International Journal of Analysis and Applications

Volume 16, Number 5 (2018), 733-750

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-733

CUBIC GRAPHS WITH APPLICATION

SHEIKH RASHID1, NAVEED YAQOOB2,∗, MUHAMMAD AKRAM3, MUHAMMAD

GULISTAN1

1Department of Mathematics, Hazara University, Mansehra, Pakistan

2Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Zulfi, Saudi Arabia

3Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan

∗Corresponding author: nayaqoob@ymail.com

Abstract. We introduce certain concepts, including cubic graphs, internal cubic graphs, external cubic

graphs, and illustrate these concepts by examples. We deal with fundamental operations, Cartesian product,

composition, union and join of cubic graphs. We discuss some results of internal cubic graphs and external

cubic graphs. We also describe an application of cubic graphs.

1. Introduction

Cubic sets are one of the real generalizations of fuzzy sets [27] provided by Jun et al. [9–11, 15, 26] during

the last five years. They developed cubic set theory in many directions and for more detail about cubic sets

one can see [12]. Kang and Kim [13] studied mappings of cubic sets. Muhiuddin et al. [18] presented the

idea of stable cubic sets.

Fuzzy graphs were studied by Rosenfeld [23] and give a few theoretical ideas in spite of the fact that the

fundamental thought was presented by Kauffmann [14] in 1973. Bhattacharya [6] gave some remarks on

fuzzy graphs. A book written by Mordeson and Nair [17] is devoted especially to the study of fuzzy graphs

Received 2018-03-13; accepted 2018-05-22; published 2018-09-05.

2010 Mathematics Subject Classification. 68R10, 05C72.

Key words and phrases. cubic sets; cubic graphs; internal and external cubic graphs.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

733

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-733


Int. J. Anal. Appl. 16 (5) (2018) 734

and fuzzy hypergraphs. Akram et al. gave the idea of interval-valued fuzzy graphs [1, 2], intuitionistic fuzzy

graphs [3] and bipolar fuzzy graphs [4, 5]. Borzooei and Rashmanlou [7] studied Cayley interval-valued

fuzzy threshold graphs. Buckley [8] introduced self-centered graphs. Sunitha et al. [25] characterized g-self

centered fuzzy graphs. Mishra et al. [16] studied coherent category of interval-valued intuitionistic fuzzy

graphs. Pal et al. [19] and Pramanik et al. [21, 22] added some useful results to the theory of interval-valued

fuzzy graphs. Parvathi et al. [20] provided some different operations on intuitionistic fuzzy graphs and Sahoo

and Pal [24] studied product of intuitionistic fuzzy graphs.

In this paper we study some operations on cubic graphs. Internal and external cubic graphs are studied

with some example. We provided some conditions for union and join of external and internal cubic graphs.

2. Preliminaries

Here we recall some basic helping material from the existing literature.

Definition 2.1. A graph is denoted by Ω∗ = (P,Q), where P denotes the set of vertices of Ω∗ and Q stands

for the set of edges of Ω∗.

Definition 2.2. [12] Let T be a non-empty set. By a cubic set in T we mean a structure

Λ = {〈t,$̃Λ(t),µΛ(t)〉 |t ∈ T}

in which $̃Λ is an interval-valued fuzzy set in T and µΛ is a fuzzy set in T.

A cubic set Λ = {〈t,$̃Λ(t),µΛ(t)〉 |t ∈ T} is simply denoted by Λ = 〈$̃Λ,µΛ〉.

Definition 2.3. [12] Let T be a non-empty set. A cubic set Λ = 〈$̃Λ,µΛ〉 in T is said to be an internal

cubic (resp., external cubic) set if

$−Λ (t) ≤ µΛ(t) ≤ $
+
Λ (t) (resp., µΛ(t) /∈ ($

−
Λ (t),$

+
Λ (t)))

for all t ∈ T.

