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REGULAR INTERVAL-VALUED INTUITIONISTIC FUZZY SOFT GRAPH

Dr.N.Sarala1

Research Supervisor, Associate Professor,

Department of Mathematics,

A.D.M College for Women (Autonomous), Nagapattinam,

Affiliated to Bharathidasan University, Thiruchirupalli,Tamilnadu, India

[email protected]

R.Deepa2

Research Scholar (Part Time),


A.D.M College for Women (Autonomous), Nagapattinam,

Affiliated to Bharathidasan University, Thiruchirupalli, Tamilnadu, India

[email protected]

R.Deepa3

Associate Professor,


E.G.S Pillay Engineering College (Autonomous), Nagapattinam,

Tamilnadu, India

ABSTRACT

In this paper, we introduce regular interval-valued intuitionistic fuzzy soft graphs

and investigate some of their attributes. We talk about f-morphism on an interval-valued

intuitionistic fuzzy soft graph and regular interval-valued intuitionistic fuzzy soft graphs. (2, u)-

regular and totally (2, u) regular interval-valued intuitionistic fuzzy soft graphs.

KEYWORDS :Intutionistic fuzzy soft graph, f-morphism,(2,u) regular soft graph

1. INTRODUCTION

In 1965, zadeh [9] introduced the concept of fuzzy set as a method of finding uncertainty. In

1975, Rosenfeld [7] introduced the concept of fuzzy graphs. Yeh and Bang [8] also introduced

fuzzy graphs independently. Fuzzy graphs are useful to represent relationships which deal with

uncertainty and it differs greatly from classical graphs. It has numerous applications to

problems in computer science, electrical engineering, system analysis, operation research,

economics, networking routing, transportation, etc. interval-valued Fuzzy Graphs are defined

by Akram and Dudec in 2011. Atanassov [5] introduced the concept of intuitionistic fuzzy

relations and intuitionistic Fuzzy Graph. In fact interval-valued intuitionistic fuzzy graphs and

interval-valued intuitionistic fuzzy graphs are two different models that extend theory of fuzzy

graph S.N.Mishra and A.Pal [6] introduces the product of interval values intuitionistic fuzzy

graph.

2. PRILIMINARIES

We start this section by reviewing some fundamental concepts related to FSG.

Definition 2.1: A fuzzy set of a non-empty base set 𝑋 = 𝑥1, 𝑥2, … . . 𝑥𝑛 is defined by its degree

of membership function 𝑈 ; where 𝑈: 𝑋 ⟶ 0,1 assigning to all 𝑥1 ∈ 𝑋 , the degree to

which 𝑋 ∈ 𝑈.

The International journal of analytical and experimental modal analysis

Volume XII, Issue I, January/2020

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Definition 2.2: A fuzzy graph 𝐺 = (𝑋, 𝐸) is defined as a pair of function 𝑈: 𝑋 ⟶ 0,1 and

𝑆: 𝑋 × 𝑋 ⟶ 0,1 , where 𝐸 𝑥𝑖 , 𝑥𝑗 ≤ 𝑈 𝑥𝑖 ∧ 𝑈 𝑥𝑗 , ∀ 𝑥𝑖 , 𝑥𝑗 ∈ 𝑋 × 𝑋, ∀𝑥𝑖 , 𝑥𝑗 ∈ 𝑋. Here 𝑋 and

𝐸 are known as node and link of 𝐺 = (𝑋, 𝐸) correspondingly.

Definition 2.3: Consider 𝐺1 = 𝑥1, 𝐸1 and 𝐺2 = 𝑥2, 𝐸2 are fuzzy graphs over the given set 𝑋.

The union operation of 𝐺1 and 𝐺2 is provides a fuzzy graph 𝐺3 = 𝑥3, 𝐸3 over the set 𝑋. Here 𝑥3 = 𝑥1 ∨ 𝑥2 = 𝑚𝑎𝑥 𝑈1 𝑥𝑖 , 𝑈2 𝑥𝑖 , ∀𝑥𝑖 ∈ 𝑋,

And 𝑖 = 1,2,3 …𝑛 . Similarly 𝐸3 𝑥𝑖 , 𝑥𝑗 = 𝑚𝑎𝑥 𝐸1 𝑥𝑖 , 𝑥𝑗 , 𝐸2 𝑥𝑖 , 𝑥𝑗 , ∀ 𝑥𝑖 , 𝑥𝑗 ∈ 𝑋 × 𝑋 ,

where𝑖, 𝑗 = 1,2,3 …𝑛.

