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ACCENT JOURNAL OF ECONOMICS ECOLOGY & ENGINEERING Peer Reviewed and Refereed Journal IMPACT FACTOR: 2.104 (ISSN NO. 2456-1037) Vol.03, Issue 09, Conference (IC-RASEM) Special Issue 01, September 2018 Available Online: www.ajeee.co.in/index.php/AJEEE

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AN OVERVIEW OF GRAPH LABELING AND ITS APPLICATIONS AND TYPES OF GRAPH LABELING

Priyanka Bhalerao, Assistant Professor:

Shri Vaishnav Vidyapeeth Vishvidyalaya, Indore

Sanghwi Institute of Management and Science, Indore E-Mail Address: [email protected]

Dr. Seema Bagora

Abstract - Graph theory is the study of the properties and applications of graphs. This particular field of mathematics studies and finds solutions by the study of graphs that are mathematical structures. The field of graph theory plays a vital role in various fields. The significance of graph theory is growing as research becomes more model oriented as well as illustration oriented. Graph theory is widely used to prove several mathematical theorems and models for proper understanding and further research. One of its major area of Graph Theory is graph labeling. In this review paper we go through some special types of graph labeling like Graceful labeling, Prime labeling, Edge Graceful labeling, Harmonious labeling and Graph coloring along with their properties and mathematical conditions. We also discuss a few applications of Graph labeling used in key technological fields such as the coding theory, X-Ray Crystallography, Communication Network addressing and Automatic Routing.

Keywords: Graph, Applications, Labeling, Prime, Harmonious, Graceful.

1. INTRODUCTION

Graph theory is the study of the properties and applications of graphs.

This particular field of mathematics studies and finds solutions by the study of graphs that are mathematical structures .These structures are used to model pair-wise relationships between objects by the method of graphical representations and mathematical derivations.

1.1 History of graph theory

Graph theory, as a major studied branch of mathematics has its roots around 1736, when Leonhard Euler suggested a solution to the Konigsberg Seven Bridge problem in the ancient town of Konigsberg in Prussia (modern day Kaliningrad, Russia). After the publication of this paper in 1736, Vandermonde wrote a paper on the knight problem but also dealt with certain problematic aspects of Euler’s paper.

Some definitions related to a graph:

Graph: A Graph is a pair(V,E),where V is a set of vertices and E is a collection of edges.

Directed Graph: A graph where the edges point in a direction is called a directed graph.

Loop: A loop is a special type of edge that connects a vertex to itself. Loops are not used much in street network graphs.

Degree: The number of edges which connect a node.

In-degree: Number of edges pointing to a node.

Out-degree: Number of edges going out of a node.

Connected graph: A graph is said to be connected if every two vertices in are connected.

Graph labeling: It is the field of graph theory that evaluates and further applies to the qualitative labeling of graph elements to get to desired results. These results can be in the forms of theorems, proofs, conditions, equations and axioms.

Graph labeling is becoming popular to derive useful and important mathematical

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2 models for problem solution in various fields of science and technology.

History of graph labeling:

Graph labeling begun in mid 1960s with the conjecture of Ringle (1964) and a paper by Rosa (1967). Rosa introduced β – valuation, α-valuation and other labeling as a tool to decompose complete graphs.

The β valuation was later called Graceful labeling by Golomb (1972).

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Definition 1.1. Graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. It is a strong communication between number theory and structures of graphs. It is assignment of integers either to the vertices or edges or both.

2 .TYPES OF GRAPH LABELING

Graceful labeling: A Graph is known as graceful when it’s vertices are labeled from 0 to IEI. The size of graph and this labeling induces an edges labeling from 1 to IEI. Graceful labeling means that if there is more than one common points of two distinct edges, then it is not graceful labeling.

Prime labeling: A graph G with vertices is said to admit prime labeling if it’s vertices can be labeled with distinct positive integers, not to exceeding V such that the labels of each pair of adjacent vertices are relatively prime.

Edge graceful labeling: An edge graceful labeling on a simple graph (no loops or multiple edges) on p-vertices and q- edges is a labeling of the edges by distinct integers in {1……..q} such that labeling a vertex with the sum of incident edges taken modulo p assigns all values from 0 to p-1 to the vertices.

Harmonious labeling: A harmonious labeling on a graph G is an injection from vertices to G to the group of integers modulo k, where k is the number of edges of G.

Graph coloring: Graph coloring is a sub- class of graph labeling. A vertex coloring assigns different labels to adjacent vertices. An edge coloring assigns different labels to adjacent edges.

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3 3 APPLICATIONS OF GRAPH LABELING:

The coding theory: The design of certain important classes of good non periodic codes for pulse radar and missile guidance is equivalent to labeling the complete graph in such a way that all the edge labels are distinct. The node labels then determine the time positions at which pulses are transmitted.

The X-ray crystallography: X ray diffraction is one of the most powerful techniques for characterizing the structural properties if crystalline solids, in which a beam of x-rays strikes a crystal and diffracts into many specific directions. In some cases more than one structure has the same diffraction information. This problem is mathematically equivalent to determining all labeling of the appropriate graphs which produce a pre-specified set of edge labels.

3.1 Communication Network addressing:

A communication network is composed of nodes, each of which has computing power and can transmit and receive messages over communication links, wireless or cabled. The basic network topologies are include fully connected, mesh, star, ring, tree, bus. A single network may consist of several interconnected subnets of different topologies.

Networks are further classified as Local Area Networks (LAN), e.g. inside one building, or Wide Area Networks (WAN), e.g. between buildings. It might beuseful to assign each user terminal a “node label,” subject to the constraint that all connecting “edges” (communication links) receive distinct labels. In this way, the numbers of any two communicating terminals automatically specify (by simple subtraction) the link label of the connecting path; and conversely, the path

label uniquely specifies the pair of user terminals which it interconnects.

3.2 Automatic Routing with labeling:

In any traditional network if it can be represented with a specific kind of graph topology, then labeling applied may automatically detects route with any additional information. Here the graph structure can be anything like a cycle, wheel, fan-graph, friend-graph etc. but should be fixed. Now magic labeling can be applied to the network for a magic constant which is public to network. The routes automatically detects the next node to be reached by using the magic constant, its own label and labels assigned to channels, since all these must produce magic numbers.

4. CONCLUSION

The aim of this review paper is to know about graph labeling, its types and some of its important applications. Graph- Labeling is a powerful tool that helps in various technological fields.

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