research article novel properties of fuzzy labeling...
TRANSCRIPT
Research ArticleNovel Properties of Fuzzy Labeling Graphs
A Nagoor Gani1 Muhammad Akram2 and D Rajalaxmi (a) Subahashini3
1 PG amp Research Department of Mathematics Jamal Mohamed College Trichy India2Department of Mathematics University of the Punjab New Campus PO Box No 54590 Lahore Pakistan3Department of Mathematics Saranathan College of Engineering Tiruchirappalli Tamil Nadu 620 012 India
Correspondence should be addressed to Muhammad Akram makrampucitedupk
Received 13 May 2014 Accepted 22 June 2014 Published 9 July 2014
Academic Editor Pierpaolo DrsquoUrso
Copyright copy 2014 A Nagoor Gani et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The concepts of fuzzy labeling and fuzzymagic labeling graph are introduced Fuzzymagic labeling for some graphs like path cycleand star graph is defined It is proved that every fuzzy magic graph is a fuzzy labeling graph but the converse is not true We haveshown that the removal of a fuzzy bridge from a fuzzy magic cycle with odd nodes reduces the strength of a fuzzy magic cycleSome properties related to fuzzy bridge and fuzzy cut node have also been discussed
1 Introduction
Fuzzy set is a newly emerging mathematical frameworkto exemplify the phenomenon of uncertainty in real lifetribulations It was introduced by Zadeh in 1965 and theconcepts were pioneered by various independent researchesnamely Rosenfeld [1] and Bhutani and Battou [2] during1970s Bhattacharya has established the connectivity conceptsbetween fuzzy cut nodes and fuzzy bridges entitled ldquoSomeremarks on fuzzy graphs [3]rdquo Several fuzzy analogs of graphtheoretic concepts such as paths cycles and connectednesswere explored by themThere are many problems which canbe solved with the help of the fuzzy graphs
Though it is very young it has been growing fast andhas numerous applications in various fields Further researchon fuzzy graphs has been witnessing an exponential growthboth within mathematics and in its applications in scienceand Technology A fuzzy graph is the generalization of thecrisp graph Therefore it is natural that many properties aresimilar to crisp graph and also it deviates at many places
In crisp graph a bijection 119891 119881 cup 119864 rarr 119873 thatassigns to each vertex andor edge if 119866 = (119881 119864) a uniquenatural number is called a labeling The concept of magiclabeling in crisp graph was motivated by the notion of magicsquares in number theory The notion of magic graph wasfirst introduced by Sunitha and Vijaya Kumar [4] in 1964 Hedefined a graph to be magic if it has an edge-labeling within
the range of real numbers such that the sum of the labelsaround any vertex equals some constant independent of thechoice of vertex This labeling has been studied by Stewart[5 6] who called the labeling as super magic if the labelsare consecutive integers starting from 1 Several others havestudied this labeling
Kotzig and Rosa [7] defined a magic labeling to bea total labeling in which the labels are the integers from1 to |119881(119866)| + |119864(119866)| The sum of labels on an edge andits two endpoints is constant Recently Enomoto et al [8]introduced the name super edge magic for magic labelingin the sense of Kotzig and Rosa with the added propertythat the V vertices receive the smaller labels Many otherresearchers have investigated different forms ofmagic graphsfor example see Avadayappan et al [9] Ngurah et al [10] andTrenkler [11]
In this paper Section 1 contains basic definitions and inSection 2 a new concept of fuzzy labeling and fuzzy magiclabeling has been introduced and also fuzzy star graph isdefined In Section 2 fuzzy magic labeling for some graphslike path cycle and star is defined In Section 3 someproperties and results with fuzzy bridge and fuzzy cut nodesare discussed The graphs which are considered in this paperare finite and connected
We have used standard definitions and terminologies inthis paper For graphs considered in this paper the readersare referred to [12ndash19]
Hindawi Publishing CorporationJournal of MathematicsVolume 2014 Article ID 375135 6 pageshttpdxdoiorg1011552014375135
2 Journal of Mathematics
009
011007
008
002
001
006 003
010
1 2
34
Figure 1 A fuzzy labeling graph
007
001
005 006 004
1 2 3 4
002 003
Figure 2 A fuzzy magic path graph1198980(119875) = 013
11 Preliminaries Let 119880 and 119881 be two sets Then 120588 is said tobe a fuzzy relation from 119880 into 119881 if 120588 is a fuzzy set of 119880 times 119881A fuzzy graph 119866 = (120590 120583) is a pair of functions 120590 119881 rarr [0 1]and 120583 119881 times 119881 rarr [0 1] where for all 119906 V isin 119881 we have120583(119906 V) le 120590(119906) and 120590(V) A path 119875 in a fuzzy graph is a sequenceof distinct nodes V
1 V2 V
119899such that 120583(V
119894 V119894+1) gt 0 1 le 119894 le
119899 here 119899 ge 1 is called the length of the path119875The consecutivepairs (V
119894 V119894+1) are called the edge of the path A path119875 is called
a cycle if V1= V119899and 119899 ge 3 The strength of a path 119875 is
defined as ⋀119899119894=1120583(V119894 V119894+1) Let 119866 = (120590 120583) be a fuzzy graph
The degree of a vertex V is defined as 119889(V) = sum119906 = V119906isin119881 120583(V 119906)
Let 119866 = (120590 120583) be a fuzzy graph The strong degree of a node Vis defined as the sum ofmembership values of all strong edgesincident at V It is denoted by 119889
119904(V) Also if 119873
119904(V) denote the
set of all strong neighbours of V then 119889119904(V) = sum
119906isin119873119904(V) 120583(V 119906)
An edge is called a fuzzy bridge of119866 if its removal reduces thestrength of connectedness between somepair of nodes in119866 Anode is a fuzzy cut node of 119866 = (120590 120583) if removal of it reducesthe strength of connectedness between some other pairs ofnodes
Definition 1 (see [20]) A graph119866 = (120590 120583) is said to be a fuzzylabeling graph if 120590 119881 rarr [0 1] and 120583 119881 times 119881 rarr [0 1] isbijective such that themembership value of edges and verticesare distinct and 120583(119906 V) lt 120590(119906) and 120590(V) for all 119906 V isin 119881
Example 2 (see [20]) In Figure 1 120590 and 120583 are bijective suchthat no vertices and edges receive the same membershipvalue
Definition 3 (see [20]) A fuzzy labeling graph is said to be afuzzy magic graph if 120590(119906) + 120583(119906 V) + 120590(V) has a same magicvalue for all 119906 V isin 119881 which is denoted as119898
0(119866)
Example 4 (see [20]) In Figure 2 120590(1198811)+120583(119881
1 1198812)+120590(119881
2) =
007 + 001 + 005 = 013 for all 1198811 1198812 isin 119881
Definition 5 A star in a fuzzy graph consists of two node sets119881 and 119880 with |119881| = 1 and |119880| gt 1 such that 120583(V 119906
119894) gt 0 and
120583(119906119894 119906119894+1) = 0 1 le 119894 le 119899 It is denoted by 119878
1119899
Example 6 A fuzzy star graph is shown in Figure 3
Definition 7 (see [20]) The fuzzy labeling graph 119867 = (120591 120588)is called a fuzzy labeling subgraph of119866 = (120590 120583) if 120591(119906) le 120590(119906)for all 119906 isin 119881 and 120588(119906 V) le 120583(119906 V) for all 119906 V isin 119881
2 Properties of Fuzzy Labeling Graphs
Proposition 8 For all 119899 ge 1 the path 119875119899is a fuzzy magic
graph
Proof Let 119875 be any path with length 119899 ge 1 and V1 V2 V
119899
and V1V2 V2V3 V
119899minus1V119899are the nodes and edges of 119875 Let
119911 rarr (0 1] such that one can choose 119911 = 01 if 119899 le 4 and119911 = 001 if 119899 ge 5 Such fuzzy labeling is defined as follows
When length is odd
120590120596 (V2119894minus1
) = (2119899 + 2 minus 119894) 119911 1 le 119894 le119899 + 1
2
120590120596 (V2119894) = min 120590120596 (V
2119894minus1) | 1 le 119894 le
119899 + 1
2 minus 119894 (119911)
1 le 119894 le119899 + 1
2
120583120596 (V119899minus119894+2
V119899+1minus119894
)
= max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1 minus (119894 minus 1) 119911
1 le 119894 le 119899
(1)
Case (i) 119894 is evenThen 119894 = 2119909 for any positive integer 119909 and for each edge
V119894 V119894+1
1198980 (119875) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909) + 120583120596 (V
2119909 V2119909+1
) + 120590120596 (V2119909+1
)
= min 120590120596 (V2119894minus1
) | 1 le 119894 le119899 + 1
2