Definition 2.4: [8] The FSG is defined by 4 tuple as 𝐺 = (𝐺∗, 𝐹1, 𝐹2 , 𝑋) such that

1. 𝐺∗ = 𝑁, 𝐸 is a simple graph,

2. 𝑋 is a nonempty set of attributes,

3. 𝐹1, 𝑋 is a FSS over 𝑁,

4. (𝐹2, 𝑋) is a FSS over 𝐸,

5. (𝐹1 𝑖 , 𝐹2 𝑖 ) is a fuzzy soft graph of 𝐺∗ , ∀𝑖 ∈ 𝑋 . That

is, 𝐹2 𝑖 ≤ 𝑚𝑖𝑛 𝐹1 𝑖 𝑛1 , 𝐹1 𝑖 𝑛2 , ∀𝑖 ∈ 𝑋 and 𝑛1, 𝑛2 ∈ 𝑁. Note that 𝐹2 𝑖 𝑛1𝑛2 =

0, ∀𝑛1𝑛2 ∈ 𝑁 × 𝑁 − 𝐸 and ∀𝑖 ∈ 𝑋. The fuzzy soft graph (𝐹1 𝑖 , 𝐹2 𝑖 ) is defined by

𝐻 𝑖 for simplicity.

Definition 2.5: [8] A fuzzy soft graph 𝐺 is a strong FSG if 𝐻 𝑖 is a strong fuzzy graph for

all 𝑖 ∈ 𝑋, That is, 𝐹2 𝑖 (𝑛𝑗𝑛𝑘) = 𝑚𝑖𝑛 𝐹1 𝑖 𝑛𝑗 , 𝐹1 𝑖 𝑛𝑘 for all 𝑛𝑗 𝑛𝑘 ∈ 𝐸.

Definition 2.6: [8] Let 𝐺𝑎 = (𝐺𝑎∗, 𝐹1𝑎 , 𝐹2𝑎 , 𝑋𝑎) and 𝐺𝑏 = (𝐺𝑏

∗, 𝐹1𝑏 , 𝐹2𝑏 , 𝑋𝑏) be two FSGs of 𝐺𝑎∗

and 𝐺𝑏∗

, correspondingly. The union of 𝐺𝑎 and 𝐺𝑏 , symbolized by 𝐺𝑎 ∪ 𝐺𝑏 , is a FSG

(𝐹1, 𝐹2, 𝑋𝑎 ∪ 𝑋𝑏), such that (𝐹1, 𝑋𝑎 ∪ 𝑋𝑏), is a FSS over 𝑁 = 𝑁𝑎 ∪ 𝑁𝑏 , (𝐹2, 𝑋𝑎 ∪ 𝑋𝑏) is a FSS

over 𝐸𝑎 ∪ 𝐸𝑏 , and 𝐻 𝑖 = (𝐹1 𝑖 , 𝐹2 𝑖 ) is a fuzzy soft graph for all 𝑖 ∈ 𝑋𝑎 ∪ 𝑋𝑏 given by

𝐻 𝑖 = {𝐻𝑎 𝑖 , 𝑖𝑓 𝑖 ∈ 𝑋𝑎 − 𝑋𝑏 𝐻𝑏 𝑖 , 𝑖𝑓 𝑖 ∈ 𝑋𝑎 − 𝑋𝑏

𝐻𝑎 𝑖 ∪ 𝐻𝑏 𝑖 , 𝑖𝑓 𝑖 ∈ 𝑋𝑎 ∩ 𝑋𝑏

Definition 2.7: [9]

Let 𝐺1 = (G*,𝐹1

, 𝐾1 , A) and 𝐺2

= (G*,𝐹2 ,𝐾2

, A) be two fuzzy soft graphs. A homomorphism f :

𝐺1 → 𝐺2

is a mapping f : V1→ V2 which satisfies the following conditions.

(i) 𝐹1 (a) (x) ≤ 𝐹2

(a)(f(x))

𝑲𝟏 (a)(xy) ≤ 𝑲𝟐

(a) (f (x)f(y)) for all a∈A, x, y ∈ V1, x y ∈ E

DEFINITION: 2.8

Let G = (G*, F͂ , K͂ , B) be a simple graph, Y is a non-empty set and it is defined as

Y= {y1, y2 … yn}, E ⊆ YXY, P (set of attributes) and A ⊆ P.

