{1}, z) = max (S_{\underline{B} 1}^{nL} (x_{1} y_{1}), S_{A_{2}}^{nL} (z)) \\ (S_{\underline{B} 1}^{nU \circ} S_{\underline{B} 2}^{nU}) (x_{1}, z) (y_{1}, z) = max (S_{\underline{B} 1}^{nU} (x_{1} y_{1}), S_{A_{2}}^{nU} (z)) \\ (S_{\underline{B} 1}^{n \circ} S_{\underline{B} 2}^{n}) (x_{1}, z) (y_{1}, z) = max (S_{\underline{B} 1}^{n} (x_{1} y_{1}), S_{A_{2}}^{n} (z)) \end{matrix}$ For all $z \in {\underset{˙}{V}}_{2}$ and $x_{1} y_{1} \in {\overset{?}{E}}_{1}$ . $(vii) {\begin{matrix} (S_{\underline{B} 1}^{PL \circ} S_{\underline{B} 2}^{PL}) (x_{1}, x_{2}) (y_{1}, y_{2}) \\ = min (S_{A_{2}}^{PL} (x_{2}), S_{A_{2}}^{PL} (y_{2}), S_{\underline{B} 1}^{PL} (x_{1} y_{1})) \\ (S_{\underline{B} 1}^{PU \circ} S_{\underline{B} 2}^{PU}) (x_{1}, x_{2}) (y_{1}, y_{2}) \\ = min (S_{A_{2}}^{PU} (x_{2}), S_{A_{2}}^{PU} (y_{2}), S_{\underline{B} 1}^{PU} (x_{1} y_{1})) \\ (S_{\underline{B} 1}^{P \circ} S_{\underline{B} 2}^{P}) (x_{1}, x_{2}) (y_{1}, y_{2}) \\ = min (S_{A_{2}}^{P} (x_{2}), S_{A_{2}}^{P} (y_{2}), S_{\underline{B} 1}^{P} (x_{1} y_{1})) \end{matrix}$ $(viii) {\begin{matrix} (S_{\underline{B} 1}^{nL \circ} S_{\underline{B} 2}^{nL}) (x_{1}, x_{2}) (y_{1}, y_{2}) \\ = max (S_{A_{2}}^{nL} (x_{2}), S_{A_{2}}^{nL} (y_{2}), S_{\underline{B} 1}^{nL} (x_{1} y_{1})) \\ (S_{\underline{B} 1}^{nU \circ} S_{\underline{B} 2}^{nU}) (x_{1}, x_{2}) (y_{1}, y_{2}) \\ = max (S_{A_{2}}^{nU} (x_{2}), S_{A_{2}}^{nU} (y_{2}), S_{\underline{B} 1}^{nU} (x_{1} y_{1})) \\ (S_{\underline{B} 1}^{n \circ} S_{\underline{B} 2}^{n}) (x_{1}, x_{2}) (y_{1}, y_{2}) \\ = max (S_{A_{2}}^{n} (x_{2}), S_{A_{2}}^{n} (y_{2}), S_{\underline{B} 1}^{n} (x_{1} y_{1})) \end{matrix}$ For all $(x_{1}, x_{2}) (y_{1}, y_{2}) \in {\overset{?}{E}}_{1} - {\overset{?}{E}}_{1}$ .

Proposition 2. The Composition of two CBFGs Ǧ₁ and Ǧ₂, Ǧ₁ [Ǧ₂] = $(A_{1}^{\circ} A_{2}, {\underline{B}}_{1}^{\circ} {\underline{B}}_{1})$ , is CBFG

Proof. The condition on A₁ × A₂ is obvious, we only need to verify the conditions for ${\underline{B}}_{1} \times {\underline{B}}_{2}$ .