Definition 2.4. [12] For any Λi = {〈t,$̃Λi (t),µΛi (t)〉 |t ∈ T} where i ∈ I, we define

(a) ∪P
i∈I

Λi =

{〈
t,

(
∪
i∈I
$̃Λi

)
(t),

(
∨
i∈I
µΛi

)
(t)

〉
|t ∈ T

}
(P-union)

(b) ∪R
i∈I

Λi =

{〈
t,

(
∪
i∈I
$̃Λi

)
(t),

(
∧
i∈I
µΛi

)
(t)

〉
|t ∈ T

}
(R-union)

3. Cubic graphs

We develop the theory of a cubic graph and some operations on cubic graph.



Int. J. Anal. Appl. 16 (5) (2018) 735

Definition 3.1. Let M ∗ = 〈P,Q〉 be a graph. A cubic graph of a graph M ∗ = 〈P,Q〉 , is the structure

M = 〈α,β〉 , where α = 〈$̃α,µα〉 is the cubic set representation for the vertex P and β = 〈$̃β,µβ〉 denotes

the cubic set representation for the edge Q, with

$̃α : P → D[0, 1], µα : P → [0, 1],

and $̃β : Q → D[0, 1], µβ : Q → [0, 1],

such that

$̃β(pipj) � r min{$̃α(pi),$̃α(pj)},

µβ(pipj) ≤ max{µα(pi),µα(pj)},

for all (pi,pj) ∈ Q ⊆ P ×P.

Example 3.1. Let us consider a graph Ω∗ = (P,Q) such that P = {p1,p2,p3,p4}, Q = {p1p2,p2p3,p3p4,p4p1}.

Let α be a cubic set of P and let β be a cubic set of Q defined by

P $̃α µα

p1 [0.1, 0.5] 0.7

p2 [0.3, 0.7] 0.2

p3 [0.2, 0.4] 0.2

p4 [0.1, 0.8] 0.7

Q $̃β µβ

p1p2 [0.1, 0.4] 0.4

p2p3 [0.1, 0.3] 0.1

p3p4 [0.1, 0.4] 0.5

p4p1 [0.1, 0.4] 0.3

Figure 1. Cubic graph

By routine calculations, it can be observed that the graph shown in Fig. 1 is a cubic graph.

Example 3.2. Consider a graph Ω∗ = (P,Q). Let α be a cubic set of P and let β be a cubic set of Q defined

by

µα(pi) =
$−α (pi) + $

+
α (pi)

2
and µβ(ei) =

$−β (ei) + $
+
β (ei)

2
.

Then M = 〈α,β〉 is a cubic graph of Ω∗.



Int. J. Anal. Appl. 16 (5) (2018) 736

Remark 3.1. If $̃β(pipj) = [0, 0] and µβ(pipj) = 0, then the cubic graph M = 〈α,β〉 has no edge.

Definition 3.2. Let M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 be two cubic graphs of the graphs Ω∗1 and Ω∗2,

respectively. The Cartesian product of M1 and M2 is denoted by M1 × M2 = 〈α1 ×α2,β1 ×β2〉 and is

defined as follows:

(i)


 ($̃α1 × $̃α2 )(p1,p2) = r min{$̃α1 (p1),$̃α2 (p2)}(µα1 ×µα2 )(p1,p2) = max{µα1 (p1),µα2 (p2)}

for all (p1,p2) ∈ P = P1 ×P2,

(ii)


 ($̃β1 × $̃β2 )((q,q2)(q,p2)) = r min{$̃α1 (q),$̃β2 (q2p2)}(µβ1 ×µβ2 )((q,q2)(q,p2)) = max{µα1 (q),µβ2 (q2p2)}

for all q ∈ P1, and q2p2 ∈ Q2,

(iii)


 ($̃β1 × $̃β2 )((q1,r)(p1,r)) = r min{$̃β1 (q1p1),$̃α2 (r)}(µβ1 ×µβ2 )((q1,r)(p1,r)) = max{µβ1 (q1p1),µα2 (r)}

for all r ∈ P2, and q1p1 ∈ Q1.