Also consider i)D1 is a m deg given by D1: B→Is

b→ D1 (b) =D1b, b ∈ B,



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x

D1b: 𝛾→ [0, 1], 𝛾1→D1b (y1)

(B, D1) denote an intuitionistic fuzzy soft node of the m degree

T1: B→Γs

b→T1 (b) =T1b, b∈B and T1b: Y→ [0.1]

yi→ T1b (yi)

(B, T1) denote an intuitionistic fuzzy soft node of the nm degree such that

0 ≤ D1a (yi) + T1a (yi) ≤ 1, ∀ yi ∈ Y and b ∈ B

ii) D2 is a m degree given on E and given by

D2: B→Γs (yxy)

b→D2 (b) = D2b (b) = D2b, b ∈ B and

D2b: yxy→ [0, 1]

(yi, yj)→ D2b (yi, yj)

T2 is a nm deg and defined on E by

T2: B→Γs (yxy)

b→ T2 (b) = T2b, b ∈ B

T2b: yxy→ [0, 1]

(yi, yj)→ S2b (yi, yj)

Where (B, D2) and (B, T2) are I FSG links of m deg and nm deg satisfying

a) D2b (yi, yj) ≤ min {D1b (yi), D1b (yj)}

b) T2b (yi, yj) ≤ max {T1b (yi), T1b (yj)}and

c) 0 ≤ D2b (yi, yj) + T2b (yi, yj) ≤ 1

0 ≤ D2b (yi, yj), T2b (yi, yj), F (yi, yj) ≤ 1, ∀ (yi ,yj) ∈ X

The graph G = (G*, B, Y, E) = Y, E, (B, D1), (B, T1), (B, D2), (B, T2) is known as the

Intuitionistic fuzzy soft graph.

DEFINITION: 2.9

An internal-valued intuitionistic fuzzy soft graph G = (G*, F͂ , K͂ , (x, y)) is called strong

interval valued intuitionistic fuzzy soft graph

If D-y (ab) = min (D

-x (a), D

- (b)) and

T-y (ab) = min (T

-x (a), T

-x (b))



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D+

y (ab) = min (D+

x (a), D+

x (b)) and

T+

y (ab) = max (T+

x (a), T+

x (b)) ∀ ab ∈ E.

3. REGULAR INTERVAL-VALUED INTUITIONISTIC FUZZY SOFT GRAPH

DEFINITION: 3.1

An internal-valued intuitionistic fuzzy soft graph G is said to be regular if the absolute

degree of each vertex of an interval-valued intuitionistic fuzzy soft graph is constant. If the

absolute degree of each vertex is u, then we say the graph is u-regular interval valued

intuitionistic fuzzy soft graph.

DEFINITION: 3.2

Absolute degree d (u) of any vertex u of an internal-valued intuitionistic fuzzy soft graph

G is

d(u) = | ∑u≠v,v𝜖V 𝐷B+ (u,v) - ∑ u≠v,v𝜖V 𝑇B

+( u,v) |

Absolute membership of an edge e=uv ∀ e ∈ G is defined as

d(e) = | D+

B - T+

B | , where e ∈ (D, T) ∀ e ∈ G.

Example:3.2

Let G* = (V,E) where v={a1,a2,a3,a4} and

E = {a1a2, a2a3, a3a4, a1a4, a2a4, a1a3}, parameter {e} show in figure1.

Define G (A, B) by

DA (e) = {a1|(0.4,0 .7), a2|(0.5, 0.8), a3|(0.4,0 .8), a4|(0.3,0 .6)}

DB (e) = {a1a2|(0.4, 0.6), a2a3|(0.3,0 .5), a3a4|(0.3,0 .5), a1a4|(0.3,0 .5), a2a4|(0.3,0 .6),

a1a3|(0.4, 0.7)}

TA (e) = {a1|(0.3, 0.5), a2|(0.2,0 .4), a3|(0.1, 0.3), a4|(0.4, 0.6)}

TB (e) = {a1a2|(0.3, 0.5), a2a3|(0.2,0 .4), a3a4|(0.4,0 .6), a1a4|(0.4,0 .6), a2a4|(0.4, 0.6)

, a1a3|(0.3, 0.5)} a1 a2

a4 a3

Figure 1.

Absolute degree of an internal-valued intuitionistic fuzzy soft graph



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B B

Now, absolute degree of the vertices a1, a2, a3, a4 are

d(a1) = |(0. 7 +0 .6 + 0.5) − 0.5 +0 .5 +0 .6)| = |1.8 − 1.6| =0 .2

d(a2) = |(0. 6 + 0.6 +0 .5) − 0.5 +0 .4 +0 .6)| = |1.7 − 1.5| =0 .2

d(a3) = |(0. 7 +0 .5 + 0.5) − 0.4 +0 .5 +0 .6)| = |1.7 − 1.5| = 0.2

d(a4) = |(0. 5 + 0.6 +0 .5) − 0.6 + 0.6 +0 .6)| = |1.6 − 1.8| = 0.2

Here absolute degree of each vertex is 0.2. Thus, internal-valued intuitionistic fuzzy soft graph

G is 2-regular.

