Algorithm for Edge Antimagic Labeling for Specific Classes of Graphs

Nissankara Lakshmi Prasanna & Nagalla Sudhakar
International Journal of Experimental Algorithms (IJEA), Volume (5) : Issue (1) : 2015 1
Algorithm for Edge Antimagic Labeling for Specific Classes of
Graphs
Nissankara Lakshmi Prasanna prasanna.manu@gmail.com
Research scholar of ANU & Asst.professor of Vignan’s
Lara Institution of Technology and Science/CSE
Guntur, India
Nagalla Sudhakar suds.nagalla@gmail.com
Professor & Principal of Bapatla Engineering College/CSE
Guntur, India
Abstract
Graph labeling is a remarkable field having direct and indirect involvement in resolving numerous
issues in varied fields. During this paper we tend to planned new algorithms to construct edge
antimagic vertex labeling, edge antimagic total labeling, (a, d)-edge antimagic vertex labeling, (a,
d)-edge antimagic total labeling and super edge antimagic labeling of varied classes of graphs
like paths, cycles, wheels, fans, friendship graphs. With these solutions several open issues
during this space can be solved.
Keywords: Graph Labeling, Edge Antimagic Vertex Labelling, Edge Antimagic Total Labelling,
Super Edge Antimagic Labelling, Paths, Cycles, Fan Graphs, Wheels, Friendship Graphs.
1. INTRODUCTION
A graph G = (V, E) is finite, straightforward and un-directed. G denotes graph then G includes a
vertex and edge sets. Vertex set denoted by V = V (G) and edge set E = E (G). We followed the
standard notations m = |E| and n = |V|. A typical graph theoretical notation is followed refer [13].
Labelled graphs are getting associate more and more helpful family of Mathematical Models for a
broad vary of applications [4, 17]. It’s terribly crucial impact in network communications explained
in [5]. Several latest applications presented its usage to image authentication and frequency
allocation.
Graph Labeling is a method of mapping that maps several set of elements of graph to a collection
of numbers (usually +ve or non -ve integers). The foremost complete graph labeling latest survey
is in [3] [14]. Sedlacek [9] introduce labelings that simplify the thought of a magic labeling. The
magic labelings is outlined as a bijection of graph component to set of successive integers
ranging from one, satisfying some reasonably “constant sum” property. If this Bijection involves
vertices or edges or both as graph elements to a collection of integers yielding a constant sum
known as magic constant, it'll be known as Vertex or Edge or Total Magic Labeling. Hartsfield
along with Ringel in [6] introduced the idea of an Antimagic graphs. In step with them associate
Antimagic labeling is a process of edge labeling of the graph with integers 1,2,…., m in order that
weight at every vertex is totally different from the weight at every vertex.
In [7] Bodendiek with Walter outlined the thought of an (a, d)-Antimagic labeling as edge labeling
during which the weights of vertex from an AP(arithmetic progression) ranging from a and have
common distinction d. Martin Baca, Francois Bertault and MacDougall [8] initiated the notions of
the Vertex Antimagic Total Labeling [VATL] and (a, d)-Vertex Antimagic Total Labeling [(a, d)-
VATL], and conjuncture that every regular graphs are (a, d)- VATL. In the year 2004, K.A.Sugeng
et al. [10] presented the concept of SVMTL (super vertex magic total labeling) & SEMTL (super
edge magic total labeling ). The existence of In [11] Antimagic vertex labeling of categories of
hyper graphs like Cycles, Wheels and therefore the existence and non-existence of the Antimagic

vertex labeling of Wheels are mentioned in theorems. An idea to get antimagic labeling for trees
given in [12]. In reference [2] they projected the procedure (algorithms) to build (a, d)-Antimagic
labeling, Antimagic labeling, (a, d)-vertex Antimagic total labeling of complete graphs and vertex
Antimagic total labeling that could be a generalization of many different kinds of labelings.
