In this paper, we introduce the notion of Cayley intuitionistic fuzzy graphs and investigate some of their properties. We present some interesting properties of intuitionistic fuzzy graphs in terms of algebraic structures. We discuss connectedness in Cayley intuitionistic fuzzy graphs. We also describe different types of
α
connectedness in Cayley intuitionistic fuzzy graphs.
AMS Mathematics Subject Classification : 05C99, 05C25.
1. Introduction
Graph theory has numerous applications to problems in different areas including computer science, electrical engineering, system analysis, operations research, economics, networking routing, transportation, and optimization. Pointtopoint interconnection networks for parallel and distributed systems are usually modeled by
directed graphs
(or digraphs). A digraph is a graph whose edges have direction and are called
arcs
(edges). Arrows on the arcs are used to encode the directional information: an arc from vertex (node)
x
to vertex
y
indicates that one may move from
x
to
y
but not from
y
to
x
. The Cayley graph was first considered for finite groups by Cayley in 1878. Max Dehn in his unpublished lectures on group theory from 190910 reintroduced Cayley graphs under the name Gruppenbild (group diagram), which led to the geometric group theory of today. His most important application was the solution of the word problem for the fundamental group of surfaces with genus, which is equivalent to the topological problem of deciding which closed curves on the surface contract to a point
[6
,
7]
.
Fuzzy set is a newly emerging mathematical frame work to exemplify the phenomenon of uncertainty in real life tribulations. It was introduced by Zadeh in 1965, and the concepts were pioneered by various independent researches. Kaufmann’s initial definition of a fuzzy graph
[13]
was based on Zadeh’s fuzzy relations
[24]
. Rosenfeld
[17]
introduced the fuzzy analogue of several basic graphtheoretic concepts. Mordeson and Peng
[16]
defined the concept of complement of fuzzy graph and studied some operations on fuzzy graphs. Atanassov
[8]
introduced the concept of intuitionistic fuzzy relations and intuitionistic fuzzy graphs and further were studied in
[4
,
12
,
19
,
20]
. In 1983, Atanassov
[9]
introduced the concept of intuitionistic fuzzy sets as a generalization of fuzzy sets
[23]
. Atanassov added a new component (which determines the degree of nonmembership) in the definition of fuzzy set. The fuzzy sets give the degree of membership of an element in a given set (and the nonmembership degree equals one minus the degree of membership), while intuitionistic fuzzy sets give both a degree of membership and a degree of nonmembership which are moreorless independent from each other, the only requirement is that the sum of these two degrees is not greater than 1. Akram
et al
.
[1

3]
introduced many new concepts, including intuitionistic fuzzy hypergraphs, strong intuitionistic fuzzy graphs, intuitionistic fuzzy cycles and intuitionistic fuzzy trees. Wu
[22]
discussed fuzzy digraphs. Shahzamanian
et al
.
[21]
introduced the notion of roughness in Cayley graphs and investigated several properties. Namboothiri
et al
.
[18]
studied Cayley fuzzy graphs. In this paper, we introduce the notion of Cayley intuitionistic fuzzy graphs and investigate some of their properties. We present some interesting properties of intuitionistic fuzzy graphs in terms of algebraic structures. We discuss connectedness in Cayley intuitionistic fuzzy graphs. We also describe different types of
α
 connectedness in Cayley intuitionistic fuzzy graphs.
We have used standard definitions and terminologies in this paper. For other notations, terminologies and applications not mentioned in the paper, the readers are referred to
[5