Definition: 3.3

Let G = (G*, F͂ , K͂ , (A, B)) be an internal-valued intuitionistic fuzzy soft graph on

G*= (V,E). The total degree of a vertex a1 is defined as

t d(a1) = | ∑𝑎1≠𝑎2,𝑎2𝜖𝑣 𝐷B+ (a1,a2) - ∑𝑎1≠𝑎2,𝑎2𝜖𝑣 𝑇B

+(a1,a2) | + | D+A (a1) – T+

A(a1) |

= d (a1) + | D+

A (a1) – T+

A (a1) | ∀ a1a2 ∈ E.

If each vertex of G has the same total degree u; then G is said to be totally regular interval-

valued intuitionistic fuzzy soft graph.

Definition:3.4

Let G = (G*, F͂ , K͂ , (A, B)) be an internal-valued intuitionistic fuzzy soft graph. The d2

degree of a vertex a1∈ G is d2 (a1) = | ∑ D 2+

(a1, a2) - ∑ T 2+

(a1, a2) | and summation runs over all

such a1∈ V which are distance two apart from a1.

Where

DB2+ (a1, a2) = inf {DB

+ (a1, a2), DB+ (a1, a2)}

and

TB2+ (a1, a2) = sup {TB

+ (a1, a2), TB+ (a1, a2)}

Also,

DB+ (a1, a2) = 0 and TB

+ (a1, a2) = 1, for a1 a2 ∉ 𝐸

The minimum d2-degree of G is 𝛿2 (G) = ∧ {d2 (a1): a1∈ V}.

The maximum d2-degree of G is ∆2 (G) = ⋁{d2 (a2): a2 ∈ V}.

Example:3.4

Consider G* = (V, E), where v= {a1, a2, a3, a4} and

E = {a1a2, a2a3, a3a4, a4a5, a5a1}. Define G = (G*, F͂ , K͂ , (A, B)) by

DA (e) = {a1|(0.4,0 .7), a2|(0.5,0 .8), a3|(0.4,0 .8), a4|(0. 3,0 .6), a5|(0.3,0 .7)}

DB (e) = {a1a2|(0.4,0 .6), a2a3|(0.3, 0.5), a3a4|(0.3, 0.5), a4a5|(0.3,0 .5), a5a1|(0.3,0 .7)}

TA (e) = {a1|(0.3, 0.5), a2|(0.2,0 .4), a3|(0.1,0 .3), a4|(0.4,0 .6), a5|(0.2,0 .4)}

TB (e) = {a1a2|(0.3,0 .5), a2a3|(0.2, 0.4), a3a4|(0.4, 0.7), a4a5|(0.4,0 .6), a5a1|(0.3,0 .5)}



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a1

a2 a5

a3 a4

Figure 2.

d2-degree for the vertices of an interval-valued intuitionistic fuzzy soft graph

Now,

d2 (a1) = | inf {0.6,0 .5} + inf {0.7, 0.5} – sup {0.5, 0.4} – sup {.5, .6} | =0 .1

d2 (a2) = | inf {0.5, 0.5} + inf {0.6,0 .7} – sup {0.4,0 .7} – sup {0.5, 0.5} | = 0.1

d2 (a3) = | inf {0.5, 0.5} + inf {0.5, 0.6} – sup {0.7,0 .6} – sup {0.4, 0.5} | =0 .2

d2 (a4) = | inf {0.5, 0.7} + inf {0.5, 0.5} – sup {0.6,0 .5} – sup {0.7, 0.4} | =0 .3

d2 (a5) = | inf {0.7,0 .6} + inf {0.5,0 .5} – sup {0.5, 0.5} – sup {0.6, 0.7} | =0 .1

Theorem 3.1

Even length interval-valued intuitionistic fuzzy cycle soft graph is regular or u-

regular⟺ absolute membership of e and d2 (e) for each e ∈ G is equal i.e., d(e)=d2(e) ∀ e ∈ G.

Proof

Let G = (G*, F͂ , K͂ , (A, B)) is an even length interval-valued intuitionistic fuzzy cycle soft

graph then if the absolute membership of each edge is same i.e., equal to any real number u

then d(e)=d2(e) ∀ e ∈ G thus d(a1)=2u ∀ a1 ∈G. Hence the theorem is trivially true. Now if the

absolute membership of any two adjacent edges is not equal but d2 (e) is equal then for any e ∈

G.

d (e1) = d2 (e2) = d (e3) =…….= d (e2n-1) = u1 (say)

Similarly,

d (e2) = d2 (e2) = d (e4) = d2 (e4)…….= d (e2n) = u2 (say)

Since cycle is of even length thus, there must be n number of ei’s having absolute

membership u1 and u2

Also, we know that for a cycle absolute degree of any vertex a1 is



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d(a1) = | ∑𝑎1≠𝑎2,𝑎2𝜖𝑣 𝐷B+ (a1,a2) - ∑𝑎1≠𝑎2,𝑎2𝜖𝑣 𝑇B

+(a1,a2) | = d(ei) + d(ei+1) = u1 + u2

Therefore, d (a1) = u ∀ a1∈ G so G is regular. Hence the theorem

Theorem 3.2

Cartesian product of two regular interval-valued intuitionistic fuzzy soft graph G1 and

G2 is regular if G1 is a weak regular interval-valued intuitionistic fuzzy soft graph sub graph of

G2 or vice versa.