In [15], we deal with the magic labeling of vertices and edges of a graph. Again magic labeling is
expressed in-terms of Vertex Magic Total Labeling (VMTL), Edge Total Magic Labeling (EMTL)
and Total Magic Labeling (TML). We have studied existing approaches for magic labeling and we
found some improvements can be done over existing VMTL algorithms and we design algorithm
to find EMTLs. We proposed new and enhanced algorithms for VMTL, EMTL and TML. We
applied these algorithms on different kinds of graphs like cycles, wheels, fans and friendship
graphs. In[16] we proposed new algorithms to construct vertex antimagic edge labeling, (a, d)-
vertex antimagic labeling, vertex antimagic total labeling and (a, d)-vertex antimagic total labeling
and super vertex-antimagic total labeling of various classes of graphs like paths, cycles, wheels,
Fan Graph and Friend Graphs.
In continuation to this, we have projected algorithms for EATL (Edge Antimagic Total labeling) on
different categories of graphs like cycles, paths, wheels, and fan and Friendship graphs. These
algorithms are similar all categories with minor changes because of structural variations. With
these we have to study the behaviour of the graphs with specific graph size. For given Graph with
size we will determine the attainable labelings, possible values of a and d to create (a, d) Edge
opposed magic Total Labelings and also the chance of forming SEMTL (super edge magic total
labeling).
2. PRELIMINARIES
Standard definitions of paths and wheels and cycles, fan and Friendship graphs are as follows. A
Path (Pn) could be a cycle without an edge from initial vertex to final vertex. Cycle could be a
graph wherever there's an edge amid the neighbouring vertices solely and therefore the vertex is
adjacent to final one (Fig1a). Wheel could be a Cycle with central hub, wherever all vertices of
cycle are neighbouring to that (Fig1b) Fans & Friendship graphs are variations of wheels. If a
path is linked to central hub it's a Fan (Fig1c). A Friendship graph has n triangles with 1 common
vertex known as hub and n is size of Friendship graph (Fig 1d)
FIGURE 1a: Path (P4)
FIGURE 1b:
Cycle(C4)
FIGURE 1c:
Wheel(W4)
FIGURE 1d:
Fan Graph(F4)
FIGURE 1e:
FriendshipGraph(T2)
Labeling is the method of distribution integers to graph elements is often called as a mapping
function from integers to Elements of Graph. Magic Labeling is outlined as a bijection from {1, 2,
3, 4, 5… n-1,n} to n - graph elements such the total of every element could be a magic constant
K . If the element is vertex or edge or both it's referred to as Vertex or Edge or Total magic Total
Labeling.
Let a G be a Graph with vertices v and edges e if there's a 1 to 1 function from set of integers
{1,2,3…..,n-2,n-1,n} to edges of Graph & vertices are appointed label as total of edges incident to
that. If the edge weights are completely different then it'll be EAVL ( Edge Antimagic Vertex

Labeling) .For any 2 integers a, d if the sides are assign with labels is { a , a + d ,…..…. , a+(e-
1)d }, where a is greater than 0 i.e. a>0 and d greater than equal to 0 d>=0 is termed (a, d)
EATL. If the vertices are appointed with then successive numbers then, it'll be SEATL (Super
Edge Antimagic Labeling).
Likewise for a G graph with vertices V and edges E if there's a 1 to 1 operate from set of
integers {1,2,3,4,…………, v + e} to vertex set V & edge set E of graph and edge weight is total
of labels appointed to edge & labels appointed to its (V) vertices . If edge weights completely
different then it'll be Edge Antimagic Total Labeling (EATL). For any 2 integers a, d if the
vertices appointed with labels then , wherever a is greater than 0 i.e. a>0 and d greater than
equal to 0 d>=0 is termed as (a, d) EATL. If the vertices are indicated with successive numbers
then, it'll be Super Edge Antimagic Total Labeling (SEATL). Following section provides procedure
to spot above all for various topologies of Graphs. Accumulative variety of all such potentialities is
calculated. And open issues are solved & observed properties of graphs
3. PROPOSED WORK
During this section we have a tendency to discuss algorithmic rules to spot varied features of
edge Antimagic labeling. We have a tendency to offer generalized algorithmic rule in common to
any or all forms of Graphs mentioned during this paper. As a result of topological variations every
graph structure needs some modifications to algorithmic rule.1st we have a tendency to discuss
algorithmic rule then needed changes for every kind of graph are given during this section. the
subsequent functions are utilized in designing algorithmic rule.
