A vague graph is a generalized structure of a fuzzy graph that gives more precision, flexibility and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, we define two new operation on vague graphs namely normal product and tensor product and study about the degree of a vertex in vague graphs which are obtained from two given vague graphs
G
_{1}
and
G
_{2}
using the operations cartesian product, composition, tensor product and normal product. These operations are highly utilized by computer science, geometry, algebra, number theory and operation research. In addition to the existing operations these properties will also be helpful to study large vague graph as a combination of small, vague graphs and to derive its properties from those of the smaller ones.
AMS Mathematics Subject Classification : 05C99.
1. Introduction
Graphs and hypergraphs have been applied in a large number of problemsincluding cancer detection, robotics, human cardiac functions, networking and designing. It was Zadeh
[25]
who introduced fuzzy sets and fuzzy logic into mathematics to deal with problems of uncertainty. As most of the phenomena around us involve much of ambiguity and vagueness, fuzzy logic and fuzzy mathematics have to play a crucial role in modeling real time systems with some level of uncertainty. The most important feature of a fuzzy set is that a fuzzy set A is a class of objects that satisfy a certain (or several) property. Gau and Buehrer
[5]
proposed the concept of vague set in 1993, by replacing the value of an element in a set with a subinterval of [0,1]. Namely, a truemembership function
t_{v}
(
x
) and a false membership function
f_{v}
(
x
) are used to describe the boundaries of the membership degree. The initial definition given by Kaufmann
[6]
of a fuzzy graph was based on the fuzzy relation proposed by Zadeh
[26]
. Later Rosenfeld
[15]
introduced the fuzzy analogue of several basic graphtheoretic concepts. Mordeson and Nair
[7]
defined the concept of complement of fuzzy graph and studied some operations on fuzzy graphs. Akram et al.
[2
,
3
,
4]
introduced vague hypergraphs, certain types of vague graphs and regularity in vague intersection graphs and vague line graphs . Ramakrishna
[9]
introduced the concept of vague graphs and studied some of their properties. Pal and Rashmanlou
[8]
studied irregular intervalvalued fuzzy graphs. Also, they defined antipodal intervalvalued fuzzy graphs
[10]
, balanced intervalvalued fuzzy graphs
[11]
, some properties of highly irregular intervalvalued fuzzy graphs
[12]
and a study on bipolar fuzzy graphs
[14]
. Rashmanlou and Yang Bae Jun investigated complete intervalvalued fuzzy graphs
[13]
. Samanta and Pal defined fuzzy tolerance graphs
[16]
, fuzzy threshold graphs
[17]
, fuzzy planar graphs
[18]
, fuzzy
k
competition graphs and
p
competition fuzzy graphs
[19]
, irregular bipolar fuzzy graphs
[20]
, fuzzy coloring of fuzzy graphs
[21]
. In this paper, we defined two new operation on vague graphs namely normal product and tensor product and studied about the degree of a vertex in vague graphs which are obtained from two given vague graphs
G
_{1}
and
G
_{2}
using the operations cartesian product, composition, tensor product and normal product. For further details, reader may look into
[1
,
22
,
23
,
24]
.
2. Preliminaries
By a graph
G
^{∗}
= (
V
,
E
), we mean a nontrivial, finite, connected and undirected graph without loops or multiple edges. Formally, given a graph
G
^{∗}
= (
V
,
E
), two vertices
x
,
y
∈
V
are said to be neighbors, or adjacent nodes, if
xy
∈
E
. 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
. A fuzzy graph
G
is a pair of functions
G
= (
σ
,
μ
) where
σ
is a fuzzy subset of a nonempty set
V
and
μ
:
V
×
V
→ [0,1] is a symmetric fuzzy relation on
σ
, i.e.
μ
(
uv
) ≤
σ
(
u
) ∧
σ
(
v
). The degree of a vertex
u
in fuzzy graph
G
is defined by
d_{G}
(
u
) = Σ
_{u≠v}
μ
(
uv
) = Σ
_{uv∈E}
μ
(
uv
). The order of a fuzzy graph
G
is defined by
O
(
G
) = Σ
_{u∈V}
σ
(
u
).
The main objective of this paper is to study of vague graph and this graph is based on the vague set defined below.