Proof

Let G1 and G2 be two regular interval-valued intuitionistic fuzzy soft graph then the Cartesian

product of G1 and G2 is regular if the absolute membership of each arc e of G1×G2 is equal and

this is possible if d (e) = min {d(ei),d(ei)}, where ei ∈ G1 and ej ∈ G2 for all i and j thus the

condition is necessary for regularity of G1×G2 is either of G1 or G2 be a weak regular soft sub

graph of each other. Now, let G1 is weak regular sub graph G2 then we know that each edge of

G1×G2 get interval-valued membership and non-membership as minimum of D1 and D2 and

maximum of T1 and T2 thus, if G1 is weak then D1 and T1 dominates all the arc of G1×G2.So all

the arc receive same absolute membership which imply G1×G2 is regular. Hence the theorem.

Theorem 3.3

Any interval-valued intuitionistic fuzzy soft path graph of length l is never an regular interval-

valued intuitionistic fuzzy soft graph l >1.

Proof

For any interval-valued intuitionistic fuzzy soft path graph G = (G*, F͂ , K͂ , (A, B)) either

every edge have same absolute membership or some edges have district absolute membership.

Thus when all edges receive same absolute membership then at least both the end vertices of the

path soft graph G get different absolute degree then in-vertices of the soft path graph hence G is

not regular. Similarly if some edges have district absolute membership, let d(e1) ≠ d(e2) and

both e1 and e2 are adjacent let a1 be the common vertex of e1 and e2⟹d(a1) is always greater than

other and vertices of e1 and e2 which imply G is not regular. For l=1 graph is always regular

because in this case absolute membership of an edge become the absolute degree of the vertices.

Hence the theorem.

3. (A). (2, u)-Regular and Totally (2, u) - Regular Interval-Valued Intuitionistic Fuzzy Soft

Graph

Definition: 3. (a) (1) Let G = (G*, F͂, K͂, (A, B)) be an interval-valued intuitionistic fuzzy soft

graph on G*(V, E). If d2(a2) = 2, ∀a2∈V then G is said to be (2, u)-regular interval -valued

intuitionistic fuzzy soft graph.

Example: 3. (a) (1) Consider G*(V, E) where v= {a1, a2, a3, a4} and E = {a1a2, a2a3, a3a4, a4a1,}

and {e} be parameter set .Define G = (G*, F͂, K͂, (A, B)) by

DA (e) = {a1| (0.4, 0.7), a2| (0.5, 0 .8), a3| (0.4, 0 .8), a4| (0.3, 0.6)}

DB (e) = {a1a2| (0.4, 0.6), a2a3| (0.3, 0.5), a3a4| (0.3, 0 .6), a4a1| (0.3, 0.5)}



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TA (e) = {a1| (0.3, 0.5), a2| (0.2, 0.4), a3| (0.1, 0.3), a4| (0.3, 0 .5)}

TB (e) = {a1a2| (0.3, 0 .5), a2a3| (0.2, 0 .4), a3a4| (0.3, 0.5), a4a1| (0.4, 0 .7)}

Now,

a1 a2

a3 a4

Figure 3.

(2, u)-Regular interval-valued intuitionistic fuzzy soft graph

d2 (a1) = | inf {0.6, 0 .5} + inf {0.5, 0.6} – sup {0.5, 0 .4} – sup {0.7, 0 .5} | = 0.2,

d2 (a2) = | inf {0.5, 0.6} + inf {0.6, 0 .5} – sup {0.4, 0.5} – sup {0.5, 0.7} | =0.2,

d2 (a3) = | inf {0.6, 0.5} + inf {0.5, 0.6} – sup {0.5, 0 .7} – sup {0.4, 0 .5} | =0.2,

d2(a4) = | inf {0.5, 0.6} + inf {0.6,0 .5} – sup {0.7, 0.5} – sup {0.5,0 .4} | =0 .2,

Here d2(a1)=d2(a2)= d2(a3)= d2(a4)=0.2 thus the graph G is (2,2)-regular interval-

valued intuitionistic fuzzy soft graph.

