On Complement Of Intuitionstic Product Fuzzy Graphs
Transcript of On Complement Of Intuitionstic Product Fuzzy Graphs
Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 9 (2017), pp. 5381-5392
ยฉ Research India Publications
http://www.ripublication.com
On Complement Of Intuitionstic Product Fuzzy
Graphs
Y. Vaishnaw1 and Sanjay Sharma2
1Department of Mathematics Govt. College Bhakhara, Dhamtari ( C. G. ) India.
2Department of Applied Mathematics, Bhilai Institute of technology, Durg ( C. G. ) India.
Abstract
The notion of product fuzzy graph has generalized for intuitionistic product
fuzzy graph and definition of complement and ring sum of two intuitionistic
product fuzzy graph have provided. Further some properties of ring sum of
intuitionistic product fuzzy graph have discussed.
AMS Subject Classification: 05C72
Keywords: Fuzzy graph, product fuzzy graph, complement of fuzzy graph
INTRODUCTION
In 1975, Rosenfeld [4] discussed the concept of fuzzy graphs whose basic idea was
introduced by Kauffmann in 1973. The fuzzy relations between fuzzy sets were also
considered by Rosenfeld and he developed the structure of fuzzy graphs, obtaining
analogs of several graph theoretical concepts. In [4] Rosendfeld and Yeh and Beng
independently developed the theory of fuzzy graph. A fuzzy graph is a pair G:(ฯ, ฮผ),
where ฯ is a fuzzy subset of V and ฮผ is fuzzy relation on V such that, ฮผ (xy) โค ฯ (x) โ
ฯ (y) for all x,y โV where ฯ: V โ [0,1] and ฮผ: V ร V โ [0,1].
Mohideen Ismail S. et al [11] have evaluated the basic definitions of intuitionistic
fuzzy graph and discussed some properties of the power of an intuitionistic fuzzy
graph and given the relationship between the index matrix of an intuitionistic fuzzy
graph and power of an intuitionistic fuzzy graph and Nagoor Gani A. et al [2010] has
5382 Y. Vaishnaw and Sanjay Sharma
introduced regular fuzzy graph.
V. Ramaswamy and Poornima B. [12] replaced โminimumโ in the definition of fuzzy
graph by โproductโ and call the resulting structure product fuzzy graph and proved
several results which are analogous to fuzzy graphs. Further they discussed a
necessary an sufficient condition for a product partial fuzzy sub graph to be the
multiplication of two product partial fuzzy sub graphs.
The notion of product fuzzy graph has generalized for intuitionistic product fuzzy
graph and definition of complement and ring sum of two intuitionistic product fuzzy
graph have provided. Further some properties of ring sum of intuitionistic product
fuzzy graph have discussed.
1. PRELIMINARY
Definition:1.1[12] Let G be a graph whose vertex set is V,๐ be a fuzzy sub set of V
and ๐ be a fuzzy sub set of VรV then the pair (๐, ๐) is called product fuzzy graph if
ฮผ(x, y) โค ฯ(x) ร ฯ(y) , โx, y โ V
Remark : If (๐, ๐) is a product fuzzy graph and since ๐(๐ฅ) and ๐(๐ฆ) are less than or
equal to 1, it follows that ฮผ(x, y) โค ฯ(x) ร ฯ(y) โค ฯ(x) โง ฯ(y) , โ x, y โ V Hence