minus 119909 (119911) +max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minus (119899 minus 2119909) 119911 + (2119899 minus 119909 + 1) 119911
= min 120590120596 (V2119894minus1
) | 1 le 119894 le119899 + 1
2
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1 + (119899 + 1) 119911
(2)
Case (ii) 119894 is oddThen 119894 = 2119909 + 1 for any positive integer 119909 and for each
edge V119894 V119894+1
1198980(119875) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909+1
) + 120583120596 (V2119909+1
V2119909+2
)
+ 120590120596 (V2119909+2
)
Journal of Mathematics 3
V 012
010
007
008
002
003
005
u1
u2
u3
Figure 3 A fuzzy star graph
= (2119899 minus 119909 + 1) 119911
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minus (119899 minus 2119909 minus 1)
+min 120590120596 (V2119894minus1
) | 1 le 119894 le119899 + 1
2
minus (119909 + 1) 119911
= min 120590120596 (V2119894minus1
) | 1 le 119894 le119899 + 1
2
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
+ (119899 + 1) 119911
(3)
When length is even
120590120596 (V2119894) = (2119899 + 2 minus 119894) 119911 1 le 119894 le
119899
2
120590120596 (V2119894minus1
) = min 120590120596 (V2119894) | 1 le 119894 le
119899
2 minus 119894 (119911)
1 le 119894 le119899 + 2
2
120583120596 (V119899minus119894+2
V119899minus119894+1
) = max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minus (119894 minus 1) 119911 1 le 119894 le 119899
(4)
Case (i) 119894 is evenThen 119894 = 2119909 for any positive integer 119909 and for each edge
V119894 V119894+1
1198980 (119875) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909) + 120583120596 (V
2119909 V2119909+1
) + 120590120596 (V2119909+1
)
= (2119899 + 2 minus 119909) 119911
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minus (119899 minus 2119909) 119911 +min 120590120596 (V2119894) | 1 le 119894 le
119899
2
minus (119909 + 1) 119911
= min 120590120596 (V2119894) | 1 le 119894 le
119899
2
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1 + (119899 + 1) 119911
(5)
Case (ii) 119894 is oddThen 119894 = 2119909 + 1 for any positive integer 119909 and for each
edge V119894 V119894+1
1198980 (119875) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909+1
) + 120583120596 (V2119909+1
V2119909+2
) + 120590120596 (V2119909+2
)
= min 120590120596 (V2119894) | 1 le 119894 le
119899
2
minus (119909 + 1) 119911 minus (119899 minus 2119909 minus 1) 119911
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
+ (2119899 minus 119909 + 1) 119911
= min 120590120596 (V2119894) | 1 le 119894 le
119899
2
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1 + (119899 + 1) 119911
(6)
Therefore in both the cases themagic value1198980(119875) is same and
unique Thus 119875119899is fuzzy magic graph for all 119899 ge 1
Proposition 9 If 119899 is odd then the cycle 119862119899is a fuzzy magic
graph
Proof Let 119862119899be any cycle with odd number of nodes and
V1 V2 V
119899and V1V2 V2V3 V
119899V1be the nodes and edges
of 119862119899 Let 119911 rarr (0 1] such that one can choose 119911 = 01 if
119899 le 3 and 119911 = 001 if 119899 ge 4 The fuzzy labeling for cycle isdefined as follows
120590120596 (V2119894) = (2119899 + 1 minus 119894) 119911 1 le 119894 le
119899 minus 1
2
120590120596 (V2119894minus1
) = min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 minus 119894 (119911)
1 le 119894 le119899 + 1
2
120583120596 (V1 V119899) =
1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
120583120596 (V119899minus119894+1
V119899minus119894) = 120583120596 (V
1 V119899) minus 119894 (119911)
1 le 119894 le 119899 minus 1
(7)
4 Journal of Mathematics
Case (i) 119894 is evenThen 119894 = 2119909 for any positive integer 119909 and for each edge
V119894 V119894+1
1198980(119862119899) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909) + 120583120596 (V
2119909 V2119909+1
)
+ 120590120596 (V2119909+1
)
= (2119899 + 1 minus 119909) 119911 +1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
minus (119899 minus 2119909) 119911
+min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2
minus (119909 + 1) 119911
=1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
+min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 + 119899 (119911)
(8)
Case (ii) 119894 is oddThen 119894 = 2119909 + 1 for any positive integer 119909 and for each
edge V119894 V119894+1
1198980(119862119899) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909+1
) + 120583120596 (V2119909+1
V2119909+2
)
+ 120590120596 (V2119909+2
)
= min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 minus (119909 + 1) 119911
+1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
minus (119899 minus 2119909 minus 1) 119911 + (2119899 minus 119909) 119911
=1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
+min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 + 119899 (119911)
(9)
Therefore from above cases 119862119899is a fuzzy magic graph if 119899 is
odd
Proposition 10 For any 119899 ge 2 star 1198781119899
is a fuzzymagic graph
Proof Let 1198781119899
be a star graph with V 1199061 1199062 119906
119899as nodes
and V1199061 V1199062 V119906
119899as edges
Let 119911 rarr (0 1] such that one can choose 119911 = 01 if 119899 le 4and 119911 = 001 if 119899 ge 5 Such a fuzzy labeling is defined asfollows
120590120596 (119906119894) = [2 (119899 + 1) minus 119894] 119911 1 le 119894 le 119899
120590120596 (V) = min 120590120596 (119906119894) | 1 le 119894 le 119899 minus 119911
120583120596 (V 119906119899minus119894) = max 120590120596 (119906
119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minus 119894 (119911) 0 le 119894 le 119899 minus 1
(10)
Case (i) 119894 is evenThen 119894 = 2119909 for any positive integer 119909 and for each edge
V 119906119894
1198980(1198781119899) = 120590120596 (V) + 120583120596 (V 119906119894) + 120590
120596 (119906119894)
= 120590120596 (V) + 120583120596 (V 1199062119909) + 120590120596 (119906
2119909)
= min 120590120596 (119906119894) | 1 le 119894 le 119899 minus (119911)
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minus (119899 minus 2119909) 119911 + [2 (119899 + 1) minus 2119909] 119911
= min 120590120596 (119906119894) | 1 le 119894 le 119899
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
+ (119899 + 1) 119911
(11)
Case (ii) 119894 is oddThen 119894 = 2119909 + 1 for any positive integer 119909 and for each
edge V 119906119894
1198980(1198781119899) = 120590120596 (V) + 120583120596 (V 119906119894) + 120590
120596 (119906119894)
= 120590120596 (V) + 120583120596 (V 1199062119909+1
) + 120590120596 (1199062119909+2
)
= min 120590120596 (119906119894) | 1 le 119894 le 119899 minus 119911
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minus (119899 minus 2119909 minus 1) 119911 + [2 (119899 minus 119909)] 119911
= min 120590120596 (119906119894) | 1 le 119894 le 119899
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
+ (119899 + 1) 119911
(12)
From the above cases one can easily verify that all star graphsare fuzzy magic graphs
Remark 11 One can observe the same labeling holds well ifwe choose the value of 119911 as 003 005 and so forth for thePropositions 8 9 and 10
Remark 12 (1) If 119866 is a fuzzy magic graph then 119889(119906) = 119889(V)for any pair of nodes 119906 and V
(2) For any fuzzy magic graph 0 le 119889119904(V) le 119889(V)
(3) Sum of the degree of all nodes in a fuzzy magic graphis equal to twice the sum of membership values of all edges(ie sum119899
119894=1119889(V119894) = 2sum
119906 = V 120583(119906 V))(4) Sum of strong degree of all nodes in a fuzzy magic
graph is equal to twice the sum of the membership values ofall strong arcs in 119866 (ie sum119899
119894=1119889119904(V119894) = 2sum
119906isin119873119904(V) 120583(V 119906))
Journal of Mathematics 5
3 Properties of Fuzzy Magic Graphs
Proposition 13 Every fuzzy magic graph is a fuzzy labelinggraph but the converse is not true
Proof This is immediate from Definition 3
Proposition 14 For every fuzzy magic graph119866 there exists atleast one fuzzy bridge
Proof Let 119866 be a fuzzy magic graph such that there existsonly one edge 120583(119909 119910) with maximum value since 120583 is bijec-tive Nowwe claim that 120583(119909 119910) is a fuzzy bridge If we removethe edge (119909 119910) from 119866 then in its subgraph we have 1205831015840infin(119909119910) lt 120583(119909 119910) which implies (119909 119910) is a fuzzy bridge
Proposition 15 Removal of a fuzzy cut node from a fuzzymagic path 119875 is also a fuzzy magic graph
