International Journal of Computer Sciences and Engineering Open Access
Research Paper Vol.-7, Issue-5, May 2019 E-ISSN: 2347-2693
An Equitable Antimagic Labeling of Graphs: Algorithmic Approach
A. Puthussery1*, I. S. Hamid2, A. Anitha3
1*Department of Science and Humanities, CHRIST (Deemed to be University), Bengaluru, India
2 Department of Mathematics, The Madura College, Madurai, India
3 Department of Mathematics, Thiagarajar College of Engineering, Madurai, India
*Corresponding Author: [email protected], Tel.: +91-96206-45200
DOI: https://doi.org/10.26438/ijcse/v7i5.846851 | Available online at: www.ijcseonline.org
Accepted: 17/May/2019, Published: 31/May/2019
Abstract— An algorithm is a sequence of computational steps that transform input data into output. Here an algorithmic approach on the process of graph labeling specially on a particular graph labeling called equitable antimagic graphs (EAG), is adopted. A graph labeling is a process in which labels-numbers or labels, have been assigned for vertices, edges or even both, subject to certain conditions. Further equitable antimagic graphs are graphs which follows a special kind of labeling called equitable antimagic labeling. There are many algorithms used to ease the process of labeling. This paper concentrates on the development and the exposition of an algorithm for analyzing some of the graphs for the equitable antimagic property. This algorithmic approach is mainly dealt to list out equitable antimagic graphs generated from a fixed number of vertices, say n and number of edges say m.
Keywords— Labeling, Antimagic labeling, Graph Algorithm, Equitable antimagic labeling, Equitable antimagic graph
I. INTRODUCTION
Graph theory is an area of discrete mathematics that has reliably high end applications based on graphs. By a graph G
= (V, E), we mean a finite, connected, undirected structure with neither loops nor multiple edges. For graph theoretic terminologies we refer to the book in [1]. An algorithm is a sequence of computational steps that transform input data into output. Here an algorithmic approach on the process of graph labeling specially on a particular graph labeling called equitable antimagic graphs (EAG), is adopted. Throughout the paper the order and size of G are denoted by n and m respectively.
The paper is organized as follows: with a brief introduction and a description of basic terminologies an algorithmic approach to ease the process of analysing a certain class of labeling called equitable antimagic labeling is introduced.
Through this algorithmic approach a list of equitable antimagic graphs from a fixed number of vertices, say n and number of edges say m is furnished.
II. RELATED WORK
Graph labeling is one of the fascinating areas of graph theory with wide-ranging applications. The concept of graph labeling was first introduced in 1960s where the vertices and edges are assigned real values or subsets of a set subject to certain conditions [2].
An enormous body of literature has grown around graph labeling in the last five decades. Over 2500 research articles are published in the field of graph labelings. A detailed survey is given in [3]. The notion of antimagic labeling of a graph was introduced in 1990 [4].
Definition 2.1: Antimagic Labeling
Antimagic labeling is defined as a bijective edge labeling f of G as , such that the vertex sum of vertices of G are all distinct, where the vertex sum of the vertex v under the labeling f is defined to be the sum of labels of edges incident at vertex v. A graph admitting antimagic labeling is called antimagic graph.
Though there are many studies done on antimagic labeling and antimagic graphs, the existence of antimagic labeling for all connected graphs is unsettled. In this connection, the
conjecture on antimagic labeling was posed in [4] that every connected graph different from is antimagic. Motivated by the irregularity strength, in 2010, the notion of equitable irregular edge-weighting was introduced [5].
Definition 2.2: Equitable Irregular Edge-weighting
For a given graph G = (V, E), a k-edge-weighting is a map
, where k is a positive integer.
For a vertex v of G, let
denote the sum of edge- weights appearing on the edges incident at v, under the edge- weighting . An k-edge weighting of G is said to be equitable irregular if
| |
, for every pair of adjacent vertices u and v in G. A graph G is said to be equitable irregular if G admits an equitable irregular edge- weighting.Motivated by the concept of antimagic labeling and equitable irregular edge-weighting, A. Puthussery and I.S. Hamid initiated a study of equitable antimagic labeling of a graph [6].