Example 3.3. Consider two cubic graphs M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 as shown in figure 2.

Figure 2. Cubic graphs M1 and M2

Then, their corresponding Cartesian product M1 × M2 is shown in figure 3.

Figure 3. Cubic graph M1 × M2

Clearly, M1 × M2 is a cubic graph.

Proposition 3.1. The Cartesian product of two cubic graphs is a cubic graph.



Int. J. Anal. Appl. 16 (5) (2018) 737

Proof. The conditions for α1 ×α2 are obvious, therefore, we verify only conditions for β1 ×β2.

Let q ∈ P1, and q2p2 ∈ Q2. Then

($̃β1 × $̃β2 )((q,q2)(q,p2)) = r min{$̃α1 (q),$̃β2 (q2p2)}

� r min{$̃α1 (q),r min{$̃α2 (q2),$̃α2 (p2)}}

= r min{r min{$̃α1 (q),$̃α2 (q2)},r min{$̃α1 (q),$̃α2 (p2)}}

= r min{($̃α1 × $̃α2 )(q,q2), ($̃α1 × $̃α2 )((q,p2)}

(µβ1 ×µβ2 )((q,q2)(q,p2)) = max{µα1 (q),µβ2 (q2p2)}

≤ max{µα1 (q), max{µα2 (q2),µα2 (p2)}}

= max{max{µα1 (q),µα2 (q2)}, max{µα1 (q),µα2 (p2)}}

= max{(µα1 ×µα2 )(q,q2), (µα1 ×µα2 )((q,p2)}

Similarly, we can prove it for r ∈ P2, and q1p1 ∈ Q1. �

Definition 3.3. Let M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 be two cubic graphs of the graphs Ω∗1 and Ω∗2,

respectively. The composition of M1 and M2 is denoted by M1[M2] = 〈α1 ◦α2,β1 ◦β2〉 and is defined as

follows:

(i)


 ($̃α1 ◦ $̃α2 )(p1,p2) = r min{$̃α1 (p1),$̃α2 (p2)}(µα1 ◦µα2 )(p1,p2) = max{µα1 (p1),µα2 (p2)}

for all (p1,p2) ∈ P = P1 ×P2,

(ii)


 ($̃β1 ◦ $̃β2 )((q,q2)(q,p2)) = r min{$̃α1 (q),$̃β2 (q2p2)}(µβ1 ◦µβ2 )((q,q2)(q,p2)) = max{µα1 (q),µβ2 (q2p2)}

for all q ∈ P1, and q2p2 ∈ Q2,

(iii)


 ($̃β1 ◦ $̃β2 )((q1,r)(p1,r)) = r min{$̃β1 (q1p1),$̃α2 (r)}(µβ1 ◦µβ2 )((q1,r)(p1,r)) = max{µβ1 (q1p1),µα2 (r)}

for all r ∈ P2, and q1p1 ∈ Q1.

(iv)


 ($̃β1 ◦ $̃β2 )((q1,q2)(p1,p2)) = r min{$̃α2 (q2),$̃α2 (p2),$̃β1 (q1p1)}(µβ1 ◦µβ2 )((q1,q2)(p1,p2)) = max{µα2 (q2),µα2 (p2),µβ1 (q1p1)}

for all q2,p2 ∈ P2, q2 6= p2 and q1p1 ∈ Q1.

Example 3.4. From Example 3.3, consider two cubic graphs M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 as shown in

figure 2. Then, their corresponding composition M1[M2] is shown in figure 4.



Int. J. Anal. Appl. 16 (5) (2018) 738

Figure 4. Cubic graph M1[M2]

Clearly, M1[M2] is a cubic graph.

Proposition 3.2. The composition of two cubic graphs is a cubic graph.