Definition 2.1
(
[5]
). A vague set on an ordinary finite nonempty set
X
is a pair (
t_{A}
,
f_{A}
), where
t_{A}
:
X
→ [0, 1],
f_{A}
:
X
→ [0, 1] are true and false membership functions, respectively such that 0 ≤
t_{A}
(
x
)+
f_{A}
(
x
) ≤ 1, for all
x
∈
X
. Note that
t_{A}
(
x
) is considered as the lower bound for degree of membership of
x
in
A
and
f_{A}
(
x
) is the lower bound for negative of membership of
x
in
A
. So, the degree of membership of
x
in the vague set
A
is characterized by interval [
t_{A}
(
x
), 1−
f_{A}
(
x
)]. Let
X
and
Y
be ordinary finite nonempty sets. We call a vague relation to be a vague subset of
X
×
Y
, that is an expression
R
defined by
where
t_{R}
:
X
×
Y
→ [0, 1],
f_{R}
:
X
×
Y
→ [0, 1], which satisfies the condition 0 ≤
t_{R}
(
x
,
y
) +
f_{R}
(
x
,
y
) ≤ 1, for all (
x
,
y
) ∈
X
×
Y
.
Definition 2.2
(
[9]
). Let
G
^{∗}
= (
V
,
E
) be a crisp graph. A pair
G
= (
A
,
B
) is called a vague graph on a crisp graph
G
^{∗}
= (
V
,
E
), where
A
= (
t_{A}
,
f_{A}
) is a vague set on
V
and
B
= (
t_{B}
,
f_{B}
) is a vague set on
E
⊆
V
×
V
such that
for each edge
xy
∈
E
.
If
G
is a vague graph, then the order of
G
is defined and denoted as
and the size of
G
is
The open degree of a vertex
u
in a vague graph
G
= (
A
,
B
) is defined as
d
(
u
) =
d^{t}
(
u
),
d^{f}
(
u
)) where
If all the vertices have the same open neighborhood degree
n
, then
G
is called an
n
regular vague graph.
Definition 2.3.
Let
G
_{1}
= (
A
_{1}
,
B
_{1}
) and
G
_{2}
= (
A
_{2}
,
B
_{2}
) be two vague graphs of
= (
V
_{1}
,
E
_{1}
) and
= (
V
_{2}
,
E
_{2}
) respectively.
(1) The cartesian product
G
_{1}
×
G
_{2}
of
G
_{1}
and
G
_{2}
is defined as pair (
A
_{1}
×
A
_{2}
,
B
_{1}
×
B
_{2}
) such that
(2) The composition
G
_{1}
◦
G
_{2}
of
G
_{1}
and
G
_{2}
is defined as pair (
A
_{1}
◦
A
_{2}
,
B
_{1}
◦
B
_{2}
) such that

for all (u1;u2)(v1;v2) ∈E◦−E,
where
E
^{◦}
=
E
∪ {(
u
_{1}
,
u
_{2}
)(
v
_{1}
,
v
_{2}
) 
u
_{1}
v
_{1}
∈
E
_{1}
,
u
_{2}
≠
v
_{2}
}.
3. Degree of vertices in vague graphs
Operation in fuzzy graph is a great tool to consider large fuzzy graph as a combination of small fuzzy graphs and to derive its properties from those of the smaller ones. Also, they are conveniently used in many combinatorial applications. In various situations they present a suitable construction means. For example in partition theory we deal with complex objects. A typical such object is a fuzzy graph and a fuzzy hypergraph with large chromatic number that we do not know how to compute exactly the chromatic number of these graphs. In such cases, these operations have the main role in solving problems. Hence, in this section, at first we define two new operations on vague graphs namely normal product and tensor product. Then we study about the degree of a vertex in vague graphs which are obtained from two given vague graphs
G
_{1}
and
G
_{2}
using the operations cartesian product, composition, tensor product and normal product.
Definition 3.1.
The normal product of two vague graphs
G_{i}
= (
A_{i}
,
B_{i}
) on
G_{i}
= (
V_{i}
,
E_{i}
),
i
= 1, 2 is defined as a vague graph (
A
_{1}
●
A
_{2}
,
B
_{1}
●
B
_{2}
) on
G
= (
V
,
E
) where
V
=
V
_{1}
×
V
_{2}
and
E
= {((
u
,
u
_{2}
)(
u
,
v
_{2}
)) 
u
∈
V
_{1}
,
u
_{2}
v
_{2}
∈
E
_{2}
} ∪ {((
u
_{1}
,
z
)(
v
_{1}
,
z
)) 
u
_{1}
v
_{1}
∈
E
_{1}
,
z
∈
V
_{2}
}∪{((
u
_{1}
,
u
_{2}
)(
v
_{1}
,
v
_{2}
)) 
u
_{1}
v
_{1}
∈
E
_{1}
,
u
_{2}
v
_{2}
∈
E
_{2}
} such that.

for allu1v1∈E1andu2v2∈E2.
Definition 3.2.