npx : Generates various sizes x from set of n to set of numbers that are obtainable and unused
labels.
is_Duplicate (array, size) : The function returns Boolean value if array have duplicate values other
wise false
is_RegDiff (array, size) : The functions returns change of rate if the array have adjacent values
along with a common difference given by AP else it returns negative one (-).
is_SEATL ( array) : The function returns Boolean value if is_RegDiff (array, size) function returns
true then it also returns true otherwise false
Input : Graph G with vertices V & Edges E
Output : Possible variety of total Edge Anti Magic Vertex Labeling, Edge Anti Magic Vertex
Labeling, (a, d) Edge Anti Magic Vertex Labeling with a, d values, Total Edge Anti Magic Total
Labeling, Edge Anti Magic Total Labeling, (a, d) Edge Anti Magic Total Labeling with a, d values
and checks existence of super Edge Antimagic Labeling
Algorithmic procedure for EAVL
Read Graph with size n & set labels range {1,2,3,4….,r}.
for (i=1; i<=r ;i++)
If there exist rPn & it’s not an isomorphic then set them as labels of hub and spokes.
for (j=1;j<=r;j++)
if there exist rPn, then continue process
Otherwise display “all possible assignments are checked”.
Set them as labels of edges.
For all edges calculate weight.
Wei_ght[e] = addition of labels of its vertices.
If( is_Duplicate(wei_ght, n)) EAVL_cnt++;
Set d= is_RegDiff(wei_ght, n)
If(d>=1) adEAVL_cnt++;
If(d==1) SEAVL_cnt++;
Stop.

Algorithm for EATL
Read Graph with size n and set labels range {1,2,3,4,5….r}.
For(i=1;i<=r;i++)
If there exist rPn & it’s not an isomorphic then set them as labels of hub & spokes.
for (j=1;j<=r;j++)
if there exist ar rP3 then continue process.
Otherwise display “all possible assignments are checked”.
Link1: Set them as labels of last edge, 1st vertex and 1st edge.
Previous vertex=1 current vertex=2
for (i=1;i<=r;i++)
if there exist a rp2 then continue process
else go to Link1.
Link2: Set them as labels of current vertex and current edge.
Previous vertex=current vertex Current vertex= curr_vertex+1
If current vertex=n then go to Link2.
else
reset labels assigned to previous vertex as available.
Previous vertex=prev_vertex-1 Current vertex= curr_vertex-1
For all edges calculate weight.
Wei_ght[e] = addition of labels assigned to it and labels assigned to its vertices.
If ( is_Duplicate(wei_ght, n)) EATL_cnt++;
Set d= is_RegDiff(wei_ght, n)
If (d>=1) adEATL_cnt++;
If (d==1) SEATL_cnt++;
Stop.
Modifications to be done For Path:
For EAVL Range R is {1,2,.....n}.
For EATL Range R is {1,2,.....2n-1}.
We can avoid Step 2 for Paths and set label of first edge as zero.
Modifications to be done For Cycle:
For EAT Range R is {1,2,.....2n}.
We can avoid Step 2 for Cycle.
Modifications to be done For Wheel:
For EAVL Range R is {1,2,.....n+1}.
For EATL Range R is {1,2,.....3n+1}.
Modifications to be done For Fan Graph:
For EATL Range R is {1,2,.....3n}.
Set label of first edge as zero.
Modifications to be done For Friendship Graph:
For EAVL Range R is {1,2,.....2*n+1}.
For EATL Range R is {1,2,.....5n+1}.