Proof Let 119875 be any fuzzy magic path with length 119899 Thenthere must be a fuzzy cut node if we remove that cut nodefrom 119875 then it either becomes a smaller path or disconnectedpath anyway it remains to be a path with odd or even lengthby Proposition 8 it is concluded that removal of a fuzzycut node from a fuzzy magic path 119875 is also a fuzzy magicgraph
Proposition 16 When 119899 is odd removal of a fuzzy bridge froma fuzzy magic cycle 119862
119899is a fuzzy magic graph
Proof Let 119862119899be any fuzzy magic cycle with odd nodes If we
choose any path (119906 V) then there must be at least one fuzzybridge whose removal from 119862
119899will result as a path of odd or
even length By Proposition 8 the removal of a fuzzy bridgefrom a fuzzy magic cycle 119862
119899is also a fuzzy magic graph
Remark 17 (1) Removal of a fuzzy cut node from the cycle119862119899
is also a fuzzy magic graph(2) For all fuzzy magic cycles 119862
119899with odd nodes there
exists at least one pair of nodes 119906 and V such that 119889119904(119906) =
119889119904(V)
Proposition 18 Removal of a fuzzy bridge from a fuzzymagiccycle 119862
119899will reduce the strength of the fuzzy magic cycle 119862
119899
Proof Let 119862119899be a fuzzy magic cycle with odd number of
nodes Now choose any path (119906 V) from 119862119899 and then it
is obvious that there exists at least one fuzzy bridge (119909 119910)Removal of this fuzzy bridge (119909 119910)will reduce the strength ofconnectedness between 119906 and VThis implies that the removalof fuzzy bridge from the fuzzy magic cycle 119862
119899will reduce its
strength
4 Concluding Remarks
Fuzzy graph theory is finding an increasing number ofapplications in modeling real time systems where the level ofinformation inherent in the system varies with different levelsof precision Fuzzy models are becoming useful because oftheir aim in reducing the differences between the traditional
numerical models used in engineering and sciences and thesymbolic models used in expert systems In this paper theconcept of fuzzy labeling and fuzzymagic labeling graphs hasbeen introduced We plan to extend our research work to (1)bipolar fuzzy labeling and bipolar fuzzymagic labeling graphsand (2) fuzzy labeling and fuzzy magic labeling hypergraphs
Conflict of Interests
The authors declare that they do not have any conflict ofinterests regarding the publication of this paper
References
[1] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975
[2] K R Bhutani and A Battou ldquoOn 119872-strong fuzzy graphsrdquoInformation Sciences vol 155 no 1-2 pp 103ndash109 2003
[3] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987
[4] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002
[5] B M Stewart ldquoMagic graphsrdquo Canadian Journal of Mathemat-ics vol 18 pp 1031ndash1059 1966
[6] BM Stewart ldquoSupermagic complete graphsrdquoCanadian Journalof Mathematics vol 9 pp 427ndash438 1966
[7] A Kotzig and A Rosa ldquoMagic valuations of finite graphsrdquoCanadian Mathematical Bulletin vol 13 pp 451ndash461 1970
[8] H Enomoto A S Llado T Nakamigawa and G Ringel ldquoSuperedge-magic graphsrdquo SUT Journal of Mathematics vol 34 no 2pp 105ndash109 1998
[9] S Avadayappan P Jeyanthi and R Vasuki ldquoSuper magicstrength of a graphrdquo Indian Journal of Pure and Applied Mathe-matics vol 32 no 11 pp 1621ndash1630 2001
[10] A A G Ngurah A N M Salman and L Susilowati ldquo119867-supermagic labelings of graphsrdquo Discrete Mathematics vol 310no 8 pp 1293ndash1300 2010
[11] M Trenkler ldquoSome results on magic graphsrdquo in Graphs andOther Combinatorial Topics M Fieldler Ed vol 59 of TextezurMathematik Band pp 328ndash332 Teubner Leipzig Germany1983
[12] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011
[13] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011
[14] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013
[15] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2000
[16] ANagoorGani andV T ChandrasekaranAFirst Look at FuzzyGraph Theory Allied Publishers Chennai India 2010
[17] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009
[18] S Mathew and M S Sunitha ldquoNode connectivity and arcconnectivity of a fuzzy graphrdquo Information Sciences vol 180 no4 pp 519ndash531 2010
6 Journal of Mathematics
[19] J AMacDougall andWDWallis ldquoStrong edge-magic labellingof a cycle with a chordrdquo The Australasian Journal of Combina-torics vol 28 pp 245ndash255 2003
[20] A Nagoor Gani and D Rajalaxmi (a) Subahashini ldquoPropertiesof fuzzy labeling graphrdquo Applied Mathematical Sciences vol 6no 69-72 pp 3461ndash3466 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Mathematics
009
011007
008
002
001
006 003
010
1 2
34
Figure 1 A fuzzy labeling graph
007
001
005 006 004
1 2 3 4
002 003
Figure 2 A fuzzy magic path graph1198980(119875) = 013
11 Preliminaries Let 119880 and 119881 be two sets Then 120588 is said tobe a fuzzy relation from 119880 into 119881 if 120588 is a fuzzy set of 119880 times 119881A fuzzy graph 119866 = (120590 120583) is a pair of functions 120590 119881 rarr [0 1]and 120583 119881 times 119881 rarr [0 1] where for all 119906 V isin 119881 we have120583(119906 V) le 120590(119906) and 120590(V) A path 119875 in a fuzzy graph is a sequenceof distinct nodes V
1 V2 V
119899such that 120583(V
119894 V119894+1) gt 0 1 le 119894 le
119899 here 119899 ge 1 is called the length of the path119875The consecutivepairs (V
119894 V119894+1) are called the edge of the path A path119875 is called
a cycle if V1= V119899and 119899 ge 3 The strength of a path 119875 is
defined as ⋀119899119894=1120583(V119894 V119894+1) Let 119866 = (120590 120583) be a fuzzy graph
The degree of a vertex V is defined as 119889(V) = sum119906 = V119906isin119881 120583(V 119906)
Let 119866 = (120590 120583) be a fuzzy graph The strong degree of a node Vis defined as the sum ofmembership values of all strong edgesincident at V It is denoted by 119889
119904(V) Also if 119873
119904(V) denote the
set of all strong neighbours of V then 119889119904(V) = sum
119906isin119873119904(V) 120583(V 119906)
An edge is called a fuzzy bridge of119866 if its removal reduces thestrength of connectedness between somepair of nodes in119866 Anode is a fuzzy cut node of 119866 = (120590 120583) if removal of it reducesthe strength of connectedness between some other pairs ofnodes
Definition 1 (see [20]) A graph119866 = (120590 120583) is said to be a fuzzylabeling graph if 120590 119881 rarr [0 1] and 120583 119881 times 119881 rarr [0 1] isbijective such that themembership value of edges and verticesare distinct and 120583(119906 V) lt 120590(119906) and 120590(V) for all 119906 V isin 119881
Example 2 (see [20]) In Figure 1 120590 and 120583 are bijective suchthat no vertices and edges receive the same membershipvalue
Definition 3 (see [20]) A fuzzy labeling graph is said to be afuzzy magic graph if 120590(119906) + 120583(119906 V) + 120590(V) has a same magicvalue for all 119906 V isin 119881 which is denoted as119898
0(119866)
Example 4 (see [20]) In Figure 2 120590(1198811)+120583(119881
1 1198812)+120590(119881
2) =
007 + 001 + 005 = 013 for all 1198811 1198812 isin 119881
Definition 5 A star in a fuzzy graph consists of two node sets119881 and 119880 with |119881| = 1 and |119880| gt 1 such that 120583(V 119906
119894) gt 0 and
120583(119906119894 119906119894+1) = 0 1 le 119894 le 119899 It is denoted by 119878
1119899
Example 6 A fuzzy star graph is shown in Figure 3
Definition 7 (see [20]) The fuzzy labeling graph 119867 = (120591 120588)is called a fuzzy labeling subgraph of119866 = (120590 120583) if 120591(119906) le 120590(119906)for all 119906 isin 119881 and 120588(119906 V) le 120583(119906 V) for all 119906 V isin 119881
2 Properties of Fuzzy Labeling Graphs
Proposition 8 For all 119899 ge 1 the path 119875119899is a fuzzy magic
graph
Proof Let 119875 be any path with length 119899 ge 1 and V1 V2 V
119899
and V1V2 V2V3 V
119899minus1V119899are the nodes and edges of 119875 Let
119911 rarr (0 1] such that one can choose 119911 = 01 if 119899 le 4 and119911 = 001 if 119899 ge 5 Such fuzzy labeling is defined as follows
When length is odd
120590120596 (V2119894minus1
) = (2119899 + 2 minus 119894) 119911 1 le 119894 le119899 + 1
2
120590120596 (V2119894) = min 120590120596 (V
2119894minus1) | 1 le 119894 le
119899 + 1
2 minus 119894 (119911)
1 le 119894 le119899 + 1
2
120583120596 (V119899minus119894+2
V119899+1minus119894