Assume that $x \in {\underset{˙}{V}}_{1}$ and $x_{2} y_{2} \in {\overset{?}{E}}_{2}$ $\begin{matrix} (S_{\underline{B} 1}^{PL \circ} S_{\underline{B} 2}^{PL}) (x, x_{2}) (x, y_{2}) = min (S_{A_{1}}^{PL} (x), S_{\underline{B} 2}^{PL} (x_{2} y_{2})) \leq min (S_{A_{1}}^{PL} (x), min (S_{A_{2}}^{PL} (x_{2}), S_{A_{2}}^{PL} (y_{2}))) \\ = min (min S_{A_{1}}^{PL} (x), S_{A_{2}}^{PL} (x_{2})), min (S_{A_{1}}^{PL} (x), S_{A_{2}}^{PL} (y_{2}))) = min (S_{A_{1}}^{PL \circ} S_{A_{2}}^{PL}) (x, x_{2}), (S_{A_{1}}^{PL \circ} S_{A_{2}}^{PL}) (x, y_{2}) \end{matrix}$ $\begin{matrix} (S_{\underline{B} 1}^{PU \circ} S_{\underline{B} 2}^{PU}) (x, x_{2}) (x, y_{2}) = min (S_{A_{1}}^{PU} (x), S_{\underline{B} 2}^{PU} (x_{2} y_{2})) \leq min (S_{A_{1}}^{PU} (x), min (S_{A_{2}}^{PU} (x_{2}), S_{A_{2}}^{PU} (y_{2}))) A_{1} \\ = {v_{1}, v_{2}, v_{3}, v_{4}, v_{5}} = min (S_{A_{1}}^{PU \circ} S_{A_{2}}^{PU}) (x, x_{2}), (S_{A_{1}}^{PU \circ} S_{A_{2}}^{PU}) (x, y_{2}) \end{matrix}$ $\begin{matrix} (S_{\underline{B} 1}^{P \circ} S_{\underline{B} 2}^{P}) (x, x_{2}) (x, y_{2}) = min (S_{A_{1}}^{P} (x), S_{\underline{B} 2}^{P} (x_{2} y_{2})) \leq min (S_{A_{1}}^{P} (x), min (S_{A_{2}}^{P} (x_{2}), S_{A_{2}}^{P} (y_{2}))) \\ = min (min S_{A_{1}}^{P} (x), S_{A_{2}}^{P} (x_{2})), min (S_{A_{1}}^{P} (x), S_{A_{2}}^{P} (y_{2}))) = min (S_{A_{1}}^{P \circ} S_{A_{2}}^{P}) (x, x_{2}), (S_{A_{1}}^{P \circ} S_{A_{2}}^{P}) (x, y_{2}) \end{matrix}$

And $\begin{matrix} (S_{\underline{B} 1}^{nL \circ} S_{\underline{B} 2}^{nL}) (x, x_{2}) (x, y_{2}) = max (S_{A_{1}}^{nL} (x), S_{\underline{B} 2}^{nL} (x_{2} y_{2})) \geq max (S_{A_{1}}^{nL} (x), min (S_{A_{2}}^{nL} (x_{2}), S_{A_{2}}^{nL} (y_{2}))) \\ = max (max (S_{A_{1}}^{nL} (x), S_{A_{2}}^{nL} (x_{2})), max (S_{A_{1}}^{nL} (x), S_{A_{2}}^{nL} (y_{2}))) = max (S_{A_{1}}^{nL \circ} S_{A_{2}}^{nL}) (x, x_{2}), (S_{A_{1}}^{nL \circ} S_{A_{2}}^{nL}) (x, y_{2}) \end{matrix}$ $\begin{matrix} (S_{\underline{B} 1}^{nU \circ} S_{\underline{B} 2}^{nU}) (x, x_{2}) (x, y_{2}) = max (S_{A_{1}}^{nU} (x), S_{\underline{B} 2}^{nU} (x_{2} y_{2})) \geq max (S_{A_{1}}^{nU} (x), max (S_{A_{2}}^{nU} (x_{2}), S_{A_{2}}^{nU} (y_{2}))) \\ = max (max S_{A_{1}}^{nU} (x), S_{A_{2}}^{nU} (x_{2})), max (S_{A_{1}}^{nU} (x), S_{A_{2}}^{nU} (y_{2}))) = max (S_{A_{1}}^{nU \circ} S_{A_{2}}^{nU}) (x, x_{2}), (S_{A_{1}}^{nU \circ} S_{A_{2}}^{nU}) (x, y_{2}) \end{matrix}$ $\begin{matrix} (S_{\underline{B} 1}^{n \circ} S_{\underline{B} 2}^{n}) (x, x_{2}) (x, y_{2}) = max (S_{A_{1}}^{n} (x), S_{\underline{B} 2}^{n} (x_{2} y_{2})) \geq max (S_{A_{1}}^{n} (x), max (S_{A_{2}}^{n} (x_{2}), S_{A_{2}}^{n} (y_{2}))) \\ = max (max S_{A_{1}}^{n} (x), S_{A_{2}}^{n} (x_{2})), max (S_{A_{1}}^{n} (x), S_{A_{2}}^{n} (y_{2}))) = max (S_{A_{1}}^{n \circ} S_{A_{2}}^{n}) (x, x_{2}), (S_{A_{1}}^{n \circ} S_{A_{2}}^{n}) (x, y_{2}) \end{matrix}$