(๐, ๐) is a fuzzy graph thus every product fuzzy graph is a fuzzy graph
Definition:1.2 a product fuzzy graph (๐, ๐) with crisp graph (V,E) is said to be
complete product fuzzy graph if ฮผ(x, y) = ฯ(x) ร ฯ(y) , โ x, y โ V
Definition:1.3 A product fuzzy graph (๐, ๐) with crisp graph (V,E) is said to be
strong product
fuzzy graph if ฮผ(x, y) = ฯ(x) ร ฯ(y) , โ (x, y) โ E
Definition:1.4[14] The complement of a product fuzzy graph (๐, ๐) with crisp graph
(V,E) is (๏ฟฝฬ ๏ฟฝ , ๐ ฬ ) where ๏ฟฝฬฟ๏ฟฝ = ๐
And ฮผ ฬ (x, y) = ฯ(x) ร ฯ(y) โ ฮผ(x, y)
= ฯ ฬ (x) ร ฯ ฬ (y) โ ฮผ(x, y)
It follows that (ฯฬ , ฮผ ฬ ) itself is a fuzzy graph. Also
ฮผฬ ฬ (x, y) = ฯ(x) ร ฯ(y) โ ฮผฬ (x, y)
= ฯ(x) ร ฯ(y) โ [ฯ(x) ร ฯ(y) โ ฮผ(x, y)]
= ฮผ(x, y) โ x, y โ V
Definition:1.5 [15] LetG1: (ฯ1,ฮผ1) and G2: (ฯ2, ฮผ2) are product fuzzy graphs with
On Complement Of Intuitionstic Product Fuzzy Graphs 5383
G1: (V1, E2) and G2: (V2, E2) and let G = G1 โช G2: (V1 โช V2, E1 โช E2) be the union of
G1 and G2. Then the union of two fuzzy graphs G1 and G 2 is also a product fuzzy
graph and is defined as
G = G1 โช G2: (ฯ1 โช ฯ2, ฮผ1 โช ฮผ2) Where
(ฯ1 โช ฯ2)u = {
ฯ1(u) if u โ V1 โ V2ฯ2(u) if u โ V2 โ V1
max[ฯ1(u),ฯ2(u)] if u โ V1 โฉ V2
And (ฮผ1โช ฮผ
2)uv =
{
ฮผ
1(uv) if uv โ E1 โ E2
ฮผ2(uv) ifuv โ E2 โ E1
max[ฮผ1(uv),ฮผ
2(uv)] ifuv โ E1 โฉ E2
0 otherwise
Definition:1.6 [15] Let (ฯ1, ฮผ1) be a product partial fuzzy sub graph of G1 =
(V1, X1)and (ฯ 2,ฮผ2) be a product fuzzy sub graph of G2 = (V2, X2) . Let Xโฒ denot the
set of all arcs joining the vertices of V1 and V2 . Then the join of (ฯ1, ฮผ1) and
(ฯ 2, ฮผ2) is defined as (ฯ 1 + ฯ 2, ฮผ1 + ฮผ2) where
(ฯ 1 + ฯ 2)u = {
ฯ 1(u) if u โ V1ฯ 2(u) if u โ V2
ฯ1(u) โจ ฯ2(u) if u โ V1 โฉ V2
And (ฮผ1+ ฮผ
1)uv =
{
ฮผ1(u, v) if (u, v) โ X1
ฮผ2(u, v) if (u, v) โ X2
ฮผ1(u, v) โจ ฮผ
2(u, v) if (u, v) โ X1 โฉ X2
ฯ1(u) ร ฯ2(v) if (u, v) โ Xโฒ
2. MAIN RESULTS
Definition: 2.1 An intuitionstic product fuzzy graph (IPFG), denoted by
G: {(ฯ, ฯ), (ฮผ, ฮณ)} of any crisp graph G:(V,E) is defined as follows
i) The function ฯ: V โ [0,1] and ฯ: V โ [0,1] denote the degree of
membership and nonmembership of the elements xi โ V respectively such
that
0 โค {ฯ(xi) + ฯ(xi)} โค 1 โxi โ V , (i = 1,2, โฆโฆโฆn)
ii) The function ฮผ: V ร V โ [0,1] and ฮณ: V ร V โ [0,1] are defined by
ฮผ(xi, xj) โค product[ฯ(xi),ฯ(xj)]
And ฮณ(xi, xj) โค max[ฯ(xi), ฯ(xj)]
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Definition:2.2 An intuitionstic product fuzzy graph (IPFG) is said to be strong if and
only if
ฮผ: V ร V โ [0,1] and ฮณ: V ร V โ [0,1] such that
ฮผ(xi, xj) = product[ฯ(xi),ฯ(xj)]
And ฮณ(xi, xj) = max[ฯ(xi), ฯ(xj)] for all (xi, xj) โ E.
Definition:2.3 An intuitionstic product fuzzy graph (IPFG) is said to be complete if
and only if
ฮผ: V ร V โ [0,1] and ฮณ: V ร V โ [0,1] such that
ฮผ(xi, xj) = product[ฯ(xi),ฯ(xj)]
And ฮณ(xi, xj) = max[ฯ(xi), ฯ(xj)] for all xi, xj โ V.