)
= max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1 minus (119894 minus 1) 119911
1 le 119894 le 119899
(1)
Case (i) 119894 is evenThen 119894 = 2119909 for any positive integer 119909 and for each edge
V119894 V119894+1
1198980 (119875) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909) + 120583120596 (V
2119909 V2119909+1
) + 120590120596 (V2119909+1
)
= min 120590120596 (V2119894minus1
) | 1 le 119894 le119899 + 1
2
minus 119909 (119911) +max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minus (119899 minus 2119909) 119911 + (2119899 minus 119909 + 1) 119911
= min 120590120596 (V2119894minus1
) | 1 le 119894 le119899 + 1
2
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1 + (119899 + 1) 119911
(2)
Case (ii) 119894 is oddThen 119894 = 2119909 + 1 for any positive integer 119909 and for each
edge V119894 V119894+1
1198980(119875) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909+1
) + 120583120596 (V2119909+1
V2119909+2
)
+ 120590120596 (V2119909+2
)
Journal of Mathematics 3
V 012
010
007
008
002
003
005
u1
u2
u3
Figure 3 A fuzzy star graph
= (2119899 minus 119909 + 1) 119911
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minus (119899 minus 2119909 minus 1)
+min 120590120596 (V2119894minus1
) | 1 le 119894 le119899 + 1
2
minus (119909 + 1) 119911
= min 120590120596 (V2119894minus1
) | 1 le 119894 le119899 + 1
2
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
+ (119899 + 1) 119911
(3)
When length is even
120590120596 (V2119894) = (2119899 + 2 minus 119894) 119911 1 le 119894 le
119899
2
120590120596 (V2119894minus1
) = min 120590120596 (V2119894) | 1 le 119894 le
119899
2 minus 119894 (119911)
1 le 119894 le119899 + 2
2
120583120596 (V119899minus119894+2
V119899minus119894+1
) = max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minus (119894 minus 1) 119911 1 le 119894 le 119899
(4)
Case (i) 119894 is evenThen 119894 = 2119909 for any positive integer 119909 and for each edge
V119894 V119894+1
1198980 (119875) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909) + 120583120596 (V
2119909 V2119909+1
) + 120590120596 (V2119909+1
)
= (2119899 + 2 minus 119909) 119911
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minus (119899 minus 2119909) 119911 +min 120590120596 (V2119894) | 1 le 119894 le
119899
2
minus (119909 + 1) 119911
= min 120590120596 (V2119894) | 1 le 119894 le
119899
2
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1 + (119899 + 1) 119911
(5)
Case (ii) 119894 is oddThen 119894 = 2119909 + 1 for any positive integer 119909 and for each
edge V119894 V119894+1
1198980 (119875) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909+1
) + 120583120596 (V2119909+1
V2119909+2
) + 120590120596 (V2119909+2
)
= min 120590120596 (V2119894) | 1 le 119894 le
119899
2
minus (119909 + 1) 119911 minus (119899 minus 2119909 minus 1) 119911
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
+ (2119899 minus 119909 + 1) 119911
= min 120590120596 (V2119894) | 1 le 119894 le
119899
2
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1 + (119899 + 1) 119911
(6)
Therefore in both the cases themagic value1198980(119875) is same and
unique Thus 119875119899is fuzzy magic graph for all 119899 ge 1
Proposition 9 If 119899 is odd then the cycle 119862119899is a fuzzy magic
graph
Proof Let 119862119899be any cycle with odd number of nodes and
V1 V2 V
119899and V1V2 V2V3 V
119899V1be the nodes and edges
of 119862119899 Let 119911 rarr (0 1] such that one can choose 119911 = 01 if
119899 le 3 and 119911 = 001 if 119899 ge 4 The fuzzy labeling for cycle isdefined as follows
120590120596 (V2119894) = (2119899 + 1 minus 119894) 119911 1 le 119894 le
119899 minus 1
2
120590120596 (V2119894minus1
) = min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 minus 119894 (119911)
1 le 119894 le119899 + 1
2
120583120596 (V1 V119899) =
1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
120583120596 (V119899minus119894+1
V119899minus119894) = 120583120596 (V
1 V119899) minus 119894 (119911)
1 le 119894 le 119899 minus 1
(7)
4 Journal of Mathematics
Case (i) 119894 is evenThen 119894 = 2119909 for any positive integer 119909 and for each edge
V119894 V119894+1
1198980(119862119899) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909) + 120583120596 (V
2119909 V2119909+1
)
+ 120590120596 (V2119909+1
)
= (2119899 + 1 minus 119909) 119911 +1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
minus (119899 minus 2119909) 119911
+min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2
minus (119909 + 1) 119911
=1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
+min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 + 119899 (119911)
(8)
Case (ii) 119894 is oddThen 119894 = 2119909 + 1 for any positive integer 119909 and for each
edge V119894 V119894+1
1198980(119862119899) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909+1
) + 120583120596 (V2119909+1
V2119909+2
)
+ 120590120596 (V2119909+2
)
= min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 minus (119909 + 1) 119911
+1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
minus (119899 minus 2119909 minus 1) 119911 + (2119899 minus 119909) 119911
=1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
+min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 + 119899 (119911)
(9)
Therefore from above cases 119862119899is a fuzzy magic graph if 119899 is
odd
Proposition 10 For any 119899 ge 2 star 1198781119899
is a fuzzymagic graph
Proof Let 1198781119899
be a star graph with V 1199061 1199062 119906
119899as nodes
and V1199061 V1199062 V119906
119899as edges
Let 119911 rarr (0 1] such that one can choose 119911 = 01 if 119899 le 4and 119911 = 001 if 119899 ge 5 Such a fuzzy labeling is defined asfollows
120590120596 (119906119894) = [2 (119899 + 1) minus 119894] 119911 1 le 119894 le 119899
120590120596 (V) = min 120590120596 (119906119894) | 1 le 119894 le 119899 minus 119911
120583120596 (V 119906119899minus119894) = max 120590120596 (119906
119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minus 119894 (119911) 0 le 119894 le 119899 minus 1
(10)
Case (i) 119894 is evenThen 119894 = 2119909 for any positive integer 119909 and for each edge
V 119906119894
1198980(1198781119899) = 120590120596 (V) + 120583120596 (V 119906119894) + 120590
120596 (119906119894)
= 120590120596 (V) + 120583120596 (V 1199062119909) + 120590120596 (119906
2119909)
= min 120590120596 (119906119894) | 1 le 119894 le 119899 minus (119911)
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minus (119899 minus 2119909) 119911 + [2 (119899 + 1) minus 2119909] 119911
= min 120590120596 (119906119894) | 1 le 119894 le 119899
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
+ (119899 + 1) 119911
(11)
Case (ii) 119894 is oddThen 119894 = 2119909 + 1 for any positive integer 119909 and for each
edge V 119906119894
1198980(1198781119899) = 120590120596 (V) + 120583120596 (V 119906119894) + 120590
120596 (119906119894)
= 120590120596 (V) + 120583120596 (V 1199062119909+1
) + 120590120596 (1199062119909+2
)
= min 120590120596 (119906119894) | 1 le 119894 le 119899 minus 119911
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minus (119899 minus 2119909 minus 1) 119911 + [2 (119899 minus 119909)] 119911
= min 120590120596 (119906119894) | 1 le 119894 le 119899
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
+ (119899 + 1) 119911
(12)
From the above cases one can easily verify that all star graphsare fuzzy magic graphs
Remark 11 One can observe the same labeling holds well ifwe choose the value of 119911 as 003 005 and so forth for thePropositions 8 9 and 10
Remark 12 (1) If 119866 is a fuzzy magic graph then 119889(119906) = 119889(V)for any pair of nodes 119906 and V
(2) For any fuzzy magic graph 0 le 119889119904(V) le 119889(V)
(3) Sum of the degree of all nodes in a fuzzy magic graphis equal to twice the sum of membership values of all edges(ie sum119899
119894=1119889(V119894) = 2sum
119906 = V 120583(119906 V))(4) Sum of strong degree of all nodes in a fuzzy magic
graph is equal to twice the sum of the membership values ofall strong arcs in 119866 (ie sum119899
119894=1119889119904(V119894) = 2sum
119906isin119873119904(V) 120583(V 119906))
Journal of Mathematics 5
3 Properties of Fuzzy Magic Graphs
Proposition 13 Every fuzzy magic graph is a fuzzy labelinggraph but the converse is not true
Proof This is immediate from Definition 3
Proposition 14 For every fuzzy magic graph119866 there exists atleast one fuzzy bridge
