Faculty of Sciences and Mathematics, University of Niˇs, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Filomat 26:1 (2012), 177-196 DOI: 10.2298/FIL1201177A
Strong intuitionistic fuzzy graphs
Muhammad Akram, Bijan Davvaz
Abstract
We introduce the notion of strong intuitionistic fuzzy graphs and investi-
gate some of their properties. We discuss some propositions of self comple-
mentary and self weak complementary strong intuitionistic fuzzy graphs. We
introduce the concept of intuitionistic fuzzy line graphs.
1 Introduction
In 1736. Euler first introduced the concept of graph theory. In the history of
mathematics, the solution given by Euler of the well known K¨onigsberg bridge
problem is considered to be the first theorem of graph theory. This has now become
a subject generally regarded as a branch of combinatorics. The theory of graph is
an extremely useful tool for solving combinatorial problems in different areas such
as geometry, algebra, number theory, topology, operations research, optimization
and computer science.
In 1983. Atanassov [6] introduced the concept of intuitionistic fuzzy sets as
a generalization of fuzzy sets [31]. Atanassov added a new component(which de-
termines the degree of non-membership) in the definition of fuzzy set. The fuzzy
sets give the degree of membership of an element in a given set (and the non-
membership degree equals one minus the degree of membership), while intuitionis-
tic fuzzy sets give both a degree of membership and a degree of non-membership
which are more-or-less independent from each other, the only requirement is that
the sum of these two degrees is not greater than 1. Intuitionistic fuzzy sets have
been applied in a wide variety of fields including computer science, engineering,
mathematics, medicine, chemistry and economics [5, 13].
In 1975. Rosenfeld [27] introduced the concept of fuzzy graphs. The fuzzy
relations between fuzzy sets were also considered by Rosenfeld and he developed the
2010 Mathematics Subject Classifications. 05C99.
Key words and Phrases. Intuitionistic fuzzy sets, strong intuitionistic fuzzy graphs, self com-
plementary intuitionistic fuzzy graphs, intuitionistic fuzzy line graphs, applications.
Received: May 16, 2011; Accepted: June 20, 2011
Communicated by Miroslav
´
Ciri´c
178 M. Akram, B. Davvaz
structure of fuzzy graphs, obtaining analogs of several graph theoretical concepts.
Later on, Bhattacharya [8] gave some remarks on fuzzy graphs, and some operations
on fuzzy graphs were introduced by Mordeson and Peng [20]. The complement of
a fuzzy graph was defined by Mordeson [22] and further studied by Sunitha and
Vijayakumar [30]. Mordeson [21] introduced the notion of fuzzy line graph. Bhutani
and Rosenfeld introduced the concept of M-strong fuzzy graphs in [11] and studied
some of their properties. Akram and Dudek [2] discussed interval-valued fuzzy
graphs. Atanassov [5] introduced the concept of intuitionistic fuzzy relations and
intuitionistic fuzzy graphs, and further studied in [25]. In fact, interval-valued fuzzy
graphs and intuitionistic fuzzy graphs are two different models that extend theory
of fuzzy graph. In this article, we introduce the notion of strong intuitionistic fuzzy
graphs and investigate some of their properties. We discuss some propositions of
self complementary and self weak complementary strong intuitionistic fuzzy graphs.
We study intuitionistic fuzzy line graphs. The definitions and terminologies that
we used in this paper are standard.
2 Preliminaries
In this section, we review some definitions that are necessary in the paper.
A graph is an ordered pair G
∗
= (V, E), where V is the set of vertices of G
∗
and E is the set of edges of G
∗
. Two vertices x and y in a graph G
∗
are said to
be adjacent in G
∗
if {x, y} is in an edge of G
∗
(for simplicity an edge {x, y} will be
denoted by xy). A simple graph is a graph without loops and multiple edges. A
complete graph is a simple graph in which every pair of distinct vertices is connected
by an edge. The complete graph on n vertices has n(n−1)/2 edges. We will consider
only graphs with the finite number of vertices and edges.
An isomorphism of graphs G
∗
1
and G
∗
2
is a bijection between the vertex sets of
G
∗
1
and G
∗
2
such that any two vertices v
1
and v
2
of G
∗
1
are adjacent in G
∗
1
if and only
if f(v
1
) and f(v
2
) are adjacent in G
∗
2
. Isomorphic graphs are denoted by G
∗
1
' G
∗
2
.
By a complementary graph G
∗
of a simple graph G
∗
we mean a graph having the
same vertices as G
∗
and such that two vertices are adjacent in G
∗
if and only if they
are not adjacent in G
∗
. A simple graph that is isomorphism to its complement is
called self-complementary.
Let G
∗
1
= (V
1
, E
1
) and G
∗
2
= (V
2
, E
2
) be two simple graphs, we can construct
several new graphs. The first construction called the Cartesian product of G
∗
1
and
G
∗
2
gives a graph G
∗
1
× G
∗
2
= (V, E) with V = V
1
× V
2
and
E = {(x, x
2
)(x, y
2
)|x ∈ V
1
, x
2
y
2
∈ E
2
} ∪ {(x
1
, z)(y
1
, z)|x
1
y
1
∈ E
1
, z ∈ V
2
, }.