Definition:2.4 Complement of An intuitionstic product fuzzy graph (IPFG) is defined
by
The function ฯ: V โ [0,1] and ฯ: V โ [0,1] denote the degree of
membership and nonmembership of the elements xi โ V respectively such
that
0 โค {ฯ(xi) + ฯ(xi)} โค 1 โxi โ V , (i = 1,2, โฆโฆโฆn)Then
๐ = ๐ and ๐ฬ = ๐
And ๐: ๐ ร ๐ โ [0,1] and ๐พ: ๐ ร ๐ โ [0,1] such that
๏ฟฝฬ ๏ฟฝ(๐ฅ๐, ๐ฅ๐) = ๐๐๐๐๐ข๐๐ก[๐(๐ฅ๐), ๐(๐ฅ๐)] โ ๐(๐ฅ๐ , ๐ฅ๐)
โค ๐๐๐๐๐ข๐๐ก[๐(๐ฅ๐), ๐(๐ฅ๐)]
๐๐๐๐๐ข๐๐ก[๏ฟฝฬ ๏ฟฝ(๐ฅ๐), ๐(๐ฅ๐)]
i.e. ๏ฟฝฬ ๏ฟฝ(๐ฅ๐, ๐ฅ๐) โค ๐๐๐๐๐ข๐๐ก[๐(๐ฅ๐), ๐(๐ฅ๐)]
And ๏ฟฝฬ ๏ฟฝ(๐ฅ๐, ๐ฅ๐) = ๐๐๐ฅ[๐(๐ฅ๐), ๐(๐ฅ๐)] โ ๐พ(๐ฅ๐ , ๐ฅ๐)
โค ๐๐๐ฅ[๐(๐ฅ๐), ๐(๐ฅ๐)]
= ๐๐๐ฅ[๐ฬ (๐ฅ๐), ๐ฬ (๐ฅ๐)]
i.e. ๏ฟฝฬ ๏ฟฝ(๐ฅ๐, ๐ฅ๐) โค ๐๐๐ฅ[๐ฬ (๐ฅ๐), ๐ฬ (๐ฅ๐)]
Definition:2.5 Complement of an strong intuitionstic product fuzzy graph is defined
by
๐: ๐ โ [0,1] and ๐: ๐ โ [0,1] such that ๐ = ๐ and ๐ฬ = ๐
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And ๐: ๐ ร ๐ โ [0,1] and ๐พ: ๐ ร ๐ โ [0,1] such that
๏ฟฝฬ ๏ฟฝ(๐ฅ๐, ๐ฅ๐) = ๐๐๐๐๐ข๐๐ก[๐(๐ฅ๐), ๐(๐ฅ๐)] โ ๐(๐ฅ๐ , ๐ฅ๐)
= 0
Since ๐(๐ฅ๐, ๐ฅ๐) = ๐๐๐๐๐ข๐๐ก[๐(๐ฅ๐), ๐(๐ฅ๐)] โ(๐ฅ๐, ๐ฅ๐) โ ๐ธ
And ๏ฟฝฬ ๏ฟฝ(๐ฅ๐, ๐ฅ๐) = ๐๐๐ฅ[๐(๐ฅ๐), ๐(๐ฅ๐)] โ ๐พ(๐ฅ๐ , ๐ฅ๐)
= 0
Since ๐พ(๐ฅ๐ , ๐ฅ๐) = ๐๐๐๐๐ข๐๐ก[๐(๐ฅ๐), ๐(๐ฅ๐)] โ(๐ฅ๐, ๐ฅ๐) โ ๐ธ
Definition:2.6 Complement of an complete intuitionstic product fuzzy graph is
defined by
๐: ๐ โ [0,1] and ๐: ๐ โ [0,1] such that ๐ = ๐ and ๐ฬ = ๐
And ๐: ๐ ร ๐ โ [0,1] and ๐พ: ๐ ร ๐ โ [0,1] such that
๏ฟฝฬ ๏ฟฝ(๐ฅ๐, ๐ฅ๐) = ๐๐๐๐๐ข๐๐ก[๐(๐ฅ๐), ๐(๐ฅ๐)] โ ๐(๐ฅ๐ , ๐ฅ๐)
= 0
Since ๐(๐ฅ๐ , ๐ฅ๐) = ๐๐๐๐๐ข๐๐ก[๐(๐ฅ๐), ๐(๐ฅ๐)] โ๐ฅ๐ , ๐ฅ๐ โ ๐
And ๏ฟฝฬ ๏ฟฝ(๐ฅ๐, ๐ฅ๐) = ๐๐๐ฅ[๐(๐ฅ๐), ๐(๐ฅ๐)] โ ๐พ(๐ฅ๐ , ๐ฅ๐)
= 0
Since ๐พ(๐ฅ๐, ๐ฅ๐) = ๐๐๐๐๐ข๐๐ก[๐(๐ฅ๐), ๐(๐ฅ๐)] โ๐ฅ๐, ๐ฅ๐ โ ๐.