Proof Let 119866 be a fuzzy magic graph such that there existsonly one edge 120583(119909 119910) with maximum value since 120583 is bijec-tive Nowwe claim that 120583(119909 119910) is a fuzzy bridge If we removethe edge (119909 119910) from 119866 then in its subgraph we have 1205831015840infin(119909119910) lt 120583(119909 119910) which implies (119909 119910) is a fuzzy bridge
Proposition 15 Removal of a fuzzy cut node from a fuzzymagic path 119875 is also a fuzzy magic graph
Proof Let 119875 be any fuzzy magic path with length 119899 Thenthere must be a fuzzy cut node if we remove that cut nodefrom 119875 then it either becomes a smaller path or disconnectedpath anyway it remains to be a path with odd or even lengthby Proposition 8 it is concluded that removal of a fuzzycut node from a fuzzy magic path 119875 is also a fuzzy magicgraph
Proposition 16 When 119899 is odd removal of a fuzzy bridge froma fuzzy magic cycle 119862
119899is a fuzzy magic graph
Proof Let 119862119899be any fuzzy magic cycle with odd nodes If we
choose any path (119906 V) then there must be at least one fuzzybridge whose removal from 119862
119899will result as a path of odd or
even length By Proposition 8 the removal of a fuzzy bridgefrom a fuzzy magic cycle 119862
119899is also a fuzzy magic graph
Remark 17 (1) Removal of a fuzzy cut node from the cycle119862119899
is also a fuzzy magic graph(2) For all fuzzy magic cycles 119862
119899with odd nodes there
exists at least one pair of nodes 119906 and V such that 119889119904(119906) =
119889119904(V)
Proposition 18 Removal of a fuzzy bridge from a fuzzymagiccycle 119862
119899will reduce the strength of the fuzzy magic cycle 119862
119899
Proof Let 119862119899be a fuzzy magic cycle with odd number of
nodes Now choose any path (119906 V) from 119862119899 and then it
is obvious that there exists at least one fuzzy bridge (119909 119910)Removal of this fuzzy bridge (119909 119910)will reduce the strength ofconnectedness between 119906 and VThis implies that the removalof fuzzy bridge from the fuzzy magic cycle 119862
119899will reduce its
strength
4 Concluding Remarks
Fuzzy graph theory is finding an increasing number ofapplications in modeling real time systems where the level ofinformation inherent in the system varies with different levelsof precision Fuzzy models are becoming useful because oftheir aim in reducing the differences between the traditional
numerical models used in engineering and sciences and thesymbolic models used in expert systems In this paper theconcept of fuzzy labeling and fuzzymagic labeling graphs hasbeen introduced We plan to extend our research work to (1)bipolar fuzzy labeling and bipolar fuzzymagic labeling graphsand (2) fuzzy labeling and fuzzy magic labeling hypergraphs
Conflict of Interests
The authors declare that they do not have any conflict ofinterests regarding the publication of this paper
References
[1] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975
[2] K R Bhutani and A Battou ldquoOn 119872-strong fuzzy graphsrdquoInformation Sciences vol 155 no 1-2 pp 103ndash109 2003
[3] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987
[4] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002
[5] B M Stewart ldquoMagic graphsrdquo Canadian Journal of Mathemat-ics vol 18 pp 1031ndash1059 1966
[6] BM Stewart ldquoSupermagic complete graphsrdquoCanadian Journalof Mathematics vol 9 pp 427ndash438 1966
[7] A Kotzig and A Rosa ldquoMagic valuations of finite graphsrdquoCanadian Mathematical Bulletin vol 13 pp 451ndash461 1970
[8] H Enomoto A S Llado T Nakamigawa and G Ringel ldquoSuperedge-magic graphsrdquo SUT Journal of Mathematics vol 34 no 2pp 105ndash109 1998
[9] S Avadayappan P Jeyanthi and R Vasuki ldquoSuper magicstrength of a graphrdquo Indian Journal of Pure and Applied Mathe-matics vol 32 no 11 pp 1621ndash1630 2001
[10] A A G Ngurah A N M Salman and L Susilowati ldquo119867-supermagic labelings of graphsrdquo Discrete Mathematics vol 310no 8 pp 1293ndash1300 2010
[11] M Trenkler ldquoSome results on magic graphsrdquo in Graphs andOther Combinatorial Topics M Fieldler Ed vol 59 of TextezurMathematik Band pp 328ndash332 Teubner Leipzig Germany1983
[12] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011
[13] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011
[14] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013
[15] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2000
[16] ANagoorGani andV T ChandrasekaranAFirst Look at FuzzyGraph Theory Allied Publishers Chennai India 2010
[17] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009
[18] S Mathew and M S Sunitha ldquoNode connectivity and arcconnectivity of a fuzzy graphrdquo Information Sciences vol 180 no4 pp 519ndash531 2010
6 Journal of Mathematics
[19] J AMacDougall andWDWallis ldquoStrong edge-magic labellingof a cycle with a chordrdquo The Australasian Journal of Combina-torics vol 28 pp 245ndash255 2003
[20] A Nagoor Gani and D Rajalaxmi (a) Subahashini ldquoPropertiesof fuzzy labeling graphrdquo Applied Mathematical Sciences vol 6no 69-72 pp 3461ndash3466 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 3
V 012
010
007
008
002
003
005
u1
u2
u3
Figure 3 A fuzzy star graph
= (2119899 minus 119909 + 1) 119911
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minus (119899 minus 2119909 minus 1)
+min 120590120596 (V2119894minus1
) | 1 le 119894 le119899 + 1
2
minus (119909 + 1) 119911
= min 120590120596 (V2119894minus1
) | 1 le 119894 le119899 + 1
2
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
+ (119899 + 1) 119911
(3)
When length is even
120590120596 (V2119894) = (2119899 + 2 minus 119894) 119911 1 le 119894 le
119899
2
120590120596 (V2119894minus1
) = min 120590120596 (V2119894) | 1 le 119894 le
119899
2 minus 119894 (119911)
1 le 119894 le119899 + 2
2
120583120596 (V119899minus119894+2
V119899minus119894+1
) = max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minus (119894 minus 1) 119911 1 le 119894 le 119899
(4)
Case (i) 119894 is evenThen 119894 = 2119909 for any positive integer 119909 and for each edge
V119894 V119894+1
1198980 (119875) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909) + 120583120596 (V
2119909 V2119909+1
) + 120590120596 (V2119909+1
)
= (2119899 + 2 minus 119909) 119911
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minus (119899 minus 2119909) 119911 +min 120590120596 (V2119894) | 1 le 119894 le
119899
2
minus (119909 + 1) 119911
= min 120590120596 (V2119894) | 1 le 119894 le
119899
2
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1 + (119899 + 1) 119911
(5)
Case (ii) 119894 is oddThen 119894 = 2119909 + 1 for any positive integer 119909 and for each
edge V119894 V119894+1
1198980 (119875) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909+1
) + 120583120596 (V2119909+1
V2119909+2
) + 120590120596 (V2119909+2
)
= min 120590120596 (V2119894) | 1 le 119894 le
119899
2
minus (119909 + 1) 119911 minus (119899 minus 2119909 minus 1) 119911
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1
+ (2119899 minus 119909 + 1) 119911
= min 120590120596 (V2119894) | 1 le 119894 le
119899
2
+max 120590120596 (V119894) | 1 le 119894 le 119899 + 1
minusmin 120590120596 (V119894) | 1 le 119894 le 119899 + 1 + (119899 + 1) 119911
(6)
Therefore in both the cases themagic value1198980(119875) is same and
unique Thus 119875119899is fuzzy magic graph for all 119899 ge 1
Proposition 9 If 119899 is odd then the cycle 119862119899is a fuzzy magic
graph
Proof Let 119862119899be any cycle with odd number of nodes and
V1 V2 V
119899and V1V2 V2V3 V
119899V1be the nodes and edges
of 119862119899 Let 119911 rarr (0 1] such that one can choose 119911 = 01 if
119899 le 3 and 119911 = 001 if 119899 ge 4 The fuzzy labeling for cycle isdefined as follows
120590120596 (V2119894) = (2119899 + 1 minus 119894) 119911 1 le 119894 le
119899 minus 1
2
120590120596 (V2119894minus1
) = min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 minus 119894 (119911)
1 le 119894 le119899 + 1
2
120583120596 (V1 V119899) =
1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
120583120596 (V119899minus119894+1
V119899minus119894) = 120583120596 (V
1 V119899) minus 119894 (119911)
1 le 119894 le 119899 minus 1
(7)
4 Journal of Mathematics
Case (i) 119894 is evenThen 119894 = 2119909 for any positive integer 119909 and for each edge
V119894 V119894+1
1198980(119862119899) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909) + 120583120596 (V