The tensor product of two vague graphs
G_{i}
= (
A_{i}
,
B_{i}
) on
G_{i}
= (
V_{i}
,
E_{i}
),
i
= 1, 2, is defined as a vague graph (
A
_{1}
⊗
A
_{2}
,
B
_{1}
⊗
B
_{2}
) on
G
= (
V
,
E
) where
V
=
V
_{1}
×
V
_{2}
and
E
= {(
u
_{1}
,
u
_{2}
), (
v
_{1}
,
v
_{2}
) 
u
_{1}
v
_{1}
∈
E
_{1}
,
u
_{2}
v
_{2}
∈
E
_{2}
} such that

for allu1v1∈E1andu2v2∈E2.
Now, we derive degree of a vertex in the cartesian product. By the definition of cartesian product for any vertex (
u
_{1}
,
u
_{2}
) ∈
V
_{1}
×
V
_{2}
,
Theorem 3.3.
Let G
_{1}
= (
A
_{1}
,
B
_{1}
)
and
G
_{2}
= (
A
_{2}
,
B
_{2}
)
be two vague graphs
.
If
t
_{A1}
≥
t
_{B2}
,
f
_{A1}
≤
f
_{B2}
and
t
_{A2}
≥
t
_{B1}
,
f
_{A2}
≤
f
_{B1}
then
Proof
. From the definition of a vertex in the cartesian product we have
Also we have
Hence,
d
_{G1×G2}
(
u
_{1}
,
u
_{2}
) =
d
_{G1}
(
u
_{1}
) +
d
_{G2}
(
u
_{2}
). □
Example 3.4.
Consider the vague graphs
G
_{1}
,
G
_{2}
and
G
_{1}
×
G
_{2}
as follows.
Since
t
_{A1}
≥
t
_{B2}
,
f
_{A1}
≤
f
_{B2}
,
t
_{A2}
≥
t
_{B1}
and
f
_{A2}
≤
f
_{B1}
. By Theorem 3.3, we have
So,
d
_{G1×G2}
(
u
_{1}
,
u
_{2}
) = (0.5, 1.2).
Hence,
d
_{G1×G2}
(
u
_{1}
,
v
_{2}
) = (0.5, 1.2).
Similarly, we can find the degrees of all the vertices in
G
_{1}
×
G
_{2}
. This can be verified in the
Figure 1
.
Cartesian product of G_{1} and G_{2}
Now we calculate the degree of a vertex in composition. By the definition of composition for any vertex (
u
_{1}
,
u
_{2}
) ∈
V
_{1}
×
V
_{2}
we have
Theorem 3.5.
Let G
_{1}
= (
A
_{1}
,
B
_{1}
)
and G
_{2}
= (
A
_{2}
,
B
_{2}
)
be two vague graphs. If t
_{A1}
≥
t
_{B2}
,
f
_{A1}
≤
f
_{B2}
,
t
_{A2}
≥
t
_{B1}
and f
_{A2}
≤
f
_{B1}
,
then d
_{G1◦G2}
(
u
_{1}
,
u
_{2}
) = 
V
_{2}

d
_{G1}
(
u
_{1}
) +
d
_{G2}
(
u
_{2}
)
for all
(
u
_{1}
,
u
_{2}
) ∈
V
_{1}
×
V
_{2}
.
Proof
.
Similarly we can show that
Hence,
d
_{G1◦G2}
(
u
_{1}
,
u
_{2}
) =
d
_{G2}
(
u
_{2}
) + 
V
_{2}

d
_{G1}
(
u
_{1}
). □
Example 3.6.
Consider the vague graphs
G
_{1}
,
G
_{2}
and
G
_{1}
◦
G
_{2}
as follows.
Here,
t
_{A1}
≥
t
_{B2}
,
f
_{A1}
≤
f
_{B2}
,
t
_{A2}
≥
t
_{B1}
and
f
_{A2}
≤
f
_{B1}
. By Theorem 3.5, we have
Therefore,
d
_{G1◦G2}
(
u
_{1}
,
u
_{2}
) = (0.6, 2.1).
So,
d
_{G1◦G2}
(
u
_{1}
,
v
_{2}
) = (0.6, 2.1).
In the same way, we can find the degree of all the vertices in
G
_{1}
◦
G
_{2}
. This can be verified in the
Figure 2
.
Composition of G_{1} and G_{2}
Degree of a vertex in the tensor product is as follows.
By definition of tensor product for any (
u
_{1}
,
u
_{2}
) ∈
V
_{1}
×
V
_{2}
we have
Theorem 3.7.