Definition 2.3: Equitable Antimagic Labeling
For an
graph G, an equitable antimagic labeling (EAL) is a bijective function
such that
| |
for any pair of adjacent vertices u and v of G. For a vertex v of G,denote the sum of edge-weights appearing on the edges incident at v under the edge-weighting f. A graph admitting an equitable antimagic labeling (EAL) is called an equitable antimagic graph (EAG).
It is proved in [6] that some classes of graphs such as cycles, paths, complete bipartite, Möbius ladder graph, and trees under certain conditions are equitable. These constraints which make these graphs EAG are as follows.
Theorem 2.4 A path on
vertices is equitable antimagic only when
or
.
Theorem 2.5 A cycle on
vertices is equitable antimagic graph if and only if
.
Theorem 2.6 The complete bipartite graph is an equitable antimagic graph for all r.
Theorem 2.7 The Möbius ladder is an equitable antimagic graph if and only if k is odd.
From these observations we define an equitability constant k.
As the edge labels are distributed almost equally to the vertices incident an equitability constant k of equitable antimagic labeling is found and is calculated as
⌈
⌉
.The bounds for the cyclic graphs with respect to the number of edges which make them equitable antimagic graph are depicted in the next theorem.
Theorem 2.8 If G is an equitable antimagic graph, then
⌈
⌉
.Theorem 2.9 A tree T is an equitable antimagic graph if and only if T = or .
The problem of equitable antimagic labeling can be restructured as a problem to find a hollow symmetric square matrix with almost equal row sum for pairs of adjacent vertices. In case of complete graphs, the number of edges also increases as the number of vertices increases. Thus the need for easy computation became necessary to get an equitable antimagic labeling of complete graphs.
The algorithmic approach is used to solve mathematical problems in various fields of studies if the solution requires machine intervention or calculation. With the expansion of computational techniques, hundreds of motivating problems are couched in terms of graphs. An algorithm is any well- defined computational tool that takes some value as input and produces some value as output. An algorithm is thus a sequence of computational steps that transform the input into output. More on algorithms and its complexity are given in [7], [8], [9]. The algorithmic approach is used to find an equitable antimagic labeling which is given in [10]. They proved and are not equitable antimagic graph. They also showed that for
is an equitable antimagic graph using algorithmic approach. In the next section we have the algorithm to find the set of all equitable antimagic labeling for a given number of edges with respect to number of vertices.
III.
A
LGORITHMIC APPROACH ON EQUITABLE ANTIMAGIC LABELINGIt is clear from the preliminaries that the existence of the equitability constant k plays an important role in deciding whether the mentioned graph is equitable antimagic or not.
From [6] Theorem 2.5, states that for is not an equitable antimagic graph, whereas for is equitable antimagic graph. Thus, the existence of equitable antimagic graph for a fixed n with the variations in number of edges m is needed to be verified. In other words we find all equitable constant k for which
⌊
⌋
is true.In this section an algorithmic approach is introduced to verify this result. The algorithm follows brute force approach with the help of a high-speed machine. TS 150 one Socket Server Intel Xeon E3-1225 v5 (Quad Core 3.3ghz, 4*16 GB, 1*1 TB SATA 3.5“ NHS Sata Multi Burner Raid 0,1,5 inbuilt(Raid 121i) 2gb Graphic Card GT 710 is used to run this program. The user inputs the dimension / number of vertices and number of edges for which the equitable antimagic labelings need to be established. The program calculates the equitable constant as per the number of edges for which almost equal distribution of edge label has to be found.
The algorithm is as follows:
Algorithm 3.1
Step 1: Initialize dimension n and m for which equitable antimagic labeling is to be found.
Step 2: Find
⌈
⌉
from the list.
Step 3: Find from T such that sum of row elements equal to k.
Step 4: Repeat Step 3 till .