Definition:2.7 Let ๐บ1: {(๐1, ฯ1), ๐1, ฮณ1)} and ๐บ1: {(๐2, ฯ2), ๐2, ฮณ2)} be two
intuitionstic product fuzzy graph with crisp graph ๐บ1: (๐1, ๐ธ2) and ๐บ2: (๐2, ๐ธ2) res.
Then the ring sum of two intuitionstic product fuzzy ๐บ1: (๐1, ๐1) and ๐บ2: (๐2, ๐2) is
denoted by ๐บ = ๐บ1โจ๐บ2: {(๐1โจ๐2, ฯ1โจฯ2), (๐1โจ๐2, ฮณ1โจฮณ2)} is defined as
The function ๐: ๐ โ [0,1] and ๐: ๐ โ [0,1] denote the degree of membership and
nonmembership of the elements xi โ V respectively such that
0 โค {ฯ(xi) + ฯ(xi)} โค 1 โxi โ V , (i = 1,2, โฆโฆโฆn)
(ฯ1โจฯ2)x = {
(ฯ1)x if x โ V1 โ V2 (ฯ
2)x if x โ V2 โ V1
max{ฯ1(x), ฯ2(x)} ifx โ V1 โฉ V2
(๐1โจ๐2)๐ฅ = {
(๐1)๐ฅ ๐๐ ๐ฅ โ ๐1 โ ๐2 (๐2)๐ฅ ๐๐ ๐ฅ โ ๐2 โ ๐1
๐๐๐ฅ{๐1(๐ฅ), ๐2(๐ฅ)} ๐๐๐ฅ โ ๐1 โฉ ๐2
5386 Y. Vaishnaw and Sanjay Sharma
And (๐1โจ๐2)๐ฅ๐ฆ = {๐1(๐ฅ๐ฆ) ๐๐ ๐ฅ๐ฆ โ ๐ธ1 โ ๐ธ2๐2(๐ฅ๐ฆ) ๐๐ ๐ฅ๐ฆ โ ๐ธ2 โ ๐ธ10 ๐๐กโ๐๐๐ค๐๐ ๐
(๐พ1โจ๐พ2)๐ฅ๐ฆ = {๐พ1(๐ฅ๐ฆ) ๐๐ ๐ฅ๐ฆ โ ๐ธ1 โ ๐ธ2๐พ2(๐ฅ๐ฆ) ๐๐ ๐ฅ๐ฆ โ ๐ธ2 โ ๐ธ10 ๐๐กโ๐๐๐ค๐๐ ๐
Theorem 2.8 Let G1: {(ฯ1, ฯ1), ฮผ1, ฮณ1)} and G1: {(ฯ2, ฯ2), ฮผ2, ฮณ2)} with 0 โค
{ฯ(xi) + ฯ(xi)} โค 1 โxi โ V be two intuitionstic product fuzzy graph of crisp graph
G1: (V1, E2) and G2: (V2, E2) res. Then the ring sum of two intuitionstic product fuzzy
denoted by G = G1โจG2: {(ฯ1โจฯ2, ฯ1โจฯ2), (ฮผ1โจฮผ2, ฮณ1โจฮณ
2)} is also an intuitionstic
product fuzzy graph.
Proof: Since G1: {(ฯ1, ฯ1),ฮผ1, ฮณ1)} and G1: {(ฯ2, ฯ2), ฮผ2, ฮณ2)} be two intuitionstic
product fuzzy graph with 0 โค {ฯ(xi) + ฯ(xi)} โค 1 โxi โ V of crisp graph
G1: (V1, E2) and G2: (V2, E2) res. Then we have to show that G =
G1โจG2: {(ฯ1โจฯ2, ฯ1โจฯ2), (ฮผ1โจฮผ2, ฮณ1โจฮณ
2)} for this it is sufficient to show that
i) (ฮผ1โจฮผ
2)xy โค {(ฯ1โจฯ2)x ร (ฯ1โจฯ2)y}
ii) (ฮณ1โจฮณ
2)xy โค {(ฯ1โจฯ2)xโ(ฯ1โจฯ2)y} In each case.