2119909 V2119909+1
)
+ 120590120596 (V2119909+1
)
= (2119899 + 1 minus 119909) 119911 +1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
minus (119899 minus 2119909) 119911
+min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2
minus (119909 + 1) 119911
=1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
+min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 + 119899 (119911)
(8)
Case (ii) 119894 is oddThen 119894 = 2119909 + 1 for any positive integer 119909 and for each
edge V119894 V119894+1
1198980(119862119899) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909+1
) + 120583120596 (V2119909+1
V2119909+2
)
+ 120590120596 (V2119909+2
)
= min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 minus (119909 + 1) 119911
+1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
minus (119899 minus 2119909 minus 1) 119911 + (2119899 minus 119909) 119911
=1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
+min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 + 119899 (119911)
(9)
Therefore from above cases 119862119899is a fuzzy magic graph if 119899 is
odd
Proposition 10 For any 119899 ge 2 star 1198781119899
is a fuzzymagic graph
Proof Let 1198781119899
be a star graph with V 1199061 1199062 119906
119899as nodes
and V1199061 V1199062 V119906
119899as edges
Let 119911 rarr (0 1] such that one can choose 119911 = 01 if 119899 le 4and 119911 = 001 if 119899 ge 5 Such a fuzzy labeling is defined asfollows
120590120596 (119906119894) = [2 (119899 + 1) minus 119894] 119911 1 le 119894 le 119899
120590120596 (V) = min 120590120596 (119906119894) | 1 le 119894 le 119899 minus 119911
120583120596 (V 119906119899minus119894) = max 120590120596 (119906
119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minus 119894 (119911) 0 le 119894 le 119899 minus 1
(10)
Case (i) 119894 is evenThen 119894 = 2119909 for any positive integer 119909 and for each edge
V 119906119894
1198980(1198781119899) = 120590120596 (V) + 120583120596 (V 119906119894) + 120590
120596 (119906119894)
= 120590120596 (V) + 120583120596 (V 1199062119909) + 120590120596 (119906
2119909)
= min 120590120596 (119906119894) | 1 le 119894 le 119899 minus (119911)
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minus (119899 minus 2119909) 119911 + [2 (119899 + 1) minus 2119909] 119911
= min 120590120596 (119906119894) | 1 le 119894 le 119899
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
+ (119899 + 1) 119911
(11)
Case (ii) 119894 is oddThen 119894 = 2119909 + 1 for any positive integer 119909 and for each
edge V 119906119894
1198980(1198781119899) = 120590120596 (V) + 120583120596 (V 119906119894) + 120590
120596 (119906119894)
= 120590120596 (V) + 120583120596 (V 1199062119909+1
) + 120590120596 (1199062119909+2
)
= min 120590120596 (119906119894) | 1 le 119894 le 119899 minus 119911
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minus (119899 minus 2119909 minus 1) 119911 + [2 (119899 minus 119909)] 119911
= min 120590120596 (119906119894) | 1 le 119894 le 119899
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
+ (119899 + 1) 119911
(12)
From the above cases one can easily verify that all star graphsare fuzzy magic graphs
Remark 11 One can observe the same labeling holds well ifwe choose the value of 119911 as 003 005 and so forth for thePropositions 8 9 and 10
Remark 12 (1) If 119866 is a fuzzy magic graph then 119889(119906) = 119889(V)for any pair of nodes 119906 and V
(2) For any fuzzy magic graph 0 le 119889119904(V) le 119889(V)
(3) Sum of the degree of all nodes in a fuzzy magic graphis equal to twice the sum of membership values of all edges(ie sum119899
119894=1119889(V119894) = 2sum
119906 = V 120583(119906 V))(4) Sum of strong degree of all nodes in a fuzzy magic
graph is equal to twice the sum of the membership values ofall strong arcs in 119866 (ie sum119899
119894=1119889119904(V119894) = 2sum
119906isin119873119904(V) 120583(V 119906))
Journal of Mathematics 5
3 Properties of Fuzzy Magic Graphs
Proposition 13 Every fuzzy magic graph is a fuzzy labelinggraph but the converse is not true
Proof This is immediate from Definition 3
Proposition 14 For every fuzzy magic graph119866 there exists atleast one fuzzy bridge
Proof Let 119866 be a fuzzy magic graph such that there existsonly one edge 120583(119909 119910) with maximum value since 120583 is bijec-tive Nowwe claim that 120583(119909 119910) is a fuzzy bridge If we removethe edge (119909 119910) from 119866 then in its subgraph we have 1205831015840infin(119909119910) lt 120583(119909 119910) which implies (119909 119910) is a fuzzy bridge
Proposition 15 Removal of a fuzzy cut node from a fuzzymagic path 119875 is also a fuzzy magic graph
Proof Let 119875 be any fuzzy magic path with length 119899 Thenthere must be a fuzzy cut node if we remove that cut nodefrom 119875 then it either becomes a smaller path or disconnectedpath anyway it remains to be a path with odd or even lengthby Proposition 8 it is concluded that removal of a fuzzycut node from a fuzzy magic path 119875 is also a fuzzy magicgraph
Proposition 16 When 119899 is odd removal of a fuzzy bridge froma fuzzy magic cycle 119862
119899is a fuzzy magic graph
Proof Let 119862119899be any fuzzy magic cycle with odd nodes If we
choose any path (119906 V) then there must be at least one fuzzybridge whose removal from 119862
119899will result as a path of odd or
even length By Proposition 8 the removal of a fuzzy bridgefrom a fuzzy magic cycle 119862
119899is also a fuzzy magic graph
Remark 17 (1) Removal of a fuzzy cut node from the cycle119862119899
is also a fuzzy magic graph(2) For all fuzzy magic cycles 119862
119899with odd nodes there
exists at least one pair of nodes 119906 and V such that 119889119904(119906) =
119889119904(V)
Proposition 18 Removal of a fuzzy bridge from a fuzzymagiccycle 119862
119899will reduce the strength of the fuzzy magic cycle 119862
119899
Proof Let 119862119899be a fuzzy magic cycle with odd number of
nodes Now choose any path (119906 V) from 119862119899 and then it
is obvious that there exists at least one fuzzy bridge (119909 119910)Removal of this fuzzy bridge (119909 119910)will reduce the strength ofconnectedness between 119906 and VThis implies that the removalof fuzzy bridge from the fuzzy magic cycle 119862
119899will reduce its
strength
4 Concluding Remarks
Fuzzy graph theory is finding an increasing number ofapplications in modeling real time systems where the level ofinformation inherent in the system varies with different levelsof precision Fuzzy models are becoming useful because oftheir aim in reducing the differences between the traditional
numerical models used in engineering and sciences and thesymbolic models used in expert systems In this paper theconcept of fuzzy labeling and fuzzymagic labeling graphs hasbeen introduced We plan to extend our research work to (1)bipolar fuzzy labeling and bipolar fuzzymagic labeling graphsand (2) fuzzy labeling and fuzzy magic labeling hypergraphs
Conflict of Interests
The authors declare that they do not have any conflict ofinterests regarding the publication of this paper
References
[1] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975
[2] K R Bhutani and A Battou ldquoOn 119872-strong fuzzy graphsrdquoInformation Sciences vol 155 no 1-2 pp 103ndash109 2003
[3] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987
[4] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002
[5] B M Stewart ldquoMagic graphsrdquo Canadian Journal of Mathemat-ics vol 18 pp 1031ndash1059 1966
[6] BM Stewart ldquoSupermagic complete graphsrdquoCanadian Journalof Mathematics vol 9 pp 427ndash438 1966
[7] A Kotzig and A Rosa ldquoMagic valuations of finite graphsrdquoCanadian Mathematical Bulletin vol 13 pp 451ndash461 1970
[8] H Enomoto A S Llado T Nakamigawa and G Ringel ldquoSuperedge-magic graphsrdquo SUT Journal of Mathematics vol 34 no 2pp 105ndash109 1998
[9] S Avadayappan P Jeyanthi and R Vasuki ldquoSuper magicstrength of a graphrdquo Indian Journal of Pure and Applied Mathe-matics vol 32 no 11 pp 1621ndash1630 2001
[10] A A G Ngurah A N M Salman and L Susilowati ldquo119867-supermagic labelings of graphsrdquo Discrete Mathematics vol 310no 8 pp 1293ndash1300 2010
[11] M Trenkler ldquoSome results on magic graphsrdquo in Graphs andOther Combinatorial Topics M Fieldler Ed vol 59 of TextezurMathematik Band pp 328ndash332 Teubner Leipzig Germany1983
[12] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011