Next, For $z \in {\underset{˙}{V}}_{2}$ and $x_{1} y_{1} \in \overset{?}{E} 1$ the proof is same now for $(x_{1}, x_{2}) (y_{1}, y_{2}) \in \overset{?}{E} 1 - \overset{?}{E} 1$ we have $x_{1} y_{1} \in \overset{?}{E} 1$ and x₂ ≠ y₂ $\begin{matrix} (S_{\underline{B} 1}^{PL \circ} S_{\underline{B} 2}^{PL}) (x, x_{2}) (x, y_{2}) = min (S_{A_{2}}^{PL} (x_{2}), S_{A_{2}}^{PL} (y_{2}), S_{\underline{B} 1}^{PL} (x_{1} y_{1})) \leq min (S_{A_{2}}^{PL} (x_{2}), S_{A_{2}}^{PL} (y_{2}), min (S_{A_{1}}^{PL} (x_{1}), S_{A_{1}}^{PL} (y_{1}))) \\ = min (min (S_{A_{1}}^{PL} (x_{1}), S_{A_{2}}^{PL} (x_{2})), min (S_{A_{1}}^{PL} (y_{1}), S_{A_{2}}^{PL} (y_{2}))) = min ((S_{A_{1}}^{PL \circ} S_{A_{2}}^{PL}) (x_{1}, x_{2}), (S_{A_{1}}^{PL \circ} S_{A_{2}}^{PL}) (y_{1}, y_{2}) \end{matrix}$ $\begin{matrix} (S_{\underline{B} 1}^{PU \circ} S_{\underline{B} 2}^{PU}) (x_{1}, x_{2}) (y_{1}, y_{2}) = min (S_{A_{2}}^{PU} (x_{2}), S_{A_{2}}^{PU} (y_{2}), S_{\underline{B} 1}^{PU} (x_{1} y_{1})) \leq min (S_{A_{2}}^{PU} (x_{2}), S_{A_{2}}^{PU} (y_{2}), min (S_{A_{1}}^{PU} (x_{1}), S_{A_{1}}^{PU} (y_{1}))) \\ = min (min S_{A_{1}}^{PU} (x_{1}), S_{A_{2}}^{PU} (x_{2})), min (S_{A_{1}}^{PU} (y_{1}), S_{A_{2}}^{PU} (y_{2})) = min ((S_{A_{1}}^{PU \circ} S_{A_{2}}^{PU}) (x_{1}, x_{2}), (S_{A_{1}}^{PU \circ} S_{A_{2}}^{PU}) (y_{1}, y_{2}) \end{matrix}$ $\begin{matrix} (S_{\underline{B} 1}^{P \circ} S_{\underline{B} 2}^{P}) (x_{1}, x_{2}) (y_{1}, y_{2}) = min (S_{A_{2}}^{P} (x_{2}), S_{A_{2}}^{P} (y_{2}), S_{\underline{B} 1}^{P} (x_{1} y_{1})) \leq min (S_{A_{2}}^{P} (x_{2}), S_{A_{2}}^{P} (y_{2}), min (S_{A_{1}}^{P} (x_{1}), S_{A_{1}}^{P} (y_{1}))) \\ = min (min S_{A_{1}}^{P} (x_{1}), S_{A_{2}}^{P} (x_{2})), min (S_{A_{1}}^{P} (y_{1}), S_{A_{2}}^{P} (y_{2})) = min ((S_{A_{1}}^{P \circ} S_{A_{2}}^{P}) (x_{1}, x_{2}), (S_{A_{1}}^{P \circ} S_{A_{2}}^{P}) (y_{1}, y_{2}) \end{matrix}$

And $\begin{matrix} (S_{\underline{B} 1}^{nL \circ} S_{\underline{B} 2}^{nL}) (x_{1}, x_{2}) (y_{1}, y_{2}) = max (S_{A_{2}}^{nL} (x_{2}), S_{A_{2}}^{nL} (y_{2}), \\ S_{\underline{B} 1}^{nL} (x_{1} y_{1})) \geq max (S_{A_{2}}^{nL} (x_{2}), S_{A_{2}}^{nL} (y_{2}), min (S_{A_{1}}^{nL} (x_{1}), \\ S_{A_{1}}^{nL} (y_{1}))) = max (max (S_{A_{1}}^{nL} (x_{1}), S_{A_{2}}^{nL} (x_{2})), max (S_{A_{1}}^{nL} (y_{1}), \\ S_{A_{2}}^{nL} (y_{2})) = max ((S_{A_{1}}^{nL \circ} S_{A_{2}}^{nL}) (x_{1}, x_{2}), (S_{A_{1}}^{nL \circ} S_{A_{2}}^{nL}) (y_{1}, y_{2}) \end{matrix}$ $\begin{matrix} (S_{\underline{B} 1}^{nU \circ} S_{\underline{B} 2}^{nU}) (x_{1}, x_{2}) (y_{1}, y_{2}) = max (S_{A_{2}}^{nU} (x_{2}), S_{A_{2}}^{nU} (y_{2}), \\ S_{\underline{B} 1}^{nU} (x_{1} y_{1})) \geq max (S_{A_{2}}^{nU} (x_{2}), S_{A_{2}}^{nU} (y_{2}), max (S_{A_{1}}^{nU} (x_{1}), \\ S_{A_{1}}^{nU} (y_{1}))) = max (max (S_{A_{1}}^{nU} (x_{1}), S_{A_{2}}^{nU} (x_{2})), \\ max (S_{A_{1}}^{nU} (y_{1}), S_{A_{2}}^{nU} (y_{2})) = max ((S_{A_{1}}^{nU \circ} S_{A_{2}}^{nU}) (x_{1}, x_{2}), \\ (S_{A_{1}}^{nU \circ} S_{A_{2}}^{nU}) (y_{1}, y_{2}) \end{matrix}$ $\begin{matrix} (S_{\underline{B} 1}^{n \circ} S_{\underline{B} 2}^{n}) (x_{1}, x_{2}) (y_{1}, y_{2}) = max (S_{A_{2}}^{n} (x_{2}), S_{A_{2}}^{n} (y_{2}), \\ S_{\underline{B} 1}^{n} (x_{1} y_{1})) \geq max (S_{A_{2}}^{n} (x_{2}), S_{A_{2}}^{n} (y_{2}), max (S_{A_{1}}^{n} (x_{1}), \\ S_{A_{1}}^{n} (y_{1}))) = max (max (S_{A_{1}}^{n} (x_{1}), S_{A_{2}}^{n} (x_{2})), \\ max (S_{A_{1}}^{n} (y_{1}), S_{A_{2}}^{n} (y_{2})) = max ((S_{A_{1}}^{n \circ} S_{A_{2}}^{n}) (x_{1}, x_{2}), \\ (S_{A_{1}}^{n \circ} S_{A_{2}}^{n}) (y_{1}, y_{2}) \end{matrix}$