Set labels of even edges as zero.
4. RESULTS
Here we tend to designed algorithms to supply additive variety of edge anti magic vertex or total
labelings. we tend to conjointly known doable variety of (a, d) EAVL or Edge anti magic total
labelings for various values of a and d. Also we tend to calculated so many of super Antimagic
labelings if exists. Many authors analyzed behaviour of a specific graph structure for a few a and
d values. However here these algorithms turn out all such sequence of arrangement of vertices
and edges labels. So, we will simply and visually perceive behaviour of any structure.
Observations made are as follows.

Paths:
All paths of size n>=3 is Anti magic. For Associate in Nursing instance if the path size is eight
there are 6024 (EAVL) Edge anti magic vertex labelings are possible and it's 64 (a, d) Edge anti
magic vertex labelings and fifty six among them are super edge magic. It has (3, 2) and (6, 1)
Edge anti magic vertex labelings for several sequences. Paths with any length consists several
Edge anti magic total labelings. For instance the path with length five has 233520 Edge anti
magic total labelings. it's 4680 (a, d) Edge anti magic total labelings among 1840 are super. This
sort of study is often done on any path. for each path there's no (a, d)-EAT labeling with d > 6.
Cycles:
In cycles, all cycles with size >=3 is antimagic. These algorithmic rules are estimated for many
values of n. For n=5 it resulted thirty Edge anti magic vertex labelings. It also has 10 (a, d) Edge
anti magic vertex labelings for a few values of a and d and each one of those are super. For a
cycle of size five we tend to observed 1858600 Edge anti magic total labelings. It also has 10460
(a ,d) Edge anti magic total labelings among 4980 are super. feasible values of (a, d) are (6,5),
(7,5), (8,4)etc.
In the similar way we will analyze any cycle with any given size.
Wheels:
Wheel with size≥ three, has no edge magic vertex labeling. But it has edge magic total labelings.
For given wheel size we are able to show the attainable Edge anti magic total labelings. If the
wheel size is three, then there exist 1128768 Edge anti magic total labelings. It’s 4512 (a, d) Edge
anti magic total labelings for values among 3840 are super Edge anti magic total labelings. we've
got (15,1), (16,1), (17,1), (18,1), (10,3) and (11,3) etc attainable (a, d) Edge anti magic total
labelings
Fan Graphs:
Fan Graphs are Antimagic with size three to six. Although the number of such sequences are
very less all those are (a, d) Edge anti magic vertex labelings. Fan graphs with size three, four
and five are giving 8 (a, d) Edge Anti Magic Vertex Labelings however all are super Edge anti
magic vertex labelings & all are of (3,1) Edge anti magic vertex labelings. For Fan Graphs we
tend to indicate some shocking results. If the fan size is three, then there exist 152152 Edge anti
magic total labelings. It’s 1344(a, d) Edge anti magic total labeling for various values among 672
are super Edge anti magic total labelings. we've got (14,2),(9,3) ,(16,1),(8,3) (11,3), (15,1), (12,2)
and (11,3) etc attainable (a, d) Edge anti magic total labelings.
Friendship Graphs:
Friendship graph also has edge anti magic labeling. Friendship graphs also are observed as
every (a, d) edge anti magic vertex labeling is super. Friendship graph with size 2 is having 24
edge anti magic vertex labelings and none of them is (a, d) edge anti magic vertex labeling.
However Friendship graphs with size 1,3,4,5 are producing edge anti magic vertex labelings,(a,
d) edge anti magic vertex labelings. For these graphs all are super edge magic. Friendship graph
with size three producing 192 edge anti magic vertex labelings and 48 (a, d) edge anti magic
vertex labelings and every one area unit super. Friendship graph with size four manufacturing
3072 edge anti magic vertex labelings and 2304 (a, d) edge anti magic vertex labelings and all of
them are super edge anti magic vertex labelings. Each Friendship graph could be a super edge
anti magic total labeling. For Friendship graphs with size 3 has huge number of edge anti magic
total labeling (a, d) edge anti magic total labeling for d=2. These are some observations created
by us.