7
,
11
,
21]
.
2. Preliminaries
In this section, we review some elementary concepts whose understanding is necessary fully benefit from this paper.
A digraph is a pair
G
^{∗}
= (
V
,
E
), where
V
is a finite set and
E
⊆
V
×
V
. In this paper, we will write
xy
∈
E
to mean
x
→
y
∈
E
, and if
e
=
xy
∈
E
, we say
x
and
y
are
adjacent
such that
x
is a starting node and
y
is an ending node.
The study of vertextransitive graphs has a long and rich history in discrete mathematics. Prominent examples of vertextransitive graphs are Cayley graphs which are important in both theory as well as applications, e.g., Cayley graphs are good models for interconnection networks.
Definition 2.1.
Let
G
be a finite group and let
S
be a minimal generating set of
G
. A Cayley graph (
G
,
S
) has elements of
G
as its vertices, the edgeset is given by {(
g
,
gs
) :
g
∈
G
,
s
∈
S
}. Two vertices
g
_{1}
and
g
_{2}
are adjacent if
g
_{2}
=
g
_{1}
.
s
, where
s
∈
S
. Note that a generating set
S
is minimal if
S
generates
G
but no proper subset of
S
does.
Theorem 2.2.
All Cayley graphs are vertex transitive
.
Definition 2.3.
Let (
V
, ∗) be a group and
A
be any subset of
V
. Then the Cayley graph induced by (
V
, ∗,
A
) is the graph
G
= (
V
,
R
), where
R
= {(
x
,
y
) :
x
^{−1}
y
∈
A
}.
Definition 2.4
(
[23
,
24]
). A
fuzzy subset μ
on a set
X
is a map
μ
:
X
→ [0, 1]. A
fuzzy binary relation
on
X
is a fuzzy subset
μ
on
X
×
X
.
Definition 2.5
(
[22]
). Let
V
be a finite set,
A
=<
V
,
μ_{A}
> a fuzzy set of
V
and
B
=<
V
×
V
,
μ_{B}
> a fuzzy relation on
V
, then the ordered pair (
A
,
B
) is called a
fuzzy digraph
.
Definition 2.6
(
[18]
). Let (
V
, ∗) be a group and let
μ
be a fuzzy subset of
V
. Then the fuzzy relation
R
on
V
×
V
defined by
induces a fuzzy graph
G
= (
V
,
R
), called the
Cayley fuzzy graph
induced by the (
V
, ∗,
μ
).
Definition 2.7
(
[8]
). An intuitionistic fuzzy set (IFS, for short) on a universe
X
is an object of the form
where
μ_{A}
(x) ∈ [0, 1] is called degree of membership of
x
in
A
and
ν_{A}
(
x
) ∈ [0, 1] is called degree of nonmembership of
x
in
A
, and
μ_{A}
,
ν_{A}
satisfies the following condition for all
x
∈
X
,
μ_{A}
(
x
) +
ν_{A}
(
x
) ≤ 1.
Definition 2.8.
Let
X
be intuitionistic fuzzy set. For any subset A and for
α
∈ [0, 1], {
x

μ_{A}
(
x
) ≥
α
,
ν_{A}
(
x
) ≤
α
} is called
α
 cut of
A
. It is denoted by
A_{α}
.
Definition 2.9.
Let
X
be intuitionistic fuzzy set. For any subset
A
and for
α
∈ [0, 1], {
x

μ_{A}
(
x
) >
α
,
ν_{A}
(
x
) <
α
} is called strong
α
 cut of
A
. It is denoted by
Definition 2.10.
Let
X
be intuitionistic fuzzy set. For any subset
A
of
X
, the support of
A
is the set{
x
∈
X

μ_{A}
(
x
) ≥ 0,
ν_{A}
(
x
) > 0}. It is denoted by
supp
(
A
). It can also be denoted as
Definition 2.11.
An intuitionistic fuzzy relation
R
= (
μ_{R}
(
x
,
y
),
ν_{R}
(
x
,
y
)) in a universe
X
×
X
(
R
(
X
→
X
), for short) is an intuitionistic fuzzy set of the form
where
μ_{R}
:
X
×
X
→ [0, 1] and
ν_{R}
:
X
×
X
→ [0, 1]. The intuitionistic fuzzy relation
R
satisfies
μ_{R}
(
x
,
y
) +
ν_{R}
(
x
,
y
) ≤ 1 for all
x
,
y
∈
X
.
Definition 2.12.
Let
R
be an intuitionistic fuzzy relation on universe
X
. Then
R
is called
an intuitionistic fuzzy equivalence relation
on
X
if it satisfies the following conditions:

(a)Ris intuitionistic fuzzy reflexive, i.e.,R(x,x) = (1, 0) for eachx∈X,

(b)Ris intuitionistic fuzzy symmetric, i.e.,R(x,y) =R(y,x) for anyx,y∈X,

(c)Ris intuitionistic fuzzy transitive, i.e.,
Definition 2.13.
Let
R
be an intuitionistic fuzzy relation on universe
X
. Then
R
is called
an intuitionistic fuzzy partial order relation
on
X
if it satisfies the following conditions:

(a)Ris intuitionistic fuzzy reflexive, i.e.,R(x,x) = (1, 0), for eachx∈X,

(b)Ris intuitionistic fuzzy antisymmetric, i.e., for allx,y∈X R(x,y) ≠R(y,x),

(c)Ris intuitionistic fuzzy transitive, i.e.,
Definition 2.14.
Let
R
be an intuitionistic fuzzy relation on universe
X
. Then
R
is called
an intuitionistic fuzzy linear order relation
on
X
if it satisfies the following conditions:

(a)Ris intuitionistic fuzzy partial relation,

(b) (R∨R−1)(x,y) > 0 for allx,y∈X.
3. Cayley Intuitionistic Fuzzy Graphs
In this section, we introduce Cayley intuitionistic fuzzy graphs and prove that all Cayley intuitionistic fuzzy graphs are regular.
Definition 3.1.
An
intuitionistic fuzzy digraph
of a digraph
G
^{∗}
is a pair
G
= (
A
,
B
), where
A
=<
V
,
μ_{A}
,
ν_{A}
> is an intuitionistic fuzzy set in
V
and
B
=<
V
×
V
,
μ_{B}
,
ν_{B}
> is an intuitionistic fuzzy relation on
V
such that
0 ≤
μ_{B}
(
xy
)+
ν_{B}
(
xy
) ≤ 1 for all
x
,
y
∈
V
. We note that
B
may not be symmetric relation.
Definition 3.2.
Let
G
be an intuitionistic fuzzy digraph. The indegree of a vertex
x
in
G
is defined by
ind
(
x
) = (
ind_{μ}
(
x
),
ind_{ν}
(
x
)), where
Similarly, the outdegree of a vertex
x
in
G
is defined by
outd
(
x
) = (
outd_{μ}
(
x
),
outd_{ν}
(
x
)), where
An intuitionistic fuzzy digraph in which each vertex has same outdegree
r
is called
an outregular digraph
with index of outregularity
r
. Inregular digraphs are defined similarly.
Example 3.3.
Consider an intuitionistic fuzzy digraph
G
of
G
^{∗}
= (
V
,
E
), where
V
= {
v
_{1}
,
v
_{2}
,
v
_{3}
,
v
_{4}
}, E = {
v
_{1}
v
_{2}
,
v
_{2}
v
_{3}
,
v
_{3}
v
_{4}
,
v
_{4}
v
_{1}
}. By routine computations, it is easy to see from
Fig. 1
that the intuitionistic fuzzy digraph is nether outregular digraph nor inregular digraph.
Intuitionistic fuzzy digraph
Definition 3.4.
Let (
G
, ∗) be a group and let
S
be a nonempty finite subset of
G
. Then the Cayley intuitionistic fuzzy graph
G
= (
V
,
R
) is an intuitionistic fuzzy graph with the vertex set
V
=
G
and let
A
= (
μ_{A}
,
ν_{A}
) be an intuitionistic fuzzy subset of
V
. The intuitionistic fuzzy relation
R
(
x
,
y
) on
V
is defined by
R
(
x
,
y
) = {(
μ_{A}
(
x
^{−1}
y
),
μ_{A}
(
x
^{−1}
y
))
x
,
y
∈
G
and
x
^{−1}
y
∈
S
}.
Example 3.5.
Consider the group
Z
_{3}
and take
V
=
Z
_{3}
= {0, 1, 2}. Define
μ_{A}
:
V
→ [0, 1] and
ν_{A}
:
V
→ [0, 1] by
μ_{A}
(0) =
μ_{A}
(1) =
μ_{A}
(2) = 0.5,
ν_{A}
(0) =
ν_{A}
(1) =
ν_{A}
(2) = 0.4. Then the Cayley intuitionistic fuzzy graph
G
= (
V
,
R
) induced by (
Z
_{3}
, +,
A
) is given by the following
Table 1
and
Figure 2
.
R(a,b) for Cayley intuitionistic fuzzy graph
R(a, b) for Cayley intuitionistic fuzzy graph
Cayley Intuitionistic fuzzy graph
By routine computations, it is easy to see from
Fig. 2
that
G
= (
V
,
R
) is a Cayley intuitionistic fuzzy graph, and it is regular.
We notice that Cayley intuitionistic fuzzy graphs are actually intuitionistic fuzzy digraphs. Furthermore, the relation
R
in the above definition describes the strength of each directed edge. Let
G
denote an intuitionistic fuzzy graph
G
= (
V
,
R
) induced by the triple (
V
, ∗,
A
).
Definition 3.6
(
[15]
). Let (
S
, ∗) be a semigroup. Let
A
′ = (
μ
′
_{A}
,
ν
′
_{A}
) be an intuitionistic fuzzy subset of
S
. Then
A
′ is said to be
an intuitionistic fuzzy subsemigroup
of
S
if for all
x
,
y
∈
S
,
μ_{A′}
(
xy
) ≥ min(
μ_{A′}
(
x
),
μ_{A′}
(
y
)) and
ν_{A′}
(
xy
) ≤ max(
μ_{A′}
(
x
),
ν_{A′}
(
y
)).
Theorem 3.7.
The Cayley intuitionistic fuzzy graph G is vertex transitive.
Proof
. Let
a
,
b
∈
V
. Define
ψ
:
V
→
V
by
ψ
(
x
) =
ba
^{−1}
x
∀
x
∈
V
. Clearly,