Definition 18. The union of two CBFGs Ǧ₁ and Ǧ₂ is denoted by Ǧ₁ ∪ Ǧ₂ = $(A_{1} \cup A_{2}, {\underline{B}}_{1} \cup {\underline{B}}_{2})$ of defined by $(i) {\begin{matrix} (S_{A_{1}}^{PL} \cup S_{A_{2}}^{PL}) (x) = S_{A_{1}}^{PL} (x) if x \in {\underset{˙}{V}}_{1} - {\underset{˙}{V}}_{2} \\ (S_{A_{1}}^{PL} \cup S_{A_{2}}^{PL}) (x) = S_{A_{1}}^{PL} (x) if x \in {\underset{˙}{V}}_{2} - {\underset{˙}{V}}_{1} \\ (S_{A_{1}}^{PL} \cup S_{A_{2}}^{PL}) (x) = max (S_{A_{1}}^{PL} (x), S_{A_{1}}^{PL} (x)) \\ if x \in {\underset{˙}{V}}_{1} \cap {\underset{˙}{V}}_{2} \end{matrix}$ $(ii) {\begin{matrix} (S_{A_{1}}^{PU} \cup S_{A_{2}}^{PU}) (x) = S_{A_{1}}^{PU} (x) if x \in {\underset{˙}{V}}_{1} - {\underset{˙}{V}}_{2} \\ (S_{A_{1}}^{PU} \cup S_{A_{2}}^{PU}) (x) = S_{A_{1}}^{PU} (x) if x \in {\underset{˙}{V}}_{2} - {\underset{˙}{V}}_{1} \\ (S_{A_{1}}^{PU} \cup S_{A_{2}}^{PU}) (x) = max (S_{A_{1}}^{PU} (x), S_{A_{1}}^{PU} (x)) \\ if x \in {\underset{˙}{V}}_{1} \cap {\underset{˙}{V}}_{2} \end{matrix}$ $(iii) {\begin{matrix} (S_{A_{1}}^{P} \cup S_{A_{2}}^{P}) (x) = S_{A_{1}}^{P} (x) if x \in {\underset{˙}{V}}_{1} - {\underset{˙}{V}}_{2} \\ (S_{A_{1}}^{P} \cup S_{A_{2}}^{P}) (x) = S_{A_{1}}^{P} (x) if x \in {\underset{˙}{V}}_{2} - {\underset{˙}{V}}_{1} \\ (S_{A_{1}}^{P} \cup S_{A_{2}}^{P}) (x) = max (S_{A_{1}}^{P} (x), S_{A_{1}}^{P} (x)) \\ if x \in {\underset{˙}{V}}_{1} \cap {\underset{˙}{V}}_{2} \end{matrix}$ $(iv) {\begin{matrix} (S_{A_{1}}^{nL} \cup S_{A_{2}}^{nL}) (x) = S_{A_{1}}^{nL} (x) if x \in {\underset{˙}{V}}_{1} - {\underset{˙}{V}}_{2} \\ (S_{A_{1}}^{nL} \cup S_{A_{2}}^{nL}) (x) = S_{A_{1}}^{nL} (x) if x \in {\underset{˙}{V}}_{2} - {\underset{˙}{V}}_{1} \\ (S_{A_{1}}^{nL} \cup S_{A_{2}}^{nL}) (x) = max (S_{A_{1}}^{nL} (x), S_{A_{1}}^{nL} (x)) \\ if x \in {\underset{˙}{V}}_{1} \cap {\underset{˙}{V}}_{2} \end{matrix}$ $(v) {\begin{matrix} (S_{A_{1}}^{nU} \cup S_{A_{2}}^{nU}) (x) = S_{A_{1}}^{nU} (x) if x \in {\underset{˙}{V}}_{1} - {\underset{˙}{V}}_{2} \\ (S_{A_{1}}^{nU} \cup S_{A_{2}}^{nU}) (x) = S_{A_{1}}^{nU} (x) if x \in {\underset{˙}{V}}_{2} - {\underset{˙}{V}}_{1} \\ (S_{A_{1}}^{nU} \cup S_{A_{2}}^{nU}) (x) = max (S_{A_{1}}^{nU} (x), S_{A_{1}}^{nU} (x)) \\ if x \in {\underset{˙}{V}}_{1} \cap {\underset{˙}{V}}_{2} \end{matrix}$ $(vi) {\begin{matrix} (S_{A_{1}}^{n} \cup S_{A_{2}}^{n}) (x) = S_{A_{1}}^{n} (x) if x \in {\underset{˙}{V}}_{1} - {\underset{˙}{V}}_{2} \\ (S_{A_{1}}^{n} \cup S_{A_{2}}^{n}) (x) = S_{A_{1}}^{n} (x) if x \in {\underset{˙}{V}}_{2} - {\underset{˙}{V}}_{1} \\ (S_{A_{1}}^{n} \cup S_{A_{2}}^{n}) (x) = max (S_{A_{1}}^{n} (x), S_{A_{1}}^{n} (x)) \\ if x \in {\underset{˙}{V}}_{1} \cap {\underset{˙}{V}}_{2} \end{matrix}$