5. CONCLUSIONS
In this paper, we tend to provide algorithms to enumerate all Edge Antimagic labelings on wheel
graphs, Fan Graphs, cycle Graphs and Friendship graphs. The thought of the algorithms can be
applied to alternative categories of graphs or adopted to develop algorithms for other form of
labeling. within the in the meantime, we still engaged on algorithm for other form of labeling like
edge anti magic and total, harmonious, swish etc. we tend to present the number of non

isomorphic completely different anti magic labelings on every graph for a some small size graphs.
The number of non-isomorphic labeling on larger size of the remaining graphs is still an open
problem.
6. REFERENCES
[1] Magic labelings on cycles and wheels, by Baker A., J. Sawada, 2008, COCOA LNCS, 361-
373.
[2] “Algorithms to Construct Anti Magic Labeling of Complete Graphs “, by Krishnappa H K, N K
Srinath, S Manjunath .IJCIT Volume 02– Issue 03, May 2013.
[3] J. Gallian, A dynamic survey of graph labeling, Electronic J. Combin., 14 (2007), DS6.
[4] Narsingh Deo, “Graph theory with applications to Engineering and Computer Science”, 56
(2000).
[5] “applications of graph labeling in communication networks” by N. Lakshmi prasanna, k.
Sravanthi and nagalla sudhakar’s oriental journal of Computer science & technology issn:
0974-6471 april 2014,vol. 7, no. (1):pgs. 139-145.
[6] N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, Boston-San Diego-
New York- London, 1990.
[7] R. Bodendiek and G. Walther, Arithmetisch antimagische graphen, in K. Wagner and R.
Bodendiek, Graphentheorie III, BI-Wiss. Verl. Mannheim, 1993.
[8] M. Baˇca, F. Bertault, J. MacDougall, M. Miller, R. Simanjuntak, and Slamin, Vertex
Antimagic total labelings of graphs, Discuss. Math. Graph Theory, 23 (2003) 67–83.
[9] J.Sedlacek, Problem 27 in Theory of graphs and its applications, Proc. Symp. Smolenice,
June 1963, Praha (1964), p.162.
[10]J.A.MacDougall, M.Miller, K.A.Sugeng, “Super Magic Total Labelings of graphs”, Proceeding
of the Fifteenth Australian workshop on Combinatorial Algorithms 2004, Balina, NSW, (2004),
222-229.
[11]Ko-Wei-Lih,s “The Magic labelings for Wheels, Prisms and Friendship graphs” ,1983.
[12]Krishnaa.A’s “A study of the major graph labelings of the trees”, Informatica, 15(4), 515–524,
2004.
[13]D.B. West, An Introduction to Graph Theory (Prentice-Hall, 1996).
[14]Kiki A. Sugeng’s “Survey of edge antimagic labelings of graphs”, J. Indones. Math. Soc.
(MIHMI).
[15]Nissankara Lakshmi Prasanna ,Nagalla Sudhakar, ”Algorithms For Magic Labeling On
Graphs“,Journal Of Theoretical And Applied Information Technology,10th August 2014. Vol.
66 No.1.
[16]Nissankara Lakshmi Prasanna,Nagalla Sudhakar, K. Sravanthi,” Algorithm For Vertex Anti-
Magic Total Labeling On Various Classes Of Graphs”,Contemporary Engineering Sciences,
Vol. 7, 2014, No. 19, 915 – 922, http://Dx.Doi.Org/10.12988/Ces.2014.4665.
[17]A. A. Samah, M. Z. Matondang, H. Haron, H. A. Majid & N. K. Ibrahim.” Role Model of Graph
Coloring Application in Labelled 2D Line Drawing Object”, International Journal of
Experimental Algorithms (IJEA), Volume (2), Issue (3), 2011.

Algorithm for Edge Antimagic Labeling for Specific Classes of Graphs

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