Case-1 If xy โ E1 โ E2 and x, y โ V1 โ V2 then
(ฮผ1โจฮผ
2)xy = ฮผ
1(xy)
โค [ฯ1(x) ร ฯ1(y)]
= [(ฯ1 โช ฯ2)x ร (ฯ1 โช ฯ2)y] โ x, y โ V1 โ V2
= [(ฯ1โจฯ2)x ร (ฯ1โจฯ2)y]
i.e (ฮผ1โจฮผ
2)xy โค [(ฯ1โจฯ2)x ร (ฯ1โจฯ2)y] โ x, y โ V1 โ V2
And xy โ E1 โ E2 and x, y โ V1 โ V2 then
(ฮณ1โจฮณ
2)xy = ฮณ
1(xy)
โค max[ฯ1(x), ฯ1(y)]
= max[(ฯ1 โช ฯ2)x, (ฯ1 โช ฯ2)y] โ x, y โ V1 โ V2
= max[(ฯ1โจฯ2)x, (ฯ1โจฯ2)y]
i.e (ฮณ1โจฮณ
2)xy โค max[(ฯ1โจฯ2)x, (ฯ1โจฯ2)y] โ x, y โ V1 โ V2
Case-2 If xy โ E2 โ E1 and x, y โ V2 โ V1 then
On Complement Of Intuitionstic Product Fuzzy Graphs 5387
(ฮผ1โจฮผ
2)xy = ฮผ
2(xy)
โค [ฯ2(x) ร ฯ2(y)]
= [(ฯ1 โช ฯ2)x ร (ฯ1 โช ฯ2)y] โx, y โ V2 โ V1
= [(ฯ1โจฯ2)x ร (ฯ1โจฯ2)y]
i.e (ฮผ1โจฮผ
2)xy โค [(ฯ1โจฯ2)x ร (ฯ1โจฯ2)y] โ x, y โ V2 โ V1
And xy โ E2 โ E1 and x, y โ V2 โ V1 then
(ฮณ1โจฮณ
2)xy = ฮณ
2(xy)
โค max[ฯ2(x), ฯ2(y)]
= max[(ฯ1 โช ฯ2)x, (ฯ1 โช ฯ2)y] โ x, y โ
V2 โ V1
= max[(ฯ1โจฯ2)x, (ฯ1โจฯ2)y]
i.e (ฮณ1โจฮณ
2)xy โค max[(ฯ1โจฯ2)x, (ฯ1โจฯ2)y] โ x, y โ V2 โ V1
Case-3 If xy โ E1 โ E2 and x, y โ V1 โ V2 then
(ฮผ1โจฮผ
2)xy = ฮผ
1(uv)
โค [max(ฯ1(x),ฯ2(x)) ร max(ฯ1(y),ฯ2(y))]
= [(ฯ1 โช ฯ2)x ร (ฯ1 โช ฯ2)y]
= [(ฯ1โจฯ2)x ร (ฯ1โจฯ2)y]
i.e (ฮผ1โจฮผ
2)xy โค [(ฯ1โจฯ2)x ร (ฯ1โจฯ2)y] for xy โ E1 โ E2 and x, y โ V1 โ V2
And (ฮณ1โจฮณ
2)xy = ฮณ
1(xy)
โค [max(ฯ1(x), ฯ2(x))โmax(ฯ1(y), ฯ2(y))]
= max[(ฯ1 โช ฯ2)x, (ฯ1 โช ฯ2)y]
= max[(ฯ1โจฯ2)x, (ฯ1โจฯ2)y]
i.e (ฮณ1โจฮณ
2)xy โค max[(ฯ1โจฯ2)x, (ฯ1โจฯ2)y] for xy โ E1 โ E2 and x, y โ V1 โ V2
Similarly we can show that if xy โ E2 โ E1 and x, y โ V2 โ V1then
(ฮผ1โจฮผ
2)xy โค [(ฯ1โจฯ2)x ร (ฯ1โจฯ2)y] and
(ฮณ1โจฮณ
2)xy โค max[(ฯ1โจฯ2)x, (ฯ1โจฯ2)y]
This completes the proof of the proposition.
