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
R.Deepa2
Research Scholar (Part Time),
Department of Mathematics,
A.D.M College for Women (Autonomous), Nagapattinam,
Affiliated to Bharathidasan University, Thiruchirupalli, Tamilnadu, India
R.Deepa3
Associate Professor,
Department of Mathematics,
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.
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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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