Let G
_{1}
= (
A
_{1}
,
B
_{1}
)
and G
_{2}
= (
A
_{2}
,
B
_{2}
)
be two vague graphs. If t
_{B2}
≥
t
_{B1}
and f
_{B2}
≤
f
_{B1}
then d
_{G1⊗G2}
(
u
_{1}
,
u
_{2}
) =
d
_{G1}
(
u
_{1}
)
. Also, if t
_{B1}
≥
t
_{B2}
and f
_{B1}
≤
f
_{B2}
then d
_{G1⊗⊗G2}
(
u
_{1}
,
u
_{2}
) =
d
_{G2}
(
u
_{2}
).
Proof
. Let
t
_{B2}
≥
t
_{B1}
,
f
_{B2}
≤
f
_{B1}
then we have
Hence,
d
_{G1⊗G2}
(
u
_{1}
,
u
_{2}
) =
d
_{G1}
(
u
_{1}
). Similarly if
t
_{B1}
≥
t
_{B2}
and
f
_{B1}
≤
f
_{B2}
, then
d
_{G1⊗G2}
(
u
_{1}
,
u
_{2}
) =
d
_{G2}
(
u
_{2}
). □
Example 3.8.
In this example we obtain the degree of vertices of
G
_{1}
⊗
G
_{2}
by Theorem 3.7.
Consider the vague graphs
G
_{1}
and
G
_{2}
in
Figure 3
. Here
t
_{B2}
≥
t
_{B1}
,
f
_{B2}
≤
f
_{B1}
. By Theorem 3.7 we have
So,
d
_{G1⊗G2}
(
u
_{1}
,
u
_{2}
) = (0.2, 0.5) and
d
_{G1⊗G2}
(
v
_{1}
,
v
_{2}
) = (0.2, 0.5). Similarly, we can find the degree of all the vertices in
G
_{1}
⊗
G
_{2}
. This can be verified in the
Figure 3
.
Tensor product of G_{1} and G_{2}
Finally, we derive the degree of a vertex in normal product. By the definition of normal product for any (
u
_{1}
,
u
_{2}
) ∈
V
_{1}
×
V
_{2}
we have
Theorem 3.9.
Let G
_{1}
= (
A
_{1}
,
B
_{1}
)
and G
_{2}
= (
A
_{2}
,
B
_{2}
)
be two vague graphs. If t
_{A1}
≥
t
_{B2}
,
f
_{A1}
≤
f
_{B2}
,
t
_{A2}
≥
t
_{B1}
,
f
_{A2}
≤
f
_{B1}
,
t
_{B1}
≤
t
_{B2}
and f
_{B1}
≥
f
_{B2}
then d
_{G1●G2}
(
u
_{1}
,
u
_{2}
) = 
V
_{2}

d
_{G1}
(
u
_{1}
) +
d
_{G2}
(
u
_{2}
).
Proof
.
In the same way we can show that
Hence,
d
_{G1●G2}
(
u
_{1}
,
u
_{2}
) = 
V
_{2}

d
_{G1}
(
u
_{1}
) +
d
_{G2}
(
u
_{2}
). □
Example 3.10.
In this example we obtain the degree of vertices of
G
_{1}
●
G
_{2}
by Theorem 3.9.
Consider the vague graphs
G
_{1}
and
G
_{2}
in
Figure 4
. Here
t
_{A1}
≥
t
_{B2}
,
f
_{A1}
≤
f
_{B2}
,
t
_{A2}
≥
t
_{B1}
,
f
_{A2}
≤
f
_{B1}
,
t
_{B1}
≤
t
_{B2}
and
f
_{B1}
≥
f
_{B2}
. So, by Theorem 3.9 we have
Therefore,
d
_{G1●G2}
(
u
_{1}
,
u
_{2}
) = (0.6, 2).
So,
d
_{G1●G2}
(
u
_{1}
,
v
_{2}
) = (0.6, 2).
Normal product of G_{1} and G_{2}
Similarly, we can find the degree of all the vertices in
G
_{1}
●
G
_{2}
. This can be verified in the
Figure 4
.
4. Conclusion
Graph theory has several interesting applications in system analysis, operations research, computer applications, and economics. Since most of the time the aspects of graph problems are uncertain, it is nice to deal with these aspects via the methods of fuzzy systems. It is known that fuzzy graph theory has numerous applications in modern science and engineering, neural networks, expert systems, medical diagnosis, town planning and control theory. In this paper, we have found the degree of vertices in
G
_{1}
×
G
_{2}
,
G
_{1}
◦
G
_{2}
,
G
_{1}
⊗
G
_{2}
and
G
_{1}
●
G
_{2}
in terms of the degree of vertices in
G
_{1}
and
G
_{2}
under some conditions and illustrated them through examples. This will be helpful when the graphs are very large and it can help us in studying various properties of cartesian product, composition, tensor product and normal product of two vague graphs.