5388 Y. Vaishnaw and Sanjay Sharma
Example 2.9
Fig.2.10 Fig.2.11
Fig.2.12
Table 2.13 Membership table for fig. 2.10,2.11 and 2.12
Membership degree for vertices Membership degree for edges
For Graph
๐บ1: {(๐1, ฯ1), ๐1, ฮณ1)}
For Graph
G1: {(ฯ2, ฯ2), ฮผ2, ฮณ2)}
For Graph
G โถ G1โจG2
For
Graph
G1
For Graph
G2
For Graph
G โถ G1โจG2
(๐1, ฯ1)(x)=(0.4,0.3) (๐2, ฯ2)(u)=(0.7,0.2) (๐1, ฯ1)(x)=(0.4,0.3) (0.11,0.3) (0.39,0.2) (0.11,0.3)
(๐1, ฯ1)(y)=(0.3,0.1) (๐2, ฯ2)(v)=(0.6,0.1) (๐1, ฯ1)(y)=(0.3,0.1) (0.14,0.2) (0.27,0.3) (0.14,0.2)
(๐1, ฯ1)(z)=(0.6,0.2) (๐2, ฯ2)(z)=(0.5,0.3) (๐2, ฯ2)(u)=(0.7,0.2) (0.19,0.3) (0.29,0.3) (0.19,0.3)
โโโโโ โโโโโ (๐2, ฯ2)(v)=(0.6,0.1) โโ โ โโโ
โ
(0.39,0.2)
โโโโโ โโโโโ (๐, ฯ)(z)=(0.6,0.3) โโ โ โโโ (0.27,0.3)
โโโโโ โโโโโ โโโโโ โโ โ โโโ (0.29,0.3)
(๐1, ฯ1)(x)=(0.4,0.3)
(๐1, ฯ1)(y)=(0.3,0.1)
(๐1, ฯ1)(z)=(0.6,0.2)
(0.11,0.3)
(0.19,0.3)
(0.14,0.2) (๐2, ฯ2)(z)=(0.5,0.3)
(๐2, ฯ2)(v)=(0.6,0.1)
(๐2, ฯ2)(u)=(0.7,0.2)
(0.29,0.3
)
(0.27,0.3
)
(0.39,0.2)
(๐, ฯ)(x)=(0.4,0.3)
(๐, ฯ)(y)=(0.3,0.1)
(๐, ฯ)(z)=(0.6,0.3)
(0.11,0.3)
(0.19,0.3)
(0.14,0.2) (๐, ฯ)(v)=(0.6,0.1)
(๐, ฯ)(u)=(0.7,0.2)
(0.29,0.3
)
(0.27,0.3
)
(0.39,0.2)
On Complement Of Intuitionstic Product Fuzzy Graphs 5389
Theorem: 2.14 Let G1: {(ฯ1, ฯ1),ฮผ1, ฮณ1)} and G1: {(ฯ2, ฯ2), ฮผ2, ฮณ2)} with 0 โค
{ฯ(xi) + ฯ(xi)} โค 1 โxi โ V and V1 โฉ V2 = โ be two (k1, k2) regular intuitionstic
product fuzzy graph of crisp graph G1: (V1, E2) and G2: (V2, E2) res. Then G =
G1โจG2: {(ฯ1โจฯ2, ฯ1โจฯ2), (ฮผ1โจฮผ2, ฮณ1โจฮณ
2)} is also a (k1, k2) regular intuitionstic
product fuzzy graph.
Proof: Given that G1: {(ฯ1, ฯ1),ฮผ1, ฮณ1)} and G1: {(ฯ2, ฯ2), ฮผ2, ฮณ2)} with 0 โค
{ฯ(xi) + ฯ(xi)} โค 1 โxi โ V and V1 โฉ V2 = โ be two (k1, k2) regular intuitionstic
product fuzzy graph.Then
๐๐๐(๐ฑ) = (k1, k2) โx โ V1 and
๐๐๐(๐ฒ) = (k1, k2) โy โ V2
Now given that V1 โฉ V2 = โ then
d๐๐โจ๐๐(๐ฑ) = ๐๐๐โ๐๐(๐ฑ) + โ {(ฯ
1(x) ร ฯ2(y)xyโEโฒ ), (ฯ1(x) โจ ฯ2(y)}
= ๐๐๐โ๐๐(๐ฑ) + ๐
(by Definition of ring sum of intuitionstic product fuzzy graph)
= dG1(x) โx โ V1
dG1(x) = (k1, k2)
i.e. dG1โG2(x) = (k1, k2) โx โ V1 โฆโฆโฆโฆโฆโฆโฆโฆ(1)
And dG1โG2(y) = dG1โG2
(y) + โ {(ฯ1(x) ร ฯ2(y)xyโEโฒ ), (ฯ1(x) โจ ฯ2(y)}
= dG1โG2(y) + 0
(by Definition of ring sum of intuitionstic product fuzzy graph)
= dG2(y) โy โ V2
dG2(y) = (k1, k2)
i.e. dG1โG2(y) = (k1, k2) โy โ V2 โฆโฆโฆโฆโฆโฆโฆโฆ(2)
Hence by equation (1) and (2) we have
dG1โG2(x) = dG1โG2
(y)
So G = G1โจG2: {(ฯ1โจฯ2, ฯ1โจฯ2), (ฮผ1โจฮผ2, ฮณ1โจฮณ
2)} is a (k1, k2) regular
intuitionstic product fuzzy graph.