Fig.8

Ǧ₁ Cubic bipolar fuzzy graph.

Fig.9

Ǧ₂ Cubic bipolar fuzzy graph.

Fig.10

Ǧ₁ ∪ Ǧ₂ Cubic bipolar fuzzy graph.

Example 8. Consider the graphs Ǧ₁ = $(A_{1}, {\underline{B}}_{1})$ and Ǧ₂ = $(A_{2}, {\underline{B}}_{2})$ such that A₁ = {v₁, v₂, v₃, v₄, v₅}, A₂ = {v₁, v₂, v₃, v₄, v₆} and ${\underline{B}}_{1} = {v_{1} v_{2}, v_{2} v_{3}, v_{5} v_{3}, v_{5} v_{4}, v_{4} v_{1}}$ , ${\underline{B}}_{2} = {v_{1} v_{2}, v_{2} v_{3}, v_{2} v_{6}, v_{6} v_{3}}$ . Then

Proposition 2. The union of two CBFGs is a CBFG.

Proof. Assume that Ǧ₁ = $(A_{1}, {\underline{B}}_{1})$ and Ǧ₂ = $(A_{2}, {\underline{B}}_{2})$ be two CBFGs of the graphs Ǧ₁ and Ǧ₂. We have to prove that Ǧ₁ ∪ Ǧ₂ = $(A_{1} \cup A_{2}, {\underline{B}}_{1} \cup {\underline{B}}_{2})$ is CBFG. Therefore A₁ ∪ A₂ is obviously satisfy all the conditions we have only to verify the conditions for ${\underline{B}}_{1} \cup {\underline{B}}_{2}$ .

First we assume that for $xy \in {\overset{?}{E}}_{1} \cap {\overset{?}{E}}_{2}$ Then

If $xy \in {\overset{?}{E}}_{1}$ and $xy \notin {\overset{?}{E}}_{2}$ , then

If $xy \notin {\overset{?}{E}}_{1}$ and $xy \in {\overset{?}{E}}_{2}$ , then

Definition 19. The join of two CBFGs Ǥ₁ and Ǥ₂ Ǥ₁ + Ǥ₁ = $(A_{1} + A_{2}, {\underline{B}}_{1}$ + ${\underline{B}}_{2})$ of the CBFGs Ǧ₁ and Ǧ₂ is defined as: ${\begin{matrix} (S_{\underline{B} 1}^{PL} + S_{\underline{B} 2}^{PL}) (x) = (S_{A_{1}}^{PL} (x) \cup S_{A_{2}}^{PL} (x) \\ (S_{\underline{B} 1}^{PU} + S_{\underline{B} 2}^{PU}) (x) = (S_{A_{1}}^{PU} (x) \cup S_{A_{2}}^{PU} (x) \\ (S_{\underline{B} 1}^{P} + S_{\underline{B} 2}^{P}) (x) = (S_{A_{1}}^{P} (x) \cup S_{A_{2}}^{P} (x) \end{matrix}$ ${\begin{matrix} (S_{\underline{B} 1}^{nL} + S_{\underline{B} 2}^{nL}) (x) = (S_{A_{1}}^{nL} (x) \cap S_{A_{2}}^{nL} (x) \\ (S_{\underline{B} 1}^{nU} + S_{\underline{B} 2}^{nU}) (x) = (S_{A_{1}}^{nU} (x) \cap S_{A_{2}}^{nU} (x) \\ (S_{\underline{B} 1}^{n} + S_{\underline{B} 2}^{n}) (x) = (S_{A_{1}}^{n} (x) \cap S_{A_{2}}^{n} (x) \end{matrix}$