ψ
is a bijective map. For each
x
,
y
∈
V
,
Therefore,
R
(
ψ
(
x
),
ψ
(
y
)) =
R
(
x
,
y
). Hence
ψ
is an automorphism on
G
. Also
ψ
(
a
) =
b
. Hence
G
is vertex transitive.
Theorem 3.8.
Every vertex transitive intuitionistic fuzzy graph is regular.
Proof
. Let
G
= (
V
,
R
) be any vertex transitive intuitionistic fuzzy graph. Let
u
,
v
∈
V
. Then there is an automorphism
f
on
G
such that
f
(
u
) =
v
. Note that
Hence,
G
is regular.
From Theorem 3.6 and Theorem 3.7, we have.
Theorem 3.9.
Cayley intuitionistic fuzzy graphs are inregular and outregular.
Theorem 3.10.
If for all u
,
v
∈
V
,
Then Cayley intuitionistic fuzzy graphs are regular.
Theorem 3.11.
Let G
= (
V
,
R
)
be an intuitionistic fuzzy graph
.
Then intuitionistic fuzzy relation R is reflexive if and only if R
(1, 1) = (1, 0),
that is
,
μ_{A}
(1) = 1
and
ν_{A}
(1) = 0.
Proof
.
R
is reflexive if and only if
R
(
x
,
x
) = (1, 0) for all
x
∈
V
. Now
Hence
R
is reflexive if and only if
μ_{A}
(1) = 1 and
ν_{A}
(0) = 0.
Theorem 3.12.
Let G
= (
V
,
R
)
be an intuitionistic fuzzy graph
.
Then intuition istic fuzzy relation R is symmetric if and only if
(
μ_{A}
(x),
ν_{A}
(x)) = (
μ_{A}
(
x
^{−1}
),
ν_{A}
(
x
^{−1}
))
for all x
∈
V
.
Proof
. Suppose that
R
is symmetric. Then for any
x
∈
V
,
Conversely, suppose that (
μ_{A}
(
x
),
ν_{A}
(
x
)) = (
μ_{A}
(
x
^{−1}
),
ν_{A}
(
x
^{−1}
)) for all
x
∈
V
. Then for all
x
,
y
∈
V
,
Hence
R
is symmetric.
Theorem 3.13.
An intuitionistic fuzzy relation R is anti symmetric if and only if
{
x
: (
μ_{A}
(
x
),
ν_{A}
(
x
)) = (
μ_{A}
(
x
^{−1}
),
ν_{A}
(
x
^{−1}
))} = {(1, 1)}.
Proof
. Suppose that
R
is anti symmetric and let
x
∈
V
. Then
Conversely, suppose that {
x
: (
μ_{A}
(x),
ν_{A}
(x)) = (
μ_{A}
(
x
^{−1}
),
ν_{A}
(
x
^{−1}
))} = {(1, 0)}. Then for any
x
,
y
∈
V
,
R
(
x
,
y
) =
R
(
y
,
x
) ⇔ (
μ_{A}
(
x
^{−1}
y
),
ν_{A}
(
y
^{−1}
x
)). This implies that (
μ_{A}
(
x
^{−1}
y
),
ν_{A}
(
y
^{−1}
x
)) = (
μ_{A}
((
x
^{−1}
y
)
^{−1}
),
ν_{A}
((
x
^{−1}
y
)
^{−1}
)). That is
x
^{−1}
y
= 1. Equivalently,
x
=
y
. Hence
R
is anti symmetric.
Theorem 3.14.
An intuitionistic fuzzy relation R is transitive if and only if
(
μ_{A}
,
ν_{A}
)
is an intuitionistic fuzzy subsemigroup of
(
G
, ∗).
Proof
. Suppose that
R
is transitive and let
x
,
y
∈
V
. Then
R
^{2}
≤
R
. Now for any
x
∈
V
, we have
R
(1,
x
) = (
μ_{A}
(
x
),
ν_{A}
(
x
)). This implies that ∨{
R
(1,
z
) ∧
R
(
z
,
xy
)
z
∈
V
} =
R
^{2}
(1,
xy
) ≤
R
(1,
xy
). That is ∨{
μ_{R}
(
z
) ∧
μ_{R}
(
z
^{−1}
xy
)
z
∈
V
} ≤
μ_{R}
(
xy
) and ∧{
ν_{R}
(
z
) ∨
ν_{R}
(
z
^{−1}
xy
)
z
∈
V
} ≥
ν_{R}
(
xy
). Hence
μ_{A}
(
xy
) ≥
μ_{A}
(
x
) ∧
μ_{A}
(
y
) and
ν_{A}
(
xy
) ≤
μ_{A}
(
x
) ∨
ν_{A}
(
y
). Hence (
μ_{A}
,
ν_{A}
) is an intuitionistic fuzzy subsemigroup of (
S
, ∗).
Conversely, suppose that
A
= (
μ_{A}
,
ν_{A}
) is an intuitionistic fuzzy subsemigroup of (
G
, ∗). That is, for all
x
,
y
∈
V
μ_{A}
(
xy
) ≥
μ_{A}
(
x
) ∧
μ_{A}
(
y
) and
ν_{A}
(
xy
) ≤
ν_{A}
(
x
) ∨
ν_{A}
(
y
). Then for any
x
,
y
∈
V
,
Hence
Hence
R
is transitive.
We conclude that:
Theorem 3.15.
An intuitionistic fuzzy relation R is a partial order if and only if A
= (
μ_{B}
,
ν_{B}
)
is an intuitionistic fuzzy subsemigroup of
(
V
, ∗)
satisfying