Acknowledgements
The authors are extremely grateful to the Editor in Chief Prof. Cheon Seoung Ryoo and anonymous referees for giving them many valuable comments and helpful suggestions which helps to improve the presentation of this paper.
BIO
R.A. Borzooei is Professor of Math., Shahid Beheshti University of Tehran, Tehran, Iran. His research interests include Fuzzy Graphs Theory, Fuzzy logical algebras, BLalgebras, BCK and BCIalgebras.
Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran.
email: borzooei@sbu.ac.ir
H. Rashmanlou
Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran.
email: Rashmanlou.1987@gmail.com
Abu Nayeem Sk. Md.
,
Pal M.
(2008)
The pcenter problem on fuzzy networks and reduction of cost
Iranian Journal of Fuzzy Systems
5
1 
26
Akram M.
,
Gani N.
,
Borumand Saeid A.
(2014)
Vague hypergraphs
Journal of Intelligent and Fuzzy Systems
26
647 
653
Akram M.
,
Feng F.
,
Sarwar S.
,
Jun Y.B.
(2014)
Certain types of vague graphs
University Politehnica of Bucharest Scientific Bulletin Series A
76
141 
154
Akram M.
,
Murtaza Yousaf M.
,
Dudek Wieslaw A.
2014
Regularity in vague intersection graphs and vague line graphs
Abstract and Applied Analysis
Article ID 525389
DOI : 10.1155/2014/525389
Gau W.L.
,
Buehrer D.J.
(1993)
Vague sets
IEEE Transactions on Systems, Man and Cybernetics
23
610 
614
DOI : 10.1109/21.229476
Kauffman A.
(1973)
Introduction a la Theorie des SousEmsembles Flous,1
Masson et Cie
Mordeson J.N.
,
Nair P.S.
2000
Fuzzy Graphs and Fuzzy Hypergraphs
Physica Verlag
Pal M.
,
Rashmanlou H.
(2013)
Irregular intervalvalued fuzzy graphs
Annals of Pure and Applied Mathematics
3
56 
66
Ramakrishna N.
(2009)
Vague graphs
International Journal of Computational Cognition
7
51 
58
Rashmanlou H.
,
Pal M.
(2013)
Antipodal intervalvalued fuzzy graphs
International Journal of Applications of Fuzzy Sets and Artificial Intelligence
3
107 
130
Rashmanlou H.
,
Pal M.
(2013)
Balanced intervalvalued fuzzy graph
Journal of Physical Sciences
17
43 
57
Rashmanlou H.
,
Pal M.
(2013)
Some properties of highly irregular intervalvalued fuzzy graphs
World Applied Sciences Journal
27
1756 
1773
Rashmanlou H.
,
Jun Y.B.
(2013)
Complete intervalvalued fuzzy graphs
Annals of Fuzzy Mathematics and Informatics
6
677 
687
Rashmanlou H.
,
Samanta S.
,
Pal M.
,
Borzooei R.A.
(2015)
A study on bipolar fuzzy graphs
Journal of Intelligent and Fuzzy Systems
28
571 
580
Rosenfeld A.
(1975)
Fuzzy graphs in Fuzzy Sets and Their Applications, L. A. Zadeh, K. S. Fu, and M. Shimura, Eds.
Academic Press
New York, NY, USA
77 
95
Samant S.
,
Pal M.
(2011)
Fuzzy tolerance graphs
International Journal Latest Trend Mathematics
1
57 
67
Samanta S.
,
Pal M.
(2011)
Fuzzy threshold graphs
CiiT International Journal of Fuzzy Systems
3
360 
364
Samanta S.
,
Pal M.
,
Pal A.
(2014)
New concepts of fuzzy planar graph
International Journal of Advanced Research in Articial Intelligence
3
52 
59
Samanta S.
,
Pal M.
(2013)
Fuzzy kcompetition graphs and pcompetition fuzzy graphs
Fuzzy Engineering and Information
5
191 
204
DOI : 10.1007/s1254301301406
Samanta S.
,
Pal M.
,
Pal A.
(2014)
Some more results on fuzzy kcompetition graphs
International Journal of Advanced Research in Artificial Intelligence
3
60 
67
Samanta S.
,
Pal M.
(2014)
Some more results on bipolar fuzzy sets and bipolar fuzzy intersection graphs
The Journal of Fuzzy Mathematics
22
253 
262