Definition 3.4. Let M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 be two cubic graphs of the graphs Ω∗1 and Ω∗2,

respectively. The P-union of two cubic graphs M1 and M2 is denoted by M1 ∪P M2 = 〈α1 ∪p α2,β1 ∪p β2〉

and is defined as follows:

(i) ($̃α1 ∪p $̃α2 )(p) =




$̃α1 (p) if p ∈ P1 −P2

$̃α2 (p) if p ∈ P2 −P1

r max{$̃α1 (p),$̃α2 (p)} if p ∈ P1 ∩P2

(ii) (µα1 ∪p µα2 )(p) =




µα1 (p) if p ∈ P1 −P2

µα2 (p) if p ∈ P2 −P1

max{µα1 (p),µα2 (p)} if p ∈ P1 ∩P2

(iii) ($̃β1 ∪p $̃β2 )(pipj) =




$̃β1 (pipj) if pipj ∈ Q1 −Q2

$̃β2 (pipj) if pipj ∈ Q2 −Q1

r max{$̃β1 (pipj),$̃β2 (pipj)} if pipj ∈ Q1 ∩Q2

(iv) (µβ1 ∪p µβ2 )(pipj) =




µβ1 (pipj) if pipj ∈ Q1 −Q2

µβ2 (pipj) if pipj ∈ Q2 −Q1

max{µβ1 (pipj),µβ2 (pipj)} if pipj ∈ Q1 ∩Q2

Definition 3.5. Let M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 be two cubic graphs of the graphs Ω∗1 and Ω∗2,

respectively. The R-union of two cubic graphs M1 and M2 is denoted by M1 ∪R M2 = 〈α1 ∪R α2,β1 ∪R β2〉

and is defined as follows:

(i) ($̃α1 ∪R $̃α2 )(p) =




$̃α1 (p) if p ∈ P1 −P2

$̃α2 (p) if p ∈ P2 −P1

r max{$̃α1 (p),$̃α2 (p)} if p ∈ P1 ∩P2



Int. J. Anal. Appl. 16 (5) (2018) 739

(ii) (µα1 ∪R µα2 )(p) =




µα1 (p) if p ∈ P1 −P2

µα2 (p) if p ∈ P2 −P1

min{µα1 (p),µα2 (p)} if p ∈ P1 ∩P2

(iii) ($̃β1 ∪R $̃β2 )(pipj) =




$̃β1 (pipj) if pipj ∈ Q1 −Q2

$̃β2 (pipj) if pipj ∈ Q2 −Q1

r max{$̃β1 (pipj),$̃β2 (pipj)} if pipj ∈ Q1 ∩Q2

(iv) (µβ1 ∪R µβ2 )(pipj) =




µβ1 (pipj) if pipj ∈ Q1 −Q2

µβ2 (pipj) if pipj ∈ Q2 −Q1

min{µβ1 (pipj),µβ2 (pipj)} if pipj ∈ Q1 ∩Q2

Example 3.5. Consider two cubic graphs M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 as shown in figure 5.

Figure 5. Cubic graphs M1 and M2

Then, their corresponding P-union M1 ∪P M2 is shown in figure 6.

Figure 6. Cubic graph M1 ∪P M2

Also, their corresponding R-union M1 ∪R M2 is shown in figure 7.



Int. J. Anal. Appl. 16 (5) (2018) 740

Figure 7. Cubic graph M1 ∪R M2

Clearly, M1 ∪P M2 and M1 ∪R M2 are cubic graphs.

Proposition 3.3. The P-union and R-union of two cubic graphs is a cubic graph.

Proof. Since all the conditions for α1∪pα2 are automatically satisfied therefore, we verify only conditions

for β1 ∪p β2. In the case, when qp ∈ Q1 ∩Q2. Then

($̃β1 ∪p $̃β2 )(qp) = r max{$̃β1 (qp),$̃β2 (qp)}

� r max{r min{$̃α1 (q),$̃α1 (p)},r min{$̃α2 (q),$̃α2 (p)}}

= r min{r max{$̃α1 (q),$̃α2 (q)},r max{$̃α1 (p),$̃α2 (p)}}

= r min{($̃α1 ∪p $̃α2 )(q), ($̃α1 ∪p $̃α2 )(p)}.