The composition of graphs G
∗
1
and G
∗
2
is the graph G
∗
1
[G
∗
2
] = (V
1
× V
2
, E
0
), where
E
0
= E ∪ {(x
1
, x
2
)(y
1
, y
2
)|x
1
y
1
∈ E
1
, x
2
6= y
2
}
and E is defined as in G
∗
1
× G
∗
2
. Note that G
∗
1
[G
∗
2
] 6= G
∗
2
[G
∗
1
].
Strong intuitionistic fuzzy graphs 179
The union of graphs G
∗
1
and G
∗
2
is defined as G
∗
1
∪ G
∗
2
= (V
1
∪ V
2
, E
1
∪ E
2
).
The join of G
∗
1
and G
∗
2
is the simple graph G
∗
1
+ G
∗
2
= (V
1
∪ V
2
, E
1
∪ E
2
∪ E
0
),
where E
0
is the set of all edges joining the nodes of V
1
and V
2
. In this construction
it is assumed that V
1
∩ V
2
= ∅ .
Definition 1. ([31, 32]) By a fuzzy subset µ on a set X is mean a map µ : X → [0, 1].
A map ν : X × X → [0, 1] is called a fuzzy relation on X if ν(x, y) ≤ min(µ(x), µ(y))
for all x, y ∈ X. A fuzzy relation ν is symmetric if ν(x, y) = ν(y, x) for all x, y ∈ X.
Definition 2. ([5]) A mapping A = (µ
A
, ν
A
) : X → [0, 1] × [0, 1] is called an
intuitionistic fuzzy set in X if µ
A
(x) + ν
A
(x) ≤ 1 for all x ∈ X, where the mappings
µ
A
: X → [0, 1] and ν
A
: X → [0, 1] denote the degree of membership (namely
µ
A
(x)) and the degree of non-membership (namely ν
A
(x)) of each element x ∈ X
to A, respectively.
Definition 3. ([5]) For every two intuitionistic fuzzy sets A = (µ
A
, ν
A
) and B =
(µ
B
, ν
B
) in X, we define
(A ∩ B)(x) = (min(µ
A
(x), µ
B
(x)), max(ν
A
(x), ν
B
(x))),
(A ∪ B)(x) = (max(µ
A
(x), µ
B
(x)), min(ν
A
(x), ν
B
(x))).
Definition 4. ([5]) Let X be a nonempty set. Then we call a mapping A =
(µ
A
, ν
A
) : X × X → [0, 1] × [0, 1] an intuitionistic fuzzy relation on X if µ
A
(x, y) +
ν
A
(x, y) ≤ 1 for all (x, y) ∈ X × X.
Definition 5. ([5]) Let A = (µ
A
, ν
A
) and B = (µ
B
, ν
B
) be intuitionistic fuzzy
sets on a set X. If A = (µ
A
, ν
A
) is an intuitionistic fuzzy relation on a set X,
then A = (µ
A
, ν
A
) is called an intuitionistic fuzzy relation on B = (µ
B
, ν
B
) if
µ
A
(x, y) ≤ min(µ
B
(x), µ
B
(y)) and ν
A
(x, y) ≥ max(ν
B
(x), ν
B
(y)) for all x, y ∈ X.
An intuitionistic fuzzy relation A on X is called symmetric if µ
A
(x, y) = µ
A
(y, x)
and ν
A
(x, y) = ν
A
(y, x) for all x, y ∈ X.
3 Strong intuitionistic fuzzy graphs
Throughout this paper, we denote G
∗
a crisp graph, and G an intuitionistic fuzzy
graph.
Definition 6. An intuitionistic fuzzy graph with underlying set V is defined to be
a pair G = (A, B) where
(i) the functions µ
A
: V → [0, 1] and ν
A
: V → [0, 1] denote the degree of
membership and nonmembership of the element x ∈ V , respectively such that
0 ≤ µ
A
(x) + ν
A
(x) ≤ 1 for all x ∈ V ,
(ii) the functions µ
B
: E ⊆ V × V → [0, 1] and ν
B
: E ⊆ V × V → [0, 1] are
defined by
µ
B
({x, y}) ≤ min(µ
A
(x), µ
A
(y)) and ν
B
({x, y}) ≥ max(ν
A
(x), ν
A
(y))
such that 0 ≤ µ
B
({x, y}) + ν
B
({x, y}) ≤ 1 for all {x, y} ∈ E.
180 M. Akram, B. Davvaz
We call A the intuitionistic fuzzy vertex set of V , B the intuitionistic fuzzy edge set
of G, respectively. Note that B is a symmetric intuitionistic fuzzy relation on A.
We use the notation xy for an element of E. Thus, G = (A, B) is an intuitionistic
graph of G
∗
= (V, E) if
µ
B
(xy) ≤ min(µ
A
(x), µ
A
(y)) and ν
B
(xy) ≥ max(ν
A
(x), ν
A
(y))
for all xy ∈ E.