Theorem: 3. (a) (1)

Let G = (G*, F͂, K͂, (A, B)) be a strong interval-valued intuitionistic fuzzy soft graph

on G*= (V, E) then D

+A (a1) =c1 and T

+A (a1) =c2 for all a1∈V if and only if the following

conditions are equivalent.

i) G = (G*, F͂, K͂, (A, B)) is a (2, u)-regular interval-valued intuitionistic fuzzy soft graph.

ⅱ) G = (G*, F͂, K͂, (A, B)) is a totally (2, u+c) - regular interval-valued intuitionistic fuzzy soft graph where

c = | c1 –c2 |.

Proof

Let D+

A (a1) =c1 and T+

A (a1) =c2 for all a1∈V.

Thus | D+

A (a1) - T+

A (a1) | = | c1 –c2 | = c for all a1∈V. Suppose that G = (G*, F͂ , K͂ , (A, B)) is a

(2, u)- Regular interval-valued intuitionistic fuzzy soft graph then d2 (a1) =u, for all a1∈V.

Hence,



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t d2 (a1) =d2 (a1) + | D+

A (a1) - T+A (a1) | ⟹ t d2 (a1) = u+c, ∀ a1 ∈ V.

Hence, G = (G*, F͂, K͂, (A, B)) is a totally (2, u+c))-regular interval-valued intuitionistic fuzzy

soft graph.

Thus (i) ⟹ (ii) is proved.

Suppose, G = (G*, F͂, K͂, (A, B)) is a totally (2, u+c)-regular interval-valued intuitionistic fuzzy

soft graph.

Therefore,

t d2 (a1) = u+c, ∀ a1 ∈ V

⟹ d2 (a1) + | D+

A (a1) - T+

A (a1) | = u+c, ∀ a1 ∈ V

⟹ d2 (a1) + | c1 –c2 | = u+c, ∀ a1 ∈ V

⟹ d2 (a1) + c = u+c, ∀ a1 ∈ V

⟹ d2 (a1) = u, ∀ a1 ∈ V

Hence,

G = (G*, F͂, K͂, (A, B)) is a (2, u)-regular interval-valued intuitionistic fuzzy soft graph.

Hence (i) and (ii) are equivalent. Conversely assume that (i) and (ii) are equivalent i.e., suppose

(G*, F͂, K͂, (A, B)) is (2, u)-regular interval-valued intuitionistic fuzzy soft graph and also a

totally (2, u+c))-regular interval-valued intuitionistic fuzzy soft graph.

Where, c = | c1 –c2 |.

Thus t d2 (a1) = u+c and d2 (a1) = u, ∀ a1 ∈ V

⟹ d2 (a1) + | D+

A (a1) - T+

A (a1) | = u+c and d2 (a1) = u, ∀ a1 ∈ V

⟹ | D+

A (a1) - T+

A (a1) | = c = | c1 –c2 |, ∀ a1 ∈ V

⟹ D+

A (a1) = c1 and T+

A (a1) = c2, ∀ a1 ∈ V

3. (B).Regularity on isomorphic interval-valued intuitionistic fuzzy soft graph

Definition: 3. (b) (1)

Let G1 = (G*, F͂, K͂, (A1, B1)) and G2 = (G*, F͂, K͂, (A2, B2)) be two interval-valued

intuitionistic fuzzy soft graph on (V1, E1) and (V2, E2) respectively.

A bijective function f: A1→A2 is called interval-valued intuitionistic fuzzy soft

morphism or f-morphism of interval-valued intuitionistic fuzzy soft graph if there exists some

positive real number u1 and u2 such that

(i)DA2 (f (a1)) = u1 DA1 a1 and TA2 (f(a1)) = u1 TA1 a1 , ∀ a1 ∈ V1

(ii)DB2 (f (a1), f (a2)) = u2 DB1 (a1, a2) and TB2 (f (a1), f (a2)) = u2 TB1 (a1, a2), ∀ a1 ∈ V1.



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In these cases f is called (u1, u2) f- interval-valued intuitionistic morphism on G1 over G2 when u1

= u2 = u then we say it is u-f- interval-valued intuitionistic morphism on G1 over G2.

Definition: 3. (b) (2)

A co-weak isomorphism from G1 to G2 is a map h: A1→A2 which is bijective

homomorphism that satisfies DB1 (a1, a2) = DB2 (h (a1), h (a2)) and

TB1 (a1, a2) = TB2 (h (a1), h (a2)), ∀ a1, a2 ∈ A.

A weak isomorphism from G1 to G2 is map h: A1→A2 which is bijective homomorphism that

satisfies DA1 (a1) = DA2 (h (a1)) and DA1 (a1) = TA2 (h (a1)), ∀ a1 a2 ∈ A.