(µβ1 ∪p µβ2 )(qp) = max{µβ1 (qp),µβ2 (qp)}

≤ max{max{µα1 (q),µα1 (p)}, max{µα2 (q),µα2 (p)}}

= max{max{µα1 (q),µα2 (q)}, max{µα1 (p),µα2 (p)}}

= max{(µα1 ∪p µα2 )(q), (µα1 ∪p µα2 )(p)}.

If qp ∈ Q1 and qp /∈ Q2, then

($̃β1 ∪p $̃β2 )(qp) � r min{($̃α1 ∪p $̃α2 )(q), ($̃α1 ∪p $̃α2 )(p)}

(µβ1 ∪p µβ2 )(qp) ≤ max{(µα1 ∪p µα2 )(q), (µα1 ∪p µα2 )(p)}.

If qp ∈ Q2 and qp /∈ Q1, then

($̃β1 ∪p $̃β2 )(qp) � r min{($̃α1 ∪p $̃α2 )(q), ($̃α1 ∪p $̃α2 )(p)}

(µβ1 ∪p µβ2 )(qp) ≤ max{(µα1 ∪p µα2 )(q), (µα1 ∪p µα2 )(p)}.



Int. J. Anal. Appl. 16 (5) (2018) 741

Hence the P-union of two cubic graphs is a cubic graph. The case for R-union of two cubic graphs can be

seen in a similar way. �

Definition 3.6. Let M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 be two cubic graphs of the graphs Ω∗1 and Ω∗2,

respectively. The P-join of two cubic graphs M1 and M2 is denoted by M1 +P M2 = 〈α1 +P α2,β1 +P β2〉

and is defined as follows:

(i)


 ($̃α1 +P $̃α2 )(p) = ($̃α1 ∪P $̃α2 )(p)(µα1 +P µα2 )(p) = (µα1 ∪P µα2 )(p)

for p ∈ P1 ∪P2,

(ii)


 ($̃β1 +P $̃β2 )(qp) = ($̃β1 ∪P $̃β2 )(qp)(µβ1 +P µβ2 )(qp) = (µβ1 ∪P µβ2 )(qp)

for qp ∈ Q1 ∩Q2,

(iii)


 ($̃β1 +P $̃β2 )(qp) = r min{$̃α1 (q),$̃α2 (p)}(µβ1 +P µβ2 )(qp) = min{µα1 (q),µα2 (p)}

for qp ∈ Q∗, where Q∗ is the set of all edges joining the vertices of P1 and P2.

Definition 3.7. Let M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 be two cubic graphs of the graphs Ω∗1 and Ω∗2,

respectively. The R-join of two cubic graphs M1 and M2 is denoted by M1 +R M2 = 〈α1 +R α2,β1 +R β2〉

and is defined as follows:

(i)


 ($̃α1 +R $̃α2 )(p) = ($̃α1 ∪R $̃α2 )(p)(µα1 +R µα2 )(p) = (µα1 ∪R µα2 )(p)

for p ∈ P1 ∪P2,

(ii)


 ($̃β1 +R $̃β2 )(qp) = ($̃β1 ∪R $̃β2 )(qp)(µβ1 +R µβ2 )(qp) = (µβ1 ∪R µβ2 )(qp)

for qp ∈ Q1 ∩Q2,

(iii)


 ($̃β1 +R $̃β2 )(qp) = r min{$̃α1 (q),$̃α2 (p)}(µβ1 +R µβ2 )(qp) = max{µα1 (q),µα2 (p)}

for qp ∈ Q∗, where Q∗ is the set of all edges joining the vertices of P1 and P2.

Example 3.6. Consider two cubic graphs M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 as shown in figure 8.

Figure 8. Cubic graphs M1 and M2



Int. J. Anal. Appl. 16 (5) (2018) 742

Then, their corresponding P-join M1 +P M2 is shown in figure 9.