If $x \in {\underset{˙}{V}}_{1} \cup {\underset{˙}{V}}_{2}$ .

If $xy \in {\overset{?}{E}}_{1} \cap {\overset{?}{E}}_{2}$ .

If $xy \in \overset{?}{E^{j}}$ , where $\overset{?}{E^{j}}$ represents the set of all edges joining the vertices of ${\underset{˙}{V}}_{1}$ and ${\underset{˙}{V}}_{2}$ .

Proposition 3. The join of two CBFGs is a CBFG.

Proof. Straightforward.

5 Strong cubic bipolar fuzzy graph

Definition 20. A pair $G = (A, \underline{B})$ is known as strong CBFG if $S_{\underline{B}}^{PL}$ (x) = $min (S_{A}^{PL} (x)$ , $S_{A}^{PL} ($ )), $S_{\underline{B}}^{PU} (x$ ) = $min (S_{A}^{PU} (x)$ , $S_{A}^{PU} ($ )), $S_{A}^{nL} (x$ ) = $max (S_{A}^{nL} (x)$ , $S_{A}^{nL} ($ )), $S_{A}^{nU} (x$ ) = $max (S_{A}^{nU} (x)$ , $S_{A}^{nU} ($ )) and $S_{A}^{P} (x$ ) = $min (S_{A}^{P} (x)$ , $S_{A}^{P} ($ )), $S_{A}^{n} (x$ ) = $max (S_{A}^{n} (x), S_{A}^{n} ($ )), for all $x y \in \overset{?}{E}$ .

Example 9. Let Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ be a strong CBFG where $\underset{˙}{V}$ be the set of vertices and $\overset{?}{E}$ be the set of edges. Then

Fig.11

Strong cubic bipolar fuzzy graph.

Clearly we note that Ǧis a strong CBFG.

Proposition 4. If Ǥ₁ and Ǥ₂ are two strong CBFGs, then Ǥ₁ × Ǥ₂, Ǥ₁ [Ǥ₂] and Ǥ₁ + Ǥ₂ are strong CBFGs.

Proof. The proof follows from Proposition 2.

Remark 1. The union of two strong CBFGs may or may not be a strong CBFG.

The following example supports the above Remark 1.

Example 10. The following graphs support the above remark 1.

Fig.12

Ǧ₁ Strong cubic bipolar fuzzy graph.

Fig.13

Ǧ₂ Strong cubic bipolar fuzzy graph.

Fig.14

Ǧ₁ ∪ Ǧ₂.

Proposition 5. If Ǥ₁ × Ǥ₂ is strong CBFG, then at least Ǥ₁ or Ǥ₂ must be strong CBFG.

Proof. Obvious.

Proposition 6. If Ǥ₁ [Ǥ₂]be strong, then at least Ǥ₁ or Ǥ₂must be strong CBFG.

Proof. Obvious.

Definition 21. The complement of strong CBFG Ǥ = $(A, \underline{B})$ of Ǧ = $(\underset{˙}{V}$ , $\overset{?}{E}$ ) is denoted by $\bar{A}$ = $(\bar{[S_{A}^{PL}, S_{A}^{PU}]}$ , $\bar{[S_{A}^{nL}, S_{A}^{nU}]}$ , $\bar{S_{A}^{P}}$ , $\bar{S_{A}^{n}})$ and $\underline{B}$ = $(\bar{[S_{\underline{B}}^{PL}, S_{\underline{B}}^{PU}]}$ , $\bar{[S_{\underline{B}}^{nL}, S_{\underline{B}}^{nU}]}, \bar{S_{\underline{B}}^{P}}, \bar{S_{\underline{B}}^{n}})$ and defined by

(1) $\underset{˙}{V} = \underset{˙}{V}$

(2) $\bar{S_{A}^{PL}} (x) = S_{A}^{PL} (x), \bar{S_{A}^{PU}} (x) = S_{A}^{PU} (x), \bar{S_{A}^{P}} (x) = S_{A}^{P} (x)$ and $\bar{S_{A}^{nL}} (x) = S_{A}^{nL} (x), \bar{S_{A}^{nU}} (x) = S_{A}^{nU} (x), \bar{S_{A}^{n}} (x) = S_{A}^{n} (x)$ for all $x \in \underset{˙}{V}$ .

and

Definition 22. A strong CBFG is known as self-complementary if $\bar{\bar{\overset{˘}{G}}} \approx \overset{˘}{G}$

Example 11. The following graphs is an example of complement of CBFGs.