Theorem:2.15 Let G1: {(ฯ1, ฯ1),ฮผ1, ฮณ1)} and G1: {(ฯ2, ฯ2), ฮผ2, ฮณ2)} with 0 โค
{ฯ(xi) + ฯ(xi)} โค 1 โxi โ V and V1 โฉ V2 = โ and G =
5390 Y. Vaishnaw and Sanjay Sharma
G1โจG2: {(ฯ1โจฯ2, ฯ1โจฯ2), (ฮผ1โจฮผ2, ฮณ1โจฮณ
2)} is a (k1, k2) regular intuitionstic
product fuzzy graph then G1: {(ฯ1, ฯ1),ฮผ1, ฮณ1)} and
G1: {(ฯ2, ฯ2),ฮผ2, ฮณ2)} are also (k1, k2) regular intuitionstic product fuzzy graph.
Proof: Given hat G = G1โจG2: {(ฯ1โจฯ2, ฯ1โจฯ2), (ฮผ1โจฮผ2, ฮณ1โจฮณ
2)} is a (k1, k2)
regular intuitionstic product fuzzy graph with V1 โฉ V2 = โ then we have to show that
G1: {(ฯ1, ฯ1), ฮผ1, ฮณ1)} and G1: {(ฯ2, ฯ2), ฮผ2, ฮณ2)} are also (k1, k2) regular intuitionstic
product fuzzy graph.
Now given that V1 โฉ V2 = โ then
๐๐๐โ๐๐(๐ฑ) = ๐๐๐(๐ฑ) + โ {(ฯ
1(x) ร ฯ2(y)xyโEโฒ ), (ฯ1(x) โจ ฯ2(y)}
= ๐๐๐(๐ฑ) + ๐
(by Definition of ring sum of intuitionstic product fuzzy graph)
(k1, k2) = ๐๐๐(๐ฑ) โx โ V1
Since G1โจG2 is (k1, k2) regular intuitionstic product fuzzy graph
i.e. dG1(x) = (k1, k2) โx โ V1โฆโฆโฆโฆโฆโฆโฆ(1)
And dG1โG2(y) = dG2(y) + โ {(ฯ
1(x) ร ฯ2(y)xyโEโฒ ), (ฯ1(x) โจ ฯ2(y)}
(by Definition of ring sum of intuitionstic product fuzzy graph)
dG1โG2(y) = dG2(y) + 0
dG2(y) = (k1, k2) โy โ V2
Since G1โจG2 is (k1, k2) regular intuitionistic product fuzzy graph
i.e.
dG2(y) = (k1, k2) โy โ V2 โฆโฆโฆโฆโฆโฆโฆโฆ(2)
Hence; by equation (1) and (2)
G1: {(ฯ1, ฯ1), ฮผ1, ฮณ1)} and G2: {(ฯ2, ฯ2), ฮผ2, ฮณ2)} are also (k1, k2) regular intuitionstic
product fuzzy graph.
CONCLUSION
The notion of product fuzzy graph has generalized for intuitionistic product fuzzy
graph and definition of complement and ring sum of two intuitionistic product fuzzy
graph have provided with example. Further we have proved that, If two product fuzzy
graph are (k1, k2) regular intuitionistic product fuzzy graph then ring sum of regular
On Complement Of Intuitionstic Product Fuzzy Graphs 5391
intuitionistic product fuzzy graph is also a (k1, k2) regular intuitionstic product fuzzy
graph and conversely.
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