(i)μA(1) = 1and νA(1) = 0,

(ii) {x: (μA(x),νA(x)) = (μA(x−1),νA(x−1))} = {(1, 0)}.
Theorem 3.16.
An intuitionistic fuzzy relation R is a linear order if and only if
(
μ_{B}
,
ν_{B}
)
is an intuitionistic fuzzy subsemigroup of
(
V
, ∗)
satisfying

(i)μA(1) = 1 andνA(1) = 0,

(ii) {x(μA(x),νA(x)) = (μA(x−1),νA(x−1))} = {(1, 0)}},

(ii)R2≤R,that is, {x,yμR(x,y) ≥μR◦R(x,y),νR(x,y) ≤νR◦R(x,y)‘x,y∈V},

(iv) {xμA(x) ∨μA(x−1) > 0,νA(x) ∧νA(x−1) > 0}.
Proof
. Suppose
R
is a linear order. Then by Theorem 3.15, the conditions (i),(ii) and (iii) are satisfied. For any
x
∈
V
, (
R
∨
R
^{−1}
)(1,
x
) > 0. This implies that
R
(1,
x
) ∨
R
(
x
, 1) > 0. Hence {
x
:
μ_{A}
(
x
) ∨
μ_{A}
(
x
^{−1}
) > 0,
ν_{A}
(
x
) ∧
ν_{A}
(
x
^{−1}
) > 0}. Conversely, suppose that the conditions (i), (ii) and (iii) hold. Then by Theorem 3.15,
R
is partial order. Now for any
x
,
y
∈
V
, we have (
x
^{−1}
y
), (
y
^{−1}
x
) ∈
V
. Then by condition (iv), {
x
:
μ_{A}
(
x
) ∨
μ_{A}
(
x
^{−1}
) > 0,
ν_{A}
(
x
) ∧
ν_{A}
(
x
^{−1}
) > 0}. That is
R
(1,
x
) ∨
R
(
x
, 1) > 0. Hence (
R
∨
R
^{−1}
)(
x
,
y
) > 0. Therefore
R
is linear order.
Theorem 3.17.
An intuitionistic fuzzy relation R is a equivalence relation if and only if
(
μ_{A}
,
ν_{A}
)
is an intuitionistic fuzzy sub semigroup of
(
G
, ∗)
satisfying