[13] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011
[14] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013
[15] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2000
[16] ANagoorGani andV T ChandrasekaranAFirst Look at FuzzyGraph Theory Allied Publishers Chennai India 2010
[17] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009
[18] S Mathew and M S Sunitha ldquoNode connectivity and arcconnectivity of a fuzzy graphrdquo Information Sciences vol 180 no4 pp 519ndash531 2010
6 Journal of Mathematics
[19] J AMacDougall andWDWallis ldquoStrong edge-magic labellingof a cycle with a chordrdquo The Australasian Journal of Combina-torics vol 28 pp 245ndash255 2003
[20] A Nagoor Gani and D Rajalaxmi (a) Subahashini ldquoPropertiesof fuzzy labeling graphrdquo Applied Mathematical Sciences vol 6no 69-72 pp 3461ndash3466 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Mathematics
Case (i) 119894 is evenThen 119894 = 2119909 for any positive integer 119909 and for each edge
V119894 V119894+1
1198980(119862119899) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909) + 120583120596 (V
2119909 V2119909+1
)
+ 120590120596 (V2119909+1
)
= (2119899 + 1 minus 119909) 119911 +1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
minus (119899 minus 2119909) 119911
+min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2
minus (119909 + 1) 119911
=1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
+min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 + 119899 (119911)
(8)
Case (ii) 119894 is oddThen 119894 = 2119909 + 1 for any positive integer 119909 and for each
edge V119894 V119894+1
1198980(119862119899) = 120590120596 (V
119894) + 120583120596 (V
119894 V119894+1) + 120590120596 (V
119894+1)
= 120590120596 (V2119909+1
) + 120583120596 (V2119909+1
V2119909+2
)
+ 120590120596 (V2119909+2
)
= min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 minus (119909 + 1) 119911
+1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
minus (119899 minus 2119909 minus 1) 119911 + (2119899 minus 119909) 119911
=1
2max 120590120596 (V
119894) | 1 le 119894 le 119899
+min 120590120596 (V2119894) | 1 le 119894 le
119899 minus 1
2 + 119899 (119911)
(9)
Therefore from above cases 119862119899is a fuzzy magic graph if 119899 is
odd
Proposition 10 For any 119899 ge 2 star 1198781119899
is a fuzzymagic graph
Proof Let 1198781119899
be a star graph with V 1199061 1199062 119906
119899as nodes
and V1199061 V1199062 V119906
119899as edges
Let 119911 rarr (0 1] such that one can choose 119911 = 01 if 119899 le 4and 119911 = 001 if 119899 ge 5 Such a fuzzy labeling is defined asfollows
120590120596 (119906119894) = [2 (119899 + 1) minus 119894] 119911 1 le 119894 le 119899
120590120596 (V) = min 120590120596 (119906119894) | 1 le 119894 le 119899 minus 119911
120583120596 (V 119906119899minus119894) = max 120590120596 (119906
119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minus 119894 (119911) 0 le 119894 le 119899 minus 1
(10)
Case (i) 119894 is evenThen 119894 = 2119909 for any positive integer 119909 and for each edge
V 119906119894
1198980(1198781119899) = 120590120596 (V) + 120583120596 (V 119906119894) + 120590
120596 (119906119894)
= 120590120596 (V) + 120583120596 (V 1199062119909) + 120590120596 (119906
2119909)
= min 120590120596 (119906119894) | 1 le 119894 le 119899 minus (119911)
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minus (119899 minus 2119909) 119911 + [2 (119899 + 1) minus 2119909] 119911
= min 120590120596 (119906119894) | 1 le 119894 le 119899
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
+ (119899 + 1) 119911
(11)
Case (ii) 119894 is oddThen 119894 = 2119909 + 1 for any positive integer 119909 and for each
edge V 119906119894
1198980(1198781119899) = 120590120596 (V) + 120583120596 (V 119906119894) + 120590
120596 (119906119894)
= 120590120596 (V) + 120583120596 (V 1199062119909+1
) + 120590120596 (1199062119909+2
)
= min 120590120596 (119906119894) | 1 le 119894 le 119899 minus 119911
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minus (119899 minus 2119909 minus 1) 119911 + [2 (119899 minus 119909)] 119911
= min 120590120596 (119906119894) | 1 le 119894 le 119899
+max 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
minusmin 120590120596 (119906119894) 120590120596 (V) | 1 le 119894 le 119899
+ (119899 + 1) 119911
(12)
From the above cases one can easily verify that all star graphsare fuzzy magic graphs
Remark 11 One can observe the same labeling holds well ifwe choose the value of 119911 as 003 005 and so forth for thePropositions 8 9 and 10
Remark 12 (1) If 119866 is a fuzzy magic graph then 119889(119906) = 119889(V)for any pair of nodes 119906 and V
(2) For any fuzzy magic graph 0 le 119889119904(V) le 119889(V)
(3) Sum of the degree of all nodes in a fuzzy magic graphis equal to twice the sum of membership values of all edges(ie sum119899
119894=1119889(V119894) = 2sum
119906 = V 120583(119906 V))(4) Sum of strong degree of all nodes in a fuzzy magic
graph is equal to twice the sum of the membership values ofall strong arcs in 119866 (ie sum119899
119894=1119889119904(V119894) = 2sum
119906isin119873119904(V) 120583(V 119906))
Journal of Mathematics 5
3 Properties of Fuzzy Magic Graphs
Proposition 13 Every fuzzy magic graph is a fuzzy labelinggraph but the converse is not true
Proof This is immediate from Definition 3
Proposition 14 For every fuzzy magic graph119866 there exists atleast one fuzzy bridge
Proof Let 119866 be a fuzzy magic graph such that there existsonly one edge 120583(119909 119910) with maximum value since 120583 is bijec-tive Nowwe claim that 120583(119909 119910) is a fuzzy bridge If we removethe edge (119909 119910) from 119866 then in its subgraph we have 1205831015840infin(119909119910) lt 120583(119909 119910) which implies (119909 119910) is a fuzzy bridge
Proposition 15 Removal of a fuzzy cut node from a fuzzymagic path 119875 is also a fuzzy magic graph
Proof Let 119875 be any fuzzy magic path with length 119899 Thenthere must be a fuzzy cut node if we remove that cut nodefrom 119875 then it either becomes a smaller path or disconnectedpath anyway it remains to be a path with odd or even lengthby Proposition 8 it is concluded that removal of a fuzzycut node from a fuzzy magic path 119875 is also a fuzzy magicgraph
Proposition 16 When 119899 is odd removal of a fuzzy bridge froma fuzzy magic cycle 119862
119899is a fuzzy magic graph
Proof Let 119862119899be any fuzzy magic cycle with odd nodes If we
choose any path (119906 V) then there must be at least one fuzzybridge whose removal from 119862
119899will result as a path of odd or
even length By Proposition 8 the removal of a fuzzy bridgefrom a fuzzy magic cycle 119862
119899is also a fuzzy magic graph
Remark 17 (1) Removal of a fuzzy cut node from the cycle119862119899
is also a fuzzy magic graph(2) For all fuzzy magic cycles 119862
119899with odd nodes there
exists at least one pair of nodes 119906 and V such that 119889119904(119906) =
119889119904(V)
Proposition 18 Removal of a fuzzy bridge from a fuzzymagiccycle 119862
119899will reduce the strength of the fuzzy magic cycle 119862
119899
Proof Let 119862119899be a fuzzy magic cycle with odd number of
nodes Now choose any path (119906 V) from 119862119899 and then it
is obvious that there exists at least one fuzzy bridge (119909 119910)Removal of this fuzzy bridge (119909 119910)will reduce the strength ofconnectedness between 119906 and VThis implies that the removalof fuzzy bridge from the fuzzy magic cycle 119862
119899will reduce its
strength
4 Concluding Remarks
Fuzzy graph theory is finding an increasing number ofapplications in modeling real time systems where the level ofinformation inherent in the system varies with different levelsof precision Fuzzy models are becoming useful because oftheir aim in reducing the differences between the traditional
numerical models used in engineering and sciences and thesymbolic models used in expert systems In this paper theconcept of fuzzy labeling and fuzzymagic labeling graphs hasbeen introduced We plan to extend our research work to (1)bipolar fuzzy labeling and bipolar fuzzymagic labeling graphsand (2) fuzzy labeling and fuzzy magic labeling hypergraphs
Conflict of Interests
The authors declare that they do not have any conflict ofinterests regarding the publication of this paper
References
[1] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975