Fig.15

Cubic bipolar fuzzy graph.

Fig.16

Complement of Figure 15.

Fig.17

Complement of Figure 16.

Clearly, $\bar{\bar{\overset{˘}{G}}} = \overset{˘}{G}$ . Therefore, Ǧ self-complementary.

Proposition 7. If Ǧ is self-complementary strong CBFG. Then

and

Proof. Obvious.

Proposition 8. If Ǧ is strong CBFG and if $S_{\underline{B}}^{PL} (x$ ) = $min ((S_{A}^{PL} (x)$ , $S_{A}^{PL} ($ )) $S_{\underline{B}}^{PU} (x$ ) = $min ((S_{A}^{PU} (x), S_{A}^{PU} ($ )), $S_{\underline{B}}^{P} (x$ ) = $min ((S_{A}^{P} (x), S_{A}^{P} ($ )) and $S_{\underline{B}}^{nL} (x$ ) = $max ((S_{A}^{nL} (x), S_{A}^{nL} ($ )), $S_{\underline{B}}^{nU} (x$ ) = $max ((S_{A}^{nU} (x), S_{A}^{nU} ($ )), $S_{\underline{B}}^{n} (x$ ) = $max ((S_{A}^{n} (x)$ , $S_{A}^{n} ($ )) for all $xy \in \underset{˙}{V}$ Then Ǧ is self-complementary.

Proof. Obvious.

Proposition 9. if Ǥ₁ and Ǥ₂ be strong CBFG, then Ǥ₁ ≅ Ǥ₂ iff $\bar{G_{1}} ≅ \bar{G_{2}}$ .

Proof. Straight forward.

Definition 23. A pair $D = (A, \underline{B})$ with underlying graph Ǧ = $(\underset{˙}{V}, \overset{?}{E})$ is said to be cubic bipolar fuzzy digraph where $A = {< [S_{A}^{PL}, S_{A}^{PU}], [S_{A}^{NL}, S_{A}^{NU}], S_{A}^{P}, S_{A}^{N} >}$ is an IVBFS and $\underline{B} = = {< [S_{\underline{B}}^{PL}, S_{\underline{B}}^{PU}], [S_{\underline{B}}^{NL}, S_{\underline{B}}^{NU}], S_{\underline{B}}^{P}, S_{\underline{B}}^{N} >}$ is an IVBF relation on $\overset{?}{E}$ such that $S_{\underline{B}}^{PL} (x$ $) \leq min (S_{\underline{B}}^{PL} (x), S_{\underline{B}}^{PL} ($ )), $S_{\underline{B}}^{PU} (x$ $) \leq min (S_{A}^{PU} (x), S_{A}^{PU} ($ )) with a condition $S_{\underline{B}}^{PU} (x$ $) \leq min (S_{A}^{PL} (x), S_{A}^{PL} ($ )), $S_{\underline{B}}^{NL} (x$ $) \geq max (S_{\underline{B}}^{NL} (x), S_{\underline{B}}^{NU} ($ )), $S_{\underline{B}}^{NU} (x$ $) \geq max (S_{A}^{NU} (x), S_{A}^{NU} ($ )) with a condition $S_{\underline{B}}^{NU} (x$ $) \geq max (S_{A}^{NL} (x), S_{A}^{NL} ($ )) and $S_{\underline{B}}^{P} (x$ $) \leq min (S_{A}^{P} (x), S_{A}^{P} ($ )), $S_{\underline{B}}^{N} (x$ $) \geq max (S_{A}^{N} (x), S_{A}^{N} ($ )), for all $x y \in \overset{?}{E}$ We note that $\underline{B}$ need not be symmetric.

Example 12. The graph is an example of cubic bipolar fuzzy digraph.

Fig.18

Bipolar fuzzy diagraph.

The above digraph represented by the following adjacency matrix.