(i)μA(1) = 1and νA(1) = 0,

(ii) (μA(x),νA(x)) = (μA(x−1),νA(x−1))for all x∈V.
Proof
. Proof follows from Theorem 3.15 and Theorem 3.16.
4. Cayley graphs induced by Cayley intuitionistic fuzzy graphs
Definition 4.1.
Let (
V
, ∗) be a group, let
A
be an intuitionistic fuzzy set of
V
and
G
= (
V
,
R
) be the Cayley intuitionistic fuzzy graph induced by (
V
, ∗,
A
). For any
α
∈ [0, 1], let
A_{α}
be
α
− cut of
be the strong
α
− cut of A. We define
Then it is clear that for any
α
∈ [0, 1], the Cayley intuitionistic fuzzy graph induced by (
V
, ∗,
A
) induces the Cayley graphs
Note that for any
Thus for any
α
∈ [0, 1], the Cayley intuitionistic fuzzy graph (
V
,
R
) induced by Cayley intuitionistic graphs
Remark 4.1.
Let
G
= (
V
,
R
) be any intuitionistic fuzzy graph, then
G
is connected(weakly connected, semiconnected, locally connected or quasi connected) if and only if the induce fuzzy graph
is connected(weakly connected, semiconnected, locally connected or quasi connected).
We now observe the following definition and lemma to study different types of connectedness of
G
.
Definition 4.2.
Let (
L
, ∗) be a semigroup and let
A
= (
μ_{A}
,
ν_{A}
) be an intuitionistic fuzzy subset of L. Then the subsemigroup generated by
A
is the meet of all intuitionistic fuzzy subsemigroups of
L
which contains
A
. It is denoted by <
A
>.
Example 4.3.
Consider
L
=
Z
_{3}
and
A
= (
μ_{A}
,
ν_{A}
) as in Example 3.5. Then <
A
> is given by <
μ_{A}
> (0) = 1, <
ν_{A}
> (0) = 0, and <
μ_{A}
> (
y
) = 0.5, <
ν_{A}
> (
y
) = 0.5 for
y
= 1, 2.
Lemma 4.4.
Let
(
L
, ∗)
be a semigroup and A
= (
μ_{A}
,
ν_{A}
)
be an intuitionistic fuzzy subset of L
.
Then intuitionistic fuzzy subset
<
A
>
is precisely given by
<
μ_{A}
>(
x
) = ∨{
μ_{A}
(
x
_{1}
)∧
μ_{A}
(
x
_{2}
)∧ . . . ∧
μ_{A}
(
x_{n}
) :
x
=
x
_{1}
x
_{2}
. . .
x_{n}
with
μ_{A}
(
x_{i}
) > 0
for i
= 1, 2, . . . ,
n
}, <
ν_{A}
>(
x
) = ∧{
ν_{A}
(
x
_{1}
) ∨
ν_{A}
(
x
_{2}
) ∨ . . . ∨
ν_{A}
(
x_{n}
) :
x
=
x
_{1}
x
_{2}
. . .
x_{n}
with
ν_{A}
(
xi
) > 0
for i
= 1, 2, . . . ,
n
}
for any x
∈
L
.
Proof
. Let
A
′ = (
μ
′
_{A}
,
ν
′
_{A}
) be an intuitionistic fuzzy subset of
L
defined by
μ
′
_{A}
(
x
) = ∨{
μ_{A}
(
x
_{1}
) ∧
μ_{A}
(
x
_{2}
) ∧ . . . ∧
μ_{A}
(
x_{n}
) :
x
=
x
_{1}
x
_{2}
. . .
x_{n} with μ_{A}
(
x_{i}
) > 0
for i
= 1, 2, . . . ,
n
},
ν
′
_{A}
(
x
) = ∧{
ν_{A}
(
x
_{1}
) ∨
ν_{A}
(
x
_{2}
) ∨ . . . ∨
ν_{A}
(
x_{n}
) :
x
=
x
_{1}
x
_{2}
. . .
x_{n}
with
ν_{A}
(
x_{i}
) > 0
for i
= 1, 2, . . . ,
n
}, for any
x
∈
L
. Let
x
,
y
∈
L
. If
μ_{A}
(
x
) = 0 or
μ_{A}
(
y
) = 0, then
μ_{A}
(
x
)∧
μ_{A}
(
y
) = 0 and
ν_{A}
(
x
) = 0 or
ν_{A}
(
y
) = 0, then
ν_{A}
(
x
) ∨
ν_{A}
(
y
) = 0. Therefore,
μ
′
_{A}
(
xy
) ≥
μ_{A}
(
x
) ∧
μ_{A}
(
y
) and
ν
′
_{A}
(
xy
) ≤
ν_{A}
(
x
) ∨
ν_{A}
(
y
). Again , if
μ_{A}
(
x
) ≠ 0,
ν_{A}
(x) ≠ 0, then by definition of
μ
′
_{A}
(
x
) and
ν
′
_{A}
(
x
), we have
μ
′
_{A}
(
xy
) ≥
μ_{A}
(
x
) ∧
μ_{A}
(
y
) and
ν
′
_{A}
(
xy
) ≤
ν_{A}
(
x
) ∨
ν_{A}
(
y
). Hence (
μ
′
_{A}
,
ν
′
_{A}
) is an intuitionistic fuzzy subsemigroup of
L
containing (
μ_{A}
,
ν_{A}
). Now let
L
′ be any intuitionistic fuzzy subsemigroup of
L
′ containing (
μ_{A}
,
ν_{A}
). Then for any
x
∈
L
with
x
=
x
_{1}
x
_{2}
. . .
x_{n} with μ_{A}
(
x_{i}
) > 0,
ν_{A}
(
x_{i}
) > 0, for
i
= 1, 2, . . . ,
n
, we have
μ
_{L′}
(
x_{i}
) ≥
μ
_{L′}
(
x
_{1}
) ∧
μ
_{L′}
(
x
_{2}
) ∧ . . . ∧
μ
_{L′}
(
x_{n}
) ≥
μ_{A}
(
x
_{1}
) ∧
μ_{A}
(
x
_{2}
) ∧ . . . ∧
μ_{A}
(
x_{n}
) and
ν
_{L′}
(
x_{i}
) ≤
ν
_{L′}
(
x
_{1}
) ∧
ν
_{L′}
(
x
_{2}
)∧ . . . ∧
ν
_{L′}
(
x_{n}
) ≤
ν_{A}
(
x
_{1}
) ∧
ν_{A}
(
x
_{2}
) ∧ . . . ∧
ν_{A}
(
x_{n}
). Thus
μ
_{L′}
(
x
) ≥ ∨{
μ_{A}
(
x
_{1}
) ∧
μ_{A}
(
x
_{2}
)∧ . . . ∧
μ_{A}
(
x_{n}
) :
x
=
x
_{1}
x
_{2}
. . .
x_{n} with μ_{A}
(
x_{i}
) > 0
for i
= 1, 2, . . . ,
n
},
ν