IV. RESULTS AND DISCUSSION
The Table 4.1 shown below exhibits the variety and complexity with respect to number of vertices and edges in using the mentioned algorithm. The program is run for and edges ranging from . Snapshots of output files generated running the program are given in Figure 4.1 and Figure 4.2. The labels of graphs with 6 vertices and edges ranging from in Figures 1(a) to Figure 1(i) are obtained using the above mentioned algorithm. As the number of edges differ the graph isomorphism also gets complicated. In this paper, we have looked into only the existing of equitable antimagic labeling fixing number of vertices and edges.
Table 4.1: Equitability constant with respect to number of vertices and edges
Number of vertices
Number of edges
Equitability
constant Matrix Combinations
6
7 9, 10
1307674368000 13digit number
8 12
9 15
10 18, 19
11 22
12 26
13 30, 31
14 35
15 40
Figure 4.1: EALabelings of 6 vertex with 14 edges Figure 4.2: EALabelings of 6 vertex with 11 edges
V. CONCLUSION AND FUTURE SCOPE
The field of graph theory plays a vital role in various real world applications. Again graph labeling and its various subareas like equitable antimagic labeling used in applications like crystallography, coding theory, astronomy, circuit design and many more. Many times the processes of labeling of graphs in a particular way turns out to be a
tedious task. The algorithm presented here can be adopted as a comparatively
easy way of labeling and analyzing graphs for equitable antimagic labeling. Based on the exhibited results on variety and complexity in using the algorithm with respect to number of vertices and edges [table 4.1] it can be further modified to a more refined algorithm in future.
REFERENCES
[1] G. Chartrand and Lesniak, Graphs and Digraphs, CRC Press, Boca Raton, 2005.
[2] A. Rosa, “On certain valuations of the vertices of a graph”, Theory of Graphs, Internat. Symp., pp. 349 – 355, 1966.
[3] J. A. Gallian, “A dynamic survey of graph labeling”, Electron. J.
Combin, # DS6, 2017.
[4] N. Hartsfield and G. Ringel, Pearls in Graph Theory, Academic Press, San Diego, 1990.
[5] I. S. Hamid and S. A. Kumar, “Equitable irregular edge-weighting of graphs”. SUT J. Math., 46, pp. 79 – 91, 2010.
[6] A. Puthussery and I. S. Hamid, “Equitable antimagic labeling of graphs”, submitted to Discrete Applied Mathematics, 2019.
[7] T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein,
“Introduction to Algorithms”, MIT Press, 1990.
[8] A. Levitin, “Introduction to the Design & Analysis of Algorithms”, Pearson, 2012.
[9] N. Choudhary, S. Agarwal, G. Lavania, "Smart Voting System through Facial Recognition", International Journal of Scientific Research in Computer Science and Engineering, Vol.7, Issue.2, pp.7-10, 2019.
[10] A. Puthussery, I. S. Hamid and M. Thomas, “Algorithmic approach for equitable antimagic labeling of complete graphs”, communicated Springer proceedings of International Conference of Emerging Trends in Graph Theory, 2019.
Authors Profile
A. Puthussery is an Assistant Professor at Department of Sciences and Humanities, Faculty of Engineering, CHRIST (Deemed to be University), Bengaluru, Karnataka. He has completed his MPhil in Mathematics with Graph Theory as his specialisation. Currently pursuing his doctoral studies under the supervision of Dr I. S.
Hamid at CHRIST (Deemed to be University), Bengaluru. His interest fields are Graph Theory, Combinatorics, Algorithms, Optimisation, etc.
Dr I Sahul Hamid is an Assistant Professor at Department of Mathematics, The Madura College, Madurai, Tamil Nadu. His skills and expertise are in Discrete Mathematics, Graph Theory, Combinatorics, Algorithm Analysis, etc. He has 47 articles in his credit published in well peer reviewed and Scopus indexed journals.
He has completed two DST funded major research projects and one ongoing.
Dr Anita A is an Assistant Professor at Department of Mathematics, Thiagarajar College of Engineering, Madurai, Tamil Nadu. She has published 7 articles in well reputed journals. She worked as Junior Research Fellow (JRF) under a DST Project entitled “Domination Related Functions in Graphs” from September 01, 2006 to July 31, 2007. She has the life membership in Indian Society for Technical Education and Academy of Discrete Mathematics and Applications.