We now study strong intuitionistic fuzzy graphs.
Definition 7. An intuitionistic fuzzy graph G = (A, B) is called strong intuition-
istic fuzzy graph if
µ
B
(xy) = min(µ
A
(x), µ
A
(y)) and ν
B
(xy) = max(ν
A
(x), ν
A
(y)),
for all xy ∈ E.
Example 1. Consider a graph G
∗
such that V = {x, y, z}, E = {xy, yz, zx}. Let
A be an intuitionistic fuzzy subset of V and let B be an intuitionistic fuzzy subset
of E defined by
x y z
µ
A
0.2 0.3 0.1
ν
A
0.4 0.1 0.5
xy yz xz
µ
B
0.2 0.1 0.1
ν
B
0.4 0.5 0.5
y
G
z
x
(0.1, 0.5)
(0.1, 0.5)(0.2, 0.4)
(0.1, 0.5)
(0.2, 0.4)
(0.3, 0.1)
The graph G is represented by the following adjacency matrix
A =
(0.2, 0.4) (0.2, 0.4) (0.1, 0.5)
(0.2, 0.4) (0.3, 0.1) (0.1, 0.5)
(0.1, 0.5) (0.1, 0.5) (0.1, 0.5)
.
By routine computations, it is easy to see that G is a strong intuitionistic fuzzy
graph of G
∗
.
Strong intuitionistic fuzzy graphs 181
Definition 8. Let A = (µ
A
, ν
A
) and A
0
= (µ
0
A
, ν
0
A
) be intuitionistic fuzzy subsets
of V
1
and V
2
and let B = (µ
B
, ν
B
) and B
0
= (µ
0
B
, ν
0
B
) be intuitionistic fuzzy subsets
of E
1
and E
2
, respectively. The Cartesian product of two strong intuitionistic fuzzy
graphs G
1
and G
2
of the graphs G
∗
1
and G
∗
2
is denoted by G
1
×G
2
= (A×A
0
, B ×B
0
)
and is defined as follows:
(i)
(µ
A
× µ
0
A
)(x
1
, x
2
) = min(µ
A
(x
1
), µ
0
A
(x
2
))
(ν
A
× ν
0
A
)(x
1
, x
2
) = max(ν
A
(x
1
), ν
0
A
(x
2
))
for all (x
1
, x
2
) ∈ V ,
(ii)
(µ
B
× µ
0
B
)((x, x
2
)(x, y
2
)) = min(µ
A
(x), µ
0
B
(x
2
y
2
)),
(ν
B
× ν
0
B
)((x, x
2
)(x, y
2
)) = max(ν
A
(x), ν
0
B
(x
2
y
2
))
for all x ∈ V
1
, for all x
2
y
2
∈ E
2
,
(iii)
(µ
B
× µ
0
B
)((x
1
, z)(y
1
, z)) = min(µ
B
(x
1
y
1
), µ
0
A
(z))
(ν
B
× ν
0
B
)((x
1
, z)(y
1
, z)) = max(ν
B
(x
1
y
1
), ν
0
A
(z))
for all z ∈ V
2
, for all x
1
y
1
∈ E
1
.
Proposition 1. If G
1
and G
2
are the strong intuitionistic fuzzy graphs, then G
1
×
G
2
is a strong intuitionistic fuzzy graph.
Proof. It is straightforward.
Proposition 2. If G
1
× G
2
is strong intuitionistic fuzzy graph, then at least G
1
or
G
2
must be strong.
Proof. Suppose that G
1
and G
2
are not strong intuitionistic fuzzy graphs. Then
there exist x
1
y
1
∈ E
1
and x
2
y
2
∈ E
2
such that
µ
B
1
(x
1
y
1
) < min(µ
A
1
(x), µ
A
1
(y)), µ
B
2
(x
1
y
1
) < min(µ
A
2
(x), µ
A
2
(y)) (1)
ν
B
1
(x
1
y
1
) > max(ν
A
1
(x), ν
A
1
(y)), ν
B
2
(x
1
y
1
) > max(ν
A
2
(x), ν
A
2
(y)) (2)
Assume that
µ
B
2
(x
2
y
2
) ≤ µ
B
1
(x
1
y
1
) < min(µ
A
1
(x
1
), µ
A
1
(y
1
)) ≤ µ
A
1
(x
1
) (3)
Let
E = {(x, x
2
)(x, y
2
)|x
1
∈ V
1
, x
2
y
2
∈ E
2
} ∪ {(x
1
, z)(y
1
, z)|z ∈ V
2
, x
1
y
1
∈ E
1
}.
Consider (x, x
2
)(x, y
2
) ∈ E, we have
(µ
B
1
× µ
B
2
)((x, x
2
)(x, y
2
)) = min(µ
A
1
(x), µ
B
2
(x
2
y
2
))
< min(µ
A
1
(x), µ
A
2
(x
2
), µ
A
2
(y
2
)