_{L′}
(
x
) ≤ ∧{
ν_{A}
(
x
_{1}
) ∨
ν_{A}
(
x
_{2}
) ∨ . . . ∨
ν_{A}
(
x_{n}
) :
x
=
x
_{1}
x
_{2}
. . .
x_{n} with ν_{A}
(
x_{i}
) > 0
for i
= 1, 2, . . . ,
n
}, for any
x
∈
L
. Hence
μ
_{L′}
(
x
) ≥
μ
′
_{A}
(
x
),
ν
_{L′}
(
x
) ≤
ν
′
_{A}
(
x
), for all
x
∈
L
. Thus
μ
′
_{A}
(
x
) ≤
μ
_{L′}
(
x
),
ν
′
_{A}
(
x
) ≥
μ_{A}
(
x
). Thus
A
′ = (
μ
′
_{A}
,
ν
′
_{A}
) is the meet of all intuitionistic fuzzy subsemigroup containing (
μ_{A}
,
ν_{A}
).
Theorem 4.5.
Let
(
L
, ∗)
be a semigroup and A
= (
μ_{A}
,
ν_{A}
)
be an intuitionistic fuzzy subset of L
.
Then for any
where
(<
μ_{α}
>,<
ν_{α}
>)
denotes the subsemigroup generated by
(
μ_{α}
,
ν_{α}
)
and
< (
μ
,
ν
) >
denotes intuitionistic fuzzy subsemigroup generated by
(
μ
,
ν
).
Proof
.
Therefore, (<
μ_{α}
>,<
ν_{α}
>) = (<
μ
>
_{α}
,<
ν
>
_{α}
). Similarly, we have
Remark 4.2.
Let (
L
, ∗) be a semigroup and
A
= (
μ_{A}
,
ν_{A}
) be an intuitionistic fuzzy subset of
L
. Then by Theorem 4.5, we have <
supp
(
A
) >=
supp
<
A
>.
5. Connectedness in Cayley intuitionistic fuzzy graphs
In this section, first we state the the basic Theorems which are used to prove the forthcoming Theorems.
Let
G
denotes the Cayley intuitionistic fuzzy graphs
G
= (
V
,
R
) induced by (
V
, ∗,
A
) and
G
′ = (
V
′,
R
′) be the crisp Cayley graph induced by (
V
′, ∗,
A
). Then we conclude the following results.
Theorem 5.1.
Let A be any subset of V
′
and G
′ = (
V
′,
R
′)
be the Cayley graph induced by
(
V
′, ∗,
A
).
Then G
′
is connected if and only if
<
A
> ⊇
V
−
v
_{1}
.
Theorem 5.2.
Let A be any subset of a set V
′
and G
′ = (
V
′,
R
′)
be the Cayley graph induced by the triplet
(
V
′, ∗,
A
).
Then G
′
is weakly connected if and only if
<
A
∪
A
^{−1}
>⊇
V
−
v
_{1}
,
where A
^{−1}
= {
x
^{−1}
:
x
∈
A
}.
Theorem 5.3.
Let A be any subset of a set V
′
and G
′ = (
V
′,
R
′)
be the Cayley graph induced by the triplet
(
V
′, ∗,
A
).
Then G
′
is semiconnected if and only if
<
A
> ∪ <
A
^{−1}
>⊇
V
−
v
_{1}
,
where A
^{−1}
= {
x
^{−1}
:
x
∈
A
}.
Theorem 5.4.
Let G
′ = (
V
′,
R
′)
be the Cayley graph induced by the triplet
(
V
′, ∗,
A
).
Then G
′
is locally connected if and only if
<
A
>=<
A
^{−1}
>,
where A
^{−1}
= {
x
^{−1}
:
x
∈
A
}.
Theorem 5.5.
Let G
′ = (
V
′,
R
′)
be the Cayley graph induced by the triplet
(
V
′, ∗,
A
),
where V
′
is finite
.
Then G
′
is quasi connected if and only if it is connected
.
Definition 5.6.
Let (
L
, ∗) be a group and
A
be an intuitionistic fuzzy subset of
L
. Then we define
A
^{−1}
as intuitionistic fuzzy subset of
L
given by
A
^{−1}
(
x
) =
A
(
x
^{−1}
) for all
x
∈
L
.
Theorem 5.7.
G is connected if and only if supp
<
A
>⊇
V
−
v
_{1}
.
Proof
. By Remark 4.1,4.2 and by Theorem 5.1,
Theorem 5.8.
G is weakly connected if and only if supp
(<
A
∪
A
^{−1}
>) ⊇
V
−
v
_{1}
.
Proof
. By Remark 4.1,4.2 and by Theorem 5.2,
Theorem 5.9.
G is semi connected if and only if supp
(<
A
> ∪ <
A
^{−1}
>) ⊇
V
−
v
_{1}
.
Proof
. By Remark 4.1,4.2 and by Theorem 5.3,
Theorem 5.10.
Let G is locally connected if and only if supp
(<
A
>) =
supp
(<
A
^{−1}
>).
Proof
. By Remark 4.1,4.2 and by Theorem 5.4,
Theorem 5.11.
A finite Cayley intuitionistic fuzzy graph G is quasi connected if and only if it is connected
.
Proof
. By Remark 4.1,4.2 and by Theorem 5.5,
6. Different types ofαconnectedness in Cayley intuitionistic fuzzy graphs
Definition 6.1.
The
μ
−
strength
of a path
P
=
v
_{1}
,
v
_{2}
, . . . ,
v_{n}
is defined as min(
μ
_{2}
(
v_{i}
,
v_{j}
)) for all
i
and
j
and is denoted by
S_{μ}
. The
ν
−
strength
of a path
P
=
v
_{1}
,
v
_{2}
, . . . ,
v_{n}
is defined as max(
ν
_{2}
(
v_{i}
,
v_{j}
)) for all
i
and
j
and is denoted by
S_{ν}
. The strength of
P
= {
μ
−
strength
,
μ
−
strength
}
Definition 6.2.
Let
G
be an intuitionistic fuzzy digraph. Then
G
is said to be:

(i)α−connectedif for every pair of verticesx,y∈G, there is a pathPfromxtoysuch that strength (P) ≥α,

(ii)weakly α−connectedif an intuitionistic fuzzy graph (V,R∨R−1) isα−connected,

(iii)semi α−connectedif for everyx,y∈V, there is a path of strength greater than or equal toαfromxtoyor fromytoxinG,

(iv)locally α−connectedif for every pair of verticesxandy, there is a pathPof strength greater than or equal toαfromxtoywhenever there is a pathP′ of strength greater than or equal toαfromytox,

(v)quasi α−connectedif for every pairx,y∈V, there is somez∈Vsuch that there is directed path fromztoxof strength greater than or equal toαand there is a directed path fromztoyof strength greater than or equal toα.
Remark 6.1.
Let
G
= (
V
,
R
) be any intuitionistic fuzzy graph, then
G
is
α
connected (weakly
α
connected, semi
α
connected, locally
α
connected or quasi
α
connected) if and only if the induce intuitionistic fuzzy graph
is connected(weakly connected, semiconnected, locally connected or quasi connected).
Let
G
denotes the Cayley intuitionistic fuzzy graphs
G
= (
V
,
R
) induced by (
V
, ∗,
A
). Also for any α ∈ [0, 1], then we have the following results.
Theorem 6.3.
G is α−connected if and only if
<
A
>
_{α}
⊇
V
−
v
_{1}
.
Proof
. By Remark 6.1 and by Theorems 4.5, 5.7,
Theorem 6.4.
G is weakly α
−
connected if and only if
<
A
∪
A
^{−1}
>
_{α}
⊇
V
−
v
_{1}
.
Proof
. By Remark 6.1 and by Theorems 4.5, 5.8,
Theorem 6.5.
G is semi α
−
connected if and only if
(<
A
>
_{α}
∪ <
A
^{−1}
>
_{α}
) ⊇
V
−
v
_{1}
.
Proof
. By Remark 6.1 and by Theorems 4.5, 5.9,
Theorem 6.6.
Let G is locally α
−
connected if and only if
Proof
. By Remark 6.1 and by Theorems 4.5, 5.10,
Theorem 6.7.
A finite Cayley intuitionistic fuzzy graph G is quasi α
−
connected if and only if it is α
−
connected
.
Proof
. By Remark 6.1 and by Theorems 4.5, 5.11,
This completes the proof.
Acknowledgements
The second author would like to thank the University Grants Commission, Hyderabad, for its financial support to Minor Research Project No. MRP4924/14(SERO/UGC) dated March 2014.
BIO
Dr. Akram has received MSc degrees in Mathematics and Computer Science, MPhil in(Computational) Mathematics and PhD in (Fuzzy) Mathematics. He is currently an Associate Professor in the Department of Mathematics at the University of the Punjab, Lahore. Dr. Akram’s research interests include numerical algorithms, fuzzy graphs, fuzzy algebras, and fuzzy decision support systems. He has published 3 monographs and 136 research articles in international peerreviewed journals. Some of his papers have been published in high impact journals including KnowledgeBased Systems (IF=4.104), Information Sciences (IF=3.643), Applied Soft Computing (IF=2.140), Computers & Mathematics with Applications (IF=2.069). He has been an Editorial Member of Mathematical Logical area of The Scientific World Journal (Hindawi, USA) and many other international academic journals. He has been Reviewer/Referee for 66 International Journals including Mathematical Reviews (USA) and Zentralblatt MATH (Germany). He is currently supervising several M.Phil and PhD students.
Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan.
email:makrammath@yahoo.com, m.akram@pucit.edu.pk
M.G. Karunambigai
Department of Mathematics, Sri Vasavi College, Erode638016, Tamilnadu, India.
email:karunsvc@yahoo.in
O.K. Kalaivani
Department of Mathematics, Sri Vasavi College, Erode638016, Tamilnadu, India.
email:kalaivani83@gmail.com
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