$[\begin{matrix} (\begin{matrix} ([0, 0], [0, 0], \\ (0, 0.)) \end{matrix}) & (\begin{matrix} ([0.1, 0.4], [- 0.3, - 0.1], \\ (0.2, - 0.2)) \end{matrix}) & (\begin{matrix} ([0, 0], [0, 0], \\ (0, 0)) \end{matrix}) \\ (\begin{matrix} ([0, 0], [0, 0], \\ (0, 0)) \end{matrix}) & (\begin{matrix} ([0, 0], [0, 0], \\ (0, 0)) \end{matrix}) & (\begin{matrix} ([0.1, 0.3], [- 0.2, - 0.1], \\ (0.1, - 0.2)) \end{matrix}) \\ \begin{matrix} (\begin{matrix} ([0, 0], [0, 0], \\ (0, 0)) \end{matrix}) \\ (\begin{matrix} ([0.1, 0.2], [- 0.2, - 0.1], \\ (0.1, - 0.2)) \end{matrix}) \end{matrix} & \begin{matrix} (\begin{matrix} ([0, 0], [0, 0], \\ (0, 0)) \end{matrix}) \\ (\begin{matrix} ([0, 0], [0, 0], \\ (0, 0)) \end{matrix}) \end{matrix} & \begin{matrix} (\begin{matrix} ([0, 0], [0, 0], \\ (0, 0.)) \end{matrix}) \\ (\begin{matrix} ([0, 0], [0, 0], \\ (0, 0)) \end{matrix}) \end{matrix} \end{matrix} \begin{matrix} (\begin{matrix} ([0, 0], [0, 0], \\ (0, 0)) \end{matrix}) \\ (\begin{matrix} ([0, 0], [0, 0], \\ (0, 0)) \end{matrix}) \\ \begin{matrix} (\begin{matrix} ([0.1, 0.2], [- 0.2, - 0.1], \\ (0.1, - 0.3)) \end{matrix}) \\ (\begin{matrix} ([0, 0], [0, 0], \\ (0, 0)) \end{matrix}) \end{matrix} \end{matrix}]$

6 Application

Graph models utilize extensive range of application in various areas. Item models are required to adjust more structure than simply the adjacencies between vertices. In graph behaviour studies it is a common observation, that specific people can impact thoughts of others. A directed graph, known as an influence graph can be employed to structure the behaviour. Each individual of a group is nominated by a vertex.

There is a directed edge from vertex x to vertex y when the person represented by vertex x influences the person represented by vertex y. This graph lacks loops and multiple direct edges. In influence graph an authority and strength are represented by a vertex (node) and impact of a person on another person in the social group is represented by an edge. The impact of an individual on another possess a fuzzy boundary, that results in a well representation in the fuzzy digraph that has vertex denoting a person and its membership the power of person within that group. In our work, a fuzzy impact and a cubic bipolar influence graph are employed to investigate the more influential person within social group.

6.1 Bipolar fuzzy influence graph

We follow a bipolar fuzzy influence graph of a social group depicted in Fig. 19 where the nodes demonstrating the level of strength that a person possesses and associated to a social group. The level of power or strength is explained in for of its positive and negative membership. Level of positive membership can be evaluated as amount of strength an individual possess and negative membership that how much strength is losses, e.g., Amjid possess 50% positive but 20% negative power of influency within the social group. The corners of a graph show the impact of one individual upon another individual. The level of positive membership of edges can be explained as ratio or percentage of positive and negative membership e.g. Seema’s positive influence on Abeeha is 20% while its negative influence is 10%.

Fig.19

Bipolar influence graph.

6.2 Cubic bipolar fuzzy influence graph

A human impact does not stand always positive. If two individuals have some disharmony, then the impact stands negative on another person and cubic bipolar graph is employed. Cubic bipolar fuzzy influence graph of a social group is depicted in Fig. 20 where the nodes used show the strength of a person of a social group and its level of strength is evaluated based on positive (inter valued positive) membership and negative (inter valued negative) membership. Degree of positive membership can be explained as the amount of power in future and inter valued positive membership represents the power in present that an individual possesses. Similarly, negative membership can be explained as the amount of power in future and interval valued negative membership represents in present that an individual loss. Amjid has 30% to 60% strengths for the present and 10% strengths for the future in a social group, yet he losses 20% to 60% strength for the present and 50% for the future in the same group. The edges of a graph demonstrate the impact of one person upon the other. And the level of positive membership and interval valued positive membership and negative membership and interval valued negative non-membership of edges can be explained as the percentage of positive and negative impacts for present time and future time. e.g Seema accepts 20% to 40% views of Abeeha for present and 50 % for future yet she does not accept Abeeha’s point of view upto 10% to 30% for present and 30% for future.

Fig.20

Cubic bipolar influence graph.

7 Conclusions

In this paper, we have introduced the notion of CBFG which extends the notion of cubic graph as well as bipolar fuzzy graph. The novelty and significance of new concept is demonstrated with the help of examples. We have developed many operations on CBFGs such as the union, the join, the cartesian product and the composition. Each of the operation is supported with examples. We have also utilized the concept of CBFG is a social network to examine its practicality and analyse its advantages. In near future, the concept of isomorphism and complete graphs can be developed and further the applications of CBFGs can be studied in database theory, expert system, neural system and transportation networks.

Conflict of interest

We declare that there are no conflicts of interest regarding the publication of this paper.

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