Figure 9. Cubic graph M1 +P M2

Also, their corresponding R-join M1 +R M2 is shown in figure 10.

Figure 10. Cubic graph M1 +R M2

Clearly, M1 +P M2 and M1 +R M2 are cubic graphs.

Proposition 3.4. The P-join and R-join of two cubic graphs is a cubic graph.

4. Internal and external cubic graphs

Here in this section we discuss some results related with internal and external cubic graphs.

Definition 4.1. A cubic graph M = 〈α,β〉 is said to be an

(i) internal cubic graph (IC-graph) if

µα(pi) ∈ [$−α (pi),$
+
α (pi)] and µβ(ei) ∈ [$

−
β (ei),$

+
β (ei)]



Int. J. Anal. Appl. 16 (5) (2018) 743

for each pi ∈ P and ei ∈ Q.

(ii) external cubic graph (EC-graph) if

µα(pi) /∈ ($−α (pi),$
+
α (pi)) and µβ(ei) /∈ ($

−
β (ei),$

+
β (ei))

for each pi ∈ P and ei ∈ Q.

Example 4.1. The cubic graphs M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 are internal and external cubic graphs,

respectively, as shown in figure 11.

Figure 11. IC-graph M1 and EC-graph M2

Theorem 4.1. Let {Mi = 〈αi,βi〉 |i ∈ I} be a family of IC-graphs. Then ∪P
i∈I

Mi is an IC-graph.

Proof. Since Mi is an IC-graph, we have $
−
α (p) ≤ µα(p) ≤ $+α (p) and $

−
β (e) ≤ µβ(e) ≤ $

+
β (e) for i ∈ I.

This implies that

(
∪
i∈I
$−α

)
(p) ≤

(
∨
i∈I
µα

)
(p) ≤

(
∪
i∈I
$+α

)
(p),

and

(
∪
i∈I
$−β

)
(e) ≤

(
∨
i∈I
µβ

)
(e) ≤

(
∪
i∈I
$+β

)
(e).

Hence ∪P
i∈I

Mi is an IC-graph. �

The following example shows that the R-union of IC-graphs need not be an IC-graph (EC-graph).

Example 4.2. Consider two IC-graphs M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 as shown in figure 12.



Int. J. Anal. Appl. 16 (5) (2018) 744

Figure 12. IC-graphs M1 and M2

Then, their corresponding R-union M1 ∪R M2 is shown in figure 13.

Figure 13. R-union of IC-graphs M1 and M2

It is easy to see that the cubic graph M1 ∪R M2 is neither IC-graph nor EC-graph.

We provide a condition for the R-union of two IC-graphs to be an IC-graph.

Theorem 4.2. Let M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 be IC-graphs such that

max{$−α1 (p),$
−
α2

(p)}≤ min{µα1 (p),µα2 (p)}

and

max{$−β1 (e),$
−
β2

(e)}≤ min{µβ1 (e),µβ2 (e)}

for all p ∈ P and e ∈ Q. Then the R-union of two IC-graphs M1 and M2 is an IC-graph.

Proof. Let M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 be two IC-graphs which satisfy the conditions

max{$−α1 (p),$
−
α2

(p)}≤ min{µα1 (p),µα2 (p)}

and

max{$−β1 (e),$
−
β2

(e)}≤ min{µβ1 (e),µβ2 (e)}



Int. J. Anal. Appl. 16 (5) (2018) 745

for all p ∈ P and e ∈ Q. Since µα1 (p) ∈ [$−α1 (p),$
+
α1

(p)], µβ1 (e) ∈ [$
−
β1

(e),$+β1 (e)] and µα2 (p) ∈

[$−α2 (p),$
+
α2

(p)], µβ2 (e) ∈ [$
−
β2

(e),$+β2 (e)]. This implies that

min{µα1 (p),µα2 (p)}≤ ($
+
α1
∪$+α2 )(p) and min{µβ1 (e),µβ2 (e)}≤ ($

+
β1
∪$+β2 )(e)

Thus from the given condition we get

($−α1 ∪$
−
α2

)(p) = max{$−α1 (p),$
−
α2

(p)}≤ min{µα1 (p),µα2 (p)}≤ ($
+
α1
∪$+α2 )(p),

and

($−β1 ∪$
−
β2

)(e) = max{$−β1 (e),$
−
β2

(e)}≤ min{µβ1 (e),µβ2 (e)}≤ ($
+
β1
∪$+β2 )(e).