Theorem: 3. (b) (1)

Let S be the set of all interval-valued intuitionistic fuzzy soft graphs. Now, define

the relation G1 ≈ G2 when G1 is (u1, u2) f- interval-valued intuitionistic fuzzy soft morphism on

G2 where u1, u2 are any non-zero real numbers and G1, G2 ∈ S.

Now for any identity morphism G1 over G1 is an one-one mapping and hence ′ ≈ ′ is reflexive.

Let G1 ≈ G2, then there exists a (u1, u2) -interval-valued intuitionistic fuzzy soft

morphism from G1 to G2 for some non-zero u1 and u2.

DA2 (f (a1)) = u1 DA1 a1 and TA2 (f (a1)) = u1 TA1 a1 , ∀ a1 ∈ V1

DB2 (f (a1), f (a2)) = u2 DB1 (a1, a2) and TB2 (f (a1), f (a2)) = u2 TB1 (a1, a2), ∀ a1 ,a2 ∈ V1

Consider f-1

: G1→G2. Let b1, b2 ∈ V2.

As f-1

is bijective, b1= f (a1), b2= f (a2), for some a1 a2 ∈ V1

Now

DA1(f-1

(b1)) = DA1(f-1

(f(a1) =DA1(a1) = 1

𝑢1 DA2f(a1) =

1

𝑢1 DA2(b1)

TA1(f-1

(b1)) = TA1(f-1

(f(a1) =TA1(a1) = 1

𝑢1 TA2f(a1) =

1

𝑢1 TA2(b1)

DB1(f-1

(b1), f-1

(b2)) = DB1(f-1

(f(a1), f-1

(f(a2)) =DB1(a1,a2) = 1

𝑢2 DB2 (f(a1),f(a2))

= 1

𝑢2 DB2 (b1,b2)

TB1(f-1

(b1), f-1

(b2)) = TB1(f-1

(f(a1), f-1

(f(a2)) =TB1(a1,a2) = 1

𝑢2 TB2 (f(a1),f(a2))

= 1

𝑢2 TB2 (b1,b2)

Thus there exists( 1

𝑢1 ,

1

𝑢2 ) f- interval-valued intuitionistic fuzzy soft morphism from G2 to G1.

Therefore G2 ≈ G1 and hence ‘≈’ is symmetric.

Let G1 ≈ G2 and G2 ≈ G3

Thus there exist two interval-valued intuitionistic fuzzy soft morphism say (u1, u2)-f and



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(u2, u3)-g such that f is interval-valued intuitionistic fuzzy soft morphism from G1 to G2 and g is

interval-valued intuitionistic fuzzy soft morphism from G2 to G3 for non-zero u1, u2, u3, u4.So,

DA3 g (b1) = u3 DA2 (b1) and TA3 g (b1) = u3 TA2 (b1) ,∀ b1 ∈ V2 and

DB3(g(b1),g(b2)) = u4 DB2 (b1,b2) and TB3 (g(b1),g(g2)) = u4 TB2 (b1,b2) ,∀ (b1, b2) ∈ E2.

Let h=g∘f: G1→ G3. Now,

DA3 (h (a1)) = DA3 ((g∘f) (a1)) = DA3 (g (f (a1))) = u3 DA2 (f (a1)) = u3 u1 DA1 (a1)

TA3 (h (a1)) = TA3 ((g∘f) (a1)) = TA3 (g (f (a1))) = u3 TA2 (f (a1)) = u3 u1 TA1 (a1)

DB3 (h (a1), h (a2)) = DB3 ((g∘f) (a1), (g∘f) (a2)) = DB3 (g (f (a1)), g (f (a2))) = u4 DB2 (f (a1), f (a2))

= u4 u2 DB1 (a1, a2)

TB3 (h (a1), h (a2)) = TB3 ((g∘f) (a1), (g∘f) (a2)) = TB3 (g (f (a1)), g (f (a2))) = u4 TB2 (f (a1), f (a2))

= u4 u2 TB1 (a1, a2)

Thus, there exists (u3 u1, u4 u2) h- interval-valued intuitionistic fuzzy soft morphism from G1

over G3. Therefore, G1 ≈ G3 hence ‘≈’ is transitive.

So, the relation f- interval-valued intuitionistic fuzzy soft morphism is an equivalence relation in

the collection of all interval-valued intuitionistic fuzzy soft graph.

Theorem: 3. (b) (2)

Let G1 and G2 be two IVIFSG’S such that G1 is (u1, u2) interval-valued intuitionistic

fuzzy soft morphic to G2 for some non-zero u1 and u2. The image of strong edge in G1 is strong

edge in G2 if and only if u1 = u2.