[2] K R Bhutani and A Battou ldquoOn 119872-strong fuzzy graphsrdquoInformation Sciences vol 155 no 1-2 pp 103ndash109 2003
[3] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987
[4] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002
[5] B M Stewart ldquoMagic graphsrdquo Canadian Journal of Mathemat-ics vol 18 pp 1031ndash1059 1966
[6] BM Stewart ldquoSupermagic complete graphsrdquoCanadian Journalof Mathematics vol 9 pp 427ndash438 1966
[7] A Kotzig and A Rosa ldquoMagic valuations of finite graphsrdquoCanadian Mathematical Bulletin vol 13 pp 451ndash461 1970
[8] H Enomoto A S Llado T Nakamigawa and G Ringel ldquoSuperedge-magic graphsrdquo SUT Journal of Mathematics vol 34 no 2pp 105ndash109 1998
[9] S Avadayappan P Jeyanthi and R Vasuki ldquoSuper magicstrength of a graphrdquo Indian Journal of Pure and Applied Mathe-matics vol 32 no 11 pp 1621ndash1630 2001
[10] A A G Ngurah A N M Salman and L Susilowati ldquo119867-supermagic labelings of graphsrdquo Discrete Mathematics vol 310no 8 pp 1293ndash1300 2010
[11] M Trenkler ldquoSome results on magic graphsrdquo in Graphs andOther Combinatorial Topics M Fieldler Ed vol 59 of TextezurMathematik Band pp 328ndash332 Teubner Leipzig Germany1983
[12] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011
[13] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011
[14] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013
[15] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2000
[16] ANagoorGani andV T ChandrasekaranAFirst Look at FuzzyGraph Theory Allied Publishers Chennai India 2010
[17] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009
[18] S Mathew and M S Sunitha ldquoNode connectivity and arcconnectivity of a fuzzy graphrdquo Information Sciences vol 180 no4 pp 519ndash531 2010
6 Journal of Mathematics
[19] J AMacDougall andWDWallis ldquoStrong edge-magic labellingof a cycle with a chordrdquo The Australasian Journal of Combina-torics vol 28 pp 245ndash255 2003
[20] A Nagoor Gani and D Rajalaxmi (a) Subahashini ldquoPropertiesof fuzzy labeling graphrdquo Applied Mathematical Sciences vol 6no 69-72 pp 3461ndash3466 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 5
3 Properties of Fuzzy Magic Graphs
Proposition 13 Every fuzzy magic graph is a fuzzy labelinggraph but the converse is not true
Proof This is immediate from Definition 3
Proposition 14 For every fuzzy magic graph119866 there exists atleast one fuzzy bridge
Proof Let 119866 be a fuzzy magic graph such that there existsonly one edge 120583(119909 119910) with maximum value since 120583 is bijec-tive Nowwe claim that 120583(119909 119910) is a fuzzy bridge If we removethe edge (119909 119910) from 119866 then in its subgraph we have 1205831015840infin(119909119910) lt 120583(119909 119910) which implies (119909 119910) is a fuzzy bridge
Proposition 15 Removal of a fuzzy cut node from a fuzzymagic path 119875 is also a fuzzy magic graph
Proof Let 119875 be any fuzzy magic path with length 119899 Thenthere must be a fuzzy cut node if we remove that cut nodefrom 119875 then it either becomes a smaller path or disconnectedpath anyway it remains to be a path with odd or even lengthby Proposition 8 it is concluded that removal of a fuzzycut node from a fuzzy magic path 119875 is also a fuzzy magicgraph
Proposition 16 When 119899 is odd removal of a fuzzy bridge froma fuzzy magic cycle 119862
119899is a fuzzy magic graph
Proof Let 119862119899be any fuzzy magic cycle with odd nodes If we
choose any path (119906 V) then there must be at least one fuzzybridge whose removal from 119862
119899will result as a path of odd or
even length By Proposition 8 the removal of a fuzzy bridgefrom a fuzzy magic cycle 119862
119899is also a fuzzy magic graph
Remark 17 (1) Removal of a fuzzy cut node from the cycle119862119899
is also a fuzzy magic graph(2) For all fuzzy magic cycles 119862
119899with odd nodes there
exists at least one pair of nodes 119906 and V such that 119889119904(119906) =
119889119904(V)
Proposition 18 Removal of a fuzzy bridge from a fuzzymagiccycle 119862
119899will reduce the strength of the fuzzy magic cycle 119862
119899
Proof Let 119862119899be a fuzzy magic cycle with odd number of
nodes Now choose any path (119906 V) from 119862119899 and then it
is obvious that there exists at least one fuzzy bridge (119909 119910)Removal of this fuzzy bridge (119909 119910)will reduce the strength ofconnectedness between 119906 and VThis implies that the removalof fuzzy bridge from the fuzzy magic cycle 119862
119899will reduce its
strength
4 Concluding Remarks
Fuzzy graph theory is finding an increasing number ofapplications in modeling real time systems where the level ofinformation inherent in the system varies with different levelsof precision Fuzzy models are becoming useful because oftheir aim in reducing the differences between the traditional
numerical models used in engineering and sciences and thesymbolic models used in expert systems In this paper theconcept of fuzzy labeling and fuzzymagic labeling graphs hasbeen introduced We plan to extend our research work to (1)bipolar fuzzy labeling and bipolar fuzzymagic labeling graphsand (2) fuzzy labeling and fuzzy magic labeling hypergraphs
Conflict of Interests
The authors declare that they do not have any conflict ofinterests regarding the publication of this paper
References
[1] A Rosenfeld ldquoFuzzy graphsrdquo in Fuzzy Sets and Their Applica-tions L A Zadeh K S Fu and M Shimura Eds pp 77ndash95Academic Press New York NY USA 1975
[2] K R Bhutani and A Battou ldquoOn 119872-strong fuzzy graphsrdquoInformation Sciences vol 155 no 1-2 pp 103ndash109 2003
[3] P Bhattacharya ldquoSome remarks on fuzzy graphsrdquo PatternRecognition Letters vol 6 no 5 pp 297ndash302 1987
[4] M S Sunitha and A Vijaya Kumar ldquoComplement of a fuzzygraphrdquo Indian Journal of Pure and Applied Mathematics vol 33no 9 pp 1451ndash1464 2002
[5] B M Stewart ldquoMagic graphsrdquo Canadian Journal of Mathemat-ics vol 18 pp 1031ndash1059 1966
[6] BM Stewart ldquoSupermagic complete graphsrdquoCanadian Journalof Mathematics vol 9 pp 427ndash438 1966
[7] A Kotzig and A Rosa ldquoMagic valuations of finite graphsrdquoCanadian Mathematical Bulletin vol 13 pp 451ndash461 1970
[8] H Enomoto A S Llado T Nakamigawa and G Ringel ldquoSuperedge-magic graphsrdquo SUT Journal of Mathematics vol 34 no 2pp 105ndash109 1998
[9] S Avadayappan P Jeyanthi and R Vasuki ldquoSuper magicstrength of a graphrdquo Indian Journal of Pure and Applied Mathe-matics vol 32 no 11 pp 1621ndash1630 2001
[10] A A G Ngurah A N M Salman and L Susilowati ldquo119867-supermagic labelings of graphsrdquo Discrete Mathematics vol 310no 8 pp 1293ndash1300 2010
[11] M Trenkler ldquoSome results on magic graphsrdquo in Graphs andOther Combinatorial Topics M Fieldler Ed vol 59 of TextezurMathematik Band pp 328ndash332 Teubner Leipzig Germany1983
[12] MAkram ldquoBipolar fuzzy graphsrdquo Information Sciences vol 181no 24 pp 5548ndash5564 2011
[13] M Akram and W A Dudek ldquoInterval-valued fuzzy graphsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp289ndash299 2011
[14] M Akram and W A Dudek ldquoIntuitionistic fuzzy hypergraphswith applicationsrdquo Information Sciences vol 218 pp 182ndash1932013
[15] J N Mordeson and P S Nair Fuzzy Graphs and FuzzyHypergraphs Physica Heidelberg Germany 2000
[16] ANagoorGani andV T ChandrasekaranAFirst Look at FuzzyGraph Theory Allied Publishers Chennai India 2010
[17] S Mathew and M S Sunitha ldquoTypes of arcs in a fuzzy graphrdquoInformation Sciences vol 179 no 11 pp 1760ndash1768 2009
[18] S Mathew and M S Sunitha ldquoNode connectivity and arcconnectivity of a fuzzy graphrdquo Information Sciences vol 180 no4 pp 519ndash531 2010
6 Journal of Mathematics
[19] J AMacDougall andWDWallis ldquoStrong edge-magic labellingof a cycle with a chordrdquo The Australasian Journal of Combina-torics vol 28 pp 245ndash255 2003
[20] A Nagoor Gani and D Rajalaxmi (a) Subahashini ldquoPropertiesof fuzzy labeling graphrdquo Applied Mathematical Sciences vol 6no 69-72 pp 3461ndash3466 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Mathematics
[19] J AMacDougall andWDWallis ldquoStrong edge-magic labellingof a cycle with a chordrdquo The Australasian Journal of Combina-torics vol 28 pp 245ndash255 2003
[20] A Nagoor Gani and D Rajalaxmi (a) Subahashini ldquoPropertiesof fuzzy labeling graphrdquo Applied Mathematical Sciences vol 6no 69-72 pp 3461ndash3466 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of