This shows that M1 ∪R M2 is an IC-graph. �

The following example shows that the P-union and R-union of EC-graphs need not be an EC-graph

(IC-graph).

Example 4.3. Consider two EC-graphs M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 as shown in figure 14.

Figure 14. EC-graphs M1 and M2

Then, their corresponding P-union M1 ∪P M2 is shown in figure 15.

Figure 15. P-union of EC-graphs M1 and M2

Also, the corresponding R-union M1 ∪R M2 is shown in figure 16.



Int. J. Anal. Appl. 16 (5) (2018) 746

Figure 16. R-union of EC-graphs M1 and M2

It is easy to see that the cubic graph M1 ∪P M2 and M1 ∪R M2 are neither EC-graph nor IC-graph.

We provide a condition for the P-union of two EC-graphs to be an EC-graph.

Theorem 4.3. Let M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 be two EC-graphs such that

min


 max{$

+
α1

(p),$−α2 (p)},

max{$−α1 (p),$
+
α2

(p)}


 > max{µα1 (p),µα2 (p)}
≥ max


 min{$

+
α1

(p),$−α2 (p)},

min{$−α1 (p),$
+
α2

(p)}




and

min


 max{$

+
β1

(e),$−β2 (e)},

max{$+β1 (e),$
−
β2

(e)}


 > max{µβ1 (e),µβ2 (e)}
≥ max


 min{$

+
β1

(e),$−β2 (e)},

min{$+β1 (e),$
−
β2

(e)}




for all p ∈ P and e ∈ Q. Then the P-union of two EC-graphs is an EC-graph.

We provide a condition for the R-union of two EC-graphs to be an EC-graph.

Theorem 4.4. Let M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 be two EC-graphs such that

min


 max{$

+
α1

(p),$−α2 (p)},

max{$−α1 (p),$
+
α2

(p)}


 > min{µα1 (p),µα2 (p)}
≥ max


 min{$

+
α1

(p),$−α2 (p)},

min{$−α1 (p),$
+
α2

(p)}






Int. J. Anal. Appl. 16 (5) (2018) 747

and

min


 max{$

+
β1

(e),$−β2 (e)},

max{$+β1 (e),$
−
β2

(e)}


 > min{µβ1 (e),µβ2 (e)}
≥ max


 min{$

+
β1

(e),$−β2 (e)},

min{$+β1 (e),$
−
β2

(e)}




for all p ∈ P and e ∈ Q. Then the R-union of two EC-graphs is an EC-graph.

Theorem 4.5. Let M = 〈α,β〉 be a cubic graph which is not an EC-graph. Then there exist pi ∈ P and

ei ∈ Q such that

µα(pi) ∈ ($−α (pi),$
+
α (pi)) and µβ(ei) ∈ ($

−
β (ei),$

+
β (ei)).

Proof. Straightforward. �

Theorem 4.6. Let M = 〈α,β〉 be a cubic graph of Ω∗. If M = 〈α,β〉 is both an IC-graph and an EC-graph,

then

µα(pi) ∈ U($̃α) ∪L($̃α)

and

µβ(ei) ∈ U($̃β) ∪L($̃β)

for all pi ∈ P and ei ∈ Q ⊆ P ×P. Where

U($̃α) = {$+α (pi)|pi ∈ P}, L($̃α) = {$
−
α (pi)|pi ∈ P}

and

U($̃β) = {$+β (ei)|ei ∈ Q},L($̃β) = {$
−
β (ei)|ei ∈ Q}.