Proof

Let (a1, a2) be strong edge in G1 such that f (a1) , f(a2)

is also strong edge in G2.

Now, as G1 ≈ G2

u2 DB1 (a1, a2) = DB2 (f (a1), f (a2)) = DA2 f (a1) ∧ DA2 f (a2) = u1 {DA1 (a1) ∧ DA1 (a2)}

= u1 DB1(a1, a2), ∀ a1 ∈ V1.

Hence, u2 DB1 (a1, a2) = u1 DB1 (a1, a2), ∀ a1 ∈ V1 ........................................................................... (1)

Similarly, u2 TB1 (a1, a2) = TB2 (f (a1), f (a2)) = TA2 f (a1) ∨ TA2 f (a2) = u1 {TA1 (a1) ∧ TA1 (a2)}

= u1 TB1 (a1,a2), ∀ a1 ∈ V1.

Hence, u2 TB1 (a1, a2) = u1 TB1 (a1, a2), ∀ a1 ∈ V1 ............................................................................. (2)

Equation (1) and (2) holds. i.e., u1 = u2. Hence the theorem.



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Theorem: 3. (b) (3)

If an IVIFSG G1 is co- weak isomorphic to IVIFSG G2 and if G1 is regular then G2 is

regular.

Proof

As IVIFSG G1 is co- weak isomorphic to IVIFSG G2, there exists a co weak

isomorphism h: G1→ G2 which is bijective that satisfies

DA1 (a1) ≤ DA2 (h (a1)) and TA1 (a1) ≥ TA2 (h (a1)).

It also satisfies,

DB1 (a1, a2) = DB2 (h (a1), h (a2)) and

TB1 (a1, a2) = TB2 (h (a1), h (a2)), ∀ a1, a2 ∈ V1.

As G1 is regular, for a1 ∈ V.

∑𝑎1≠𝑎2 𝐷B+ (𝑎1, 𝑎2) = constant.

𝑎2∈𝑣1

∑𝑎1≠𝑎2 TB+ (𝑎1, 𝑎2) = constant.

𝑎2∈𝑣1

∑ℎ(𝑎1)≠ℎ(𝑎2) 𝐷B2 (h (a1), h (a2)) = ∑𝑎1≠𝑎2 𝐷B+ (𝑎1, 𝑎2) = constant.

And ∑ℎ(𝑎1)≠ℎ(𝑎2) TB2 (h (a1), h (a2)) = ∑𝑎1≠𝑎2 TB

+ (𝑎1, 𝑎2) = constant.

Therefore G2 is regular.

Theorem: 3. (b) (4)

Let G1 and G2 be two IVIFSG. If G1 is weak isomorphic to G2 and if G1 is strong

then G2 is strong.

Proof

As an IVIFSG G1 be weak isomorphic with an IVIFSG G2, there exists a weak

isomorphic h: G1→G2 which is bijective that satisfies

DA1 (a1) = DA2 (h (a1)) and TA1 (a1) = TA2 (h (a1)),

DB1 (a1, a2) ≤ DB2 (h (a1), h (a2)) and TB1 (a1, a2) ≥ TB2 (h (a1), h (a2)) ∀ a1, a2 ∈ V1.

As G1 is strong, DB1 (a1, a2) = min DA1 (a1), DA1 (a2) and TB1 (a1, a2) = max TA1 (a1), TA1 (a2)

DB2 (h (a1), h (a2)) ≤ DB1 (a1, a2) = min {DA1 (a1), DA1 (a2)} = min {DA2 h (a1), DA2 h (a2)}

By definition, DB2 (h (a1), h (a2)) ≤ min {DA2 h (a1), DA2 h (a2)}

Therefore, DB2 (h (a1), h (a2)) = min {DA2 h (a1), DA2 h (a2)} Similarly,

TB2 (h (a1), h (a2)) ≥ TB1 (a1, a2) = max {TA1 (a1), TA1 (a2)} = max {TA2 h (a1), TA2 h (a2)}



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And by definition,TB2 (h (a1), h (a2)) ≥ max {TA2 h (a1), TA2 h (a2)}.

Therefore, TB2 (h (a1), h (a2)) = max {TA2 h (a1), TA2 h (a2)}

Thus G2 is strong.

4. Conclusion

A regular interval-valued intuitionistic fuzzy soft graph has numerous applications

in the modeling of real life system where the level of information inherited in the system varies

with respect to time and have a different level of precision and hesitation. Most of the actions in

real life are time dependent, symbolic models used in the expert system are more effective than

traditional one. In this paper, we introduced the concept of a regular interval-valued intuitionistic

fuzzy graph and obtained some properties over it. In future, we can extend this concept to bipolar

fuzzy soft graphs, hyper graphs and in some more areas of graph theory.

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