Proof. Assume that M = 〈α,β〉 is both an IC-graph and an EC-graph. Then by definition we have

µα(pi) ∈ [$−α (pi),$
+
α (pi)], µβ(ei) ∈ [$

−
β (ei),$

+
β (ei)]

and

µα(pi) /∈ ($−α (pi),$
+
α (pi)), µβ(ei) /∈ ($

−
β (ei),$

+
β (ei)).

Thus µα(pi) = $
−
α (pi) or µα(pi) = $

+
α (pi) and µβ(ei) = $

−
β (ei) or µβ(ei) = $

+
β (ei). Hence µα(pi) ∈

U($̃α) ∪L($̃α) and µβ(ei) ∈ U($̃β) ∪L($̃β) for all pi ∈ P and ei ∈ Q ⊆ P ×P. �

Consider two cubic graphs M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 in Ω∗. If we exchange µα1 by µα2 and µβ1 by

µβ2 we get the cubic graph as M̂1 =
〈
α̂1, β̂1

〉
and M̂2 =

〈
α̂2, β̂2

〉
, respectively.

For any two IC-graphs (or EC-graphs) M1 and M2, two cubic graphs M̂1 and M̂2 may not be IC-graph

and EC-graph.



Int. J. Anal. Appl. 16 (5) (2018) 748

Example 4.4. Consider two IC-graphs M1 = 〈α1,β1〉 and M2 = 〈α2,β2〉 as shown in figure 17.

Figure 17. IC-graphs M1 and M2

Then, their corresponding M̂1 and M̂2 are shown in figure 18.

Figure 18. Cubic graphs M̂1 and M̂2

It is easy to see that the cubic graphs M̂1 and M̂2 are neither IC-graph nor EC-graph. Similarly, we can

provide and example for two EC-graphs that are neither IC-graph nor EC-graph.

5. Application

Fuzzy graph theory is a platform which has wide range of applications in mathematics, computer science

etc. Cubic graph is a more general approach, which can be used in decision making very effectively.

Suppose we have a set of three countries like, P = {X, Y , Z} as a vertex set and the membership of each

member of the set denotes the strength of that country over the neighbouring country with respect to future

and present time by considering its economic strength. Now we want to observe the effect of strength of one

country at the another country with respect to economy. Let we have a cubic set for each country based on



Int. J. Anal. Appl. 16 (5) (2018) 749

certain information and data with respect to economy

α =



〈X : [0.6, 0.8], 0.9〉

〈Y : [0.5, 0.9], 0.7〉

〈Z : [0.3, 0.7], 0.8〉

where interval membership predicts the economy of a certain country for the future and the other membership

shows economy of a certain country for the present time based on certain information and data with respect

to economy. Now on the basis of α, we have the set β of edges as follows

β =



〈XY : [0.5, 0.8], 0.9〉

〈Y Z : [0.3, 0.7], 0.8〉

〈ZX : [0.3, 0.7], 0.9〉

where interval membership predicts the effect of economy of a certain country for the future and the other

membership shows the effect of economy of a certain country for the present time at the other country. The

corresponding cubic graph is shown in figure 19.

Figure 19. Cubic graph

So finally we concluded that economy of a certain country effect very much on the economy of the

neighboring countries.

6. Conclusions

Graphs are among the most ubiquitous models of both natural and human-made structures. They can

be used to model many types of relations and process dynamics in computer science, physical, biological

and social systems. We come up here with the idea of cubic graphs and we define different operations of

cubic graphs. We also provide a short application of cubic graph. In future we are planning to generalize

our notions to (1) Cubic line graphs, (2) Cubic hypergraphs, and (3) Cubic soft graphs.



Int. J. Anal. Appl. 16 (5) (2018) 750

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	1. Introduction
	2. Preliminaries
	3. Cubic graphs
	4. Internal and external cubic graphs
	5. Application
	6